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GEORGE BOOLE’S
COLLECTED LOGIGAL WORKS
VOLUME Η

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ΤΣ Α͂
GEORGE BOOLE’S

COLLECTED LOGICAL
WORKS

| VOLUME ii
THE LAWS OF THOUGHT (1854)

CHICAGO AND LONDON

mHE OPEN COURT PUBLISHING COMPANY
1916

Copyright in Great Britain under the Act of 1911

PRINTED IN GREAT BRITAIN BY NEILL AND CO., LTD., EDINBURGH.

NOTE ΤῸ ἘΠΕ NEW EDERION

BooLr’s Laws of Thought was originally published at
London and Cambridge in 1854, and this new edition of
that famous work, which is issued with the kind permission
of Mrs. Boole and her family, is an exact reproduction of
the original, including the title-page. The Errata have
of course been corrected. The numbers giving the pages
of the original have been inserted in the text in thicker
type and enclosed in square brackets, so that references to
pages of the original can at once be found in this edition.
An Index has also been added.

This new edition of the Laws of Thought forms the
second volume of Boole’s Collected Logical Works, and a
biography of Boole and account of his mathematical and
logical work is given in the introduction to the first volume.
In that introduction also is given some account of Boole’s
logical thought subsequent to the publication of the Laws
of Thought. The first volume also contains Boole’s three
publications on logic prior to the appearance of the Laws
of Thought, and all his papers on the theory of probabilities.

PHILIP: EB. JOURDAIN.

AN INVESTIGATION

OF

ΠΗ AWS OF EHO UG EL

ON WHICH ARE FOUNDED

THE MATHEMATICAL THEORIES OF LOGIC
AND PROBABILITIES.

BY

GHEORGHE BOGE 1.

PROFESSOR OF MATHEMATICS IN QUEEN’S COLLEGE, CORK.

HON DON.
WALTON AND MABERLEY,

UPPER GOWER-STREET, AND IVY-LANE, PATERNOSTER.«ROW.

CAMBRIDGE: MACMILLAN AND CO.

1854.

TO

JOHN RYALE Le p.

VICE-PRESIDENT AND PROFESSOR OF GREEK
IN QUEEN’S COLLEGE, CORK
THIS WORK IS INSCRIBED

IN TESTIMONY OF FRIENDSHIP AND ESTEEM

Ea
᾿

iit] PREFACE

THE following work is not a republication of a former
treatise by the Author, entitled The Mathematical Analysis |
of Logic. Its earlier portion is indeed devoted to the same
object, and it begins by establishing the same system of
fundamental laws, but its methods are more general, and
its range of applications far wider. It exhibits the results,
matured by some years of study and reflection, of a principle
of investigation relating to the intellectual operations, the
previous exposition of which was written within a few weeks
after its idea had been conceived.

That portion of this work which relates to Logic pre-
supposes in its reader a knowledge of the most important
terms of the science, as usually treated, and of its general
object. On these points there is no better guide than
Archbishop Whately’s Elements of Logic, or Mr. Thomson’s
Outlines of the Laws of Thought. To the former of these
treatises, the present revival of attention to this class of
studies seems in a great measure due. Some acquaintance
with the principles of Algebra is also requisite, but it is not
necessary that this application should have been carried
beyond the solution of simple equations. For the study of
those chapters which relate to the theory of probabilities,
a somewhat larger knowledge of Algebra is required, and
espe-[iv]cially of the doctrine of Elimination, and of the
solution of Equations containing more than one unknown
quantity. Preliminary information upon the subject-matter
will be found in the special treatises on Probabilities in
Lardners Cabinet Cyclopedia, and the Library of Useful

ΧΙ

ΧΙ THE LAWS OF THOUGHT

Knowledge, the former of these by Professor De Morgan,
the latter by Sir John Lubbock ; and in an interesting series
of Letters translated from the French of M. Quetelet. Other
references will be given in the work. On a first perusal the
reader may omit at his discretion, Chapters X., XIV., and
X1X., together with any of the applications which he may
deem uninviting or irrelevant.

In different parts of the work, and especially in the notes
to the concluding chapter, will be found references to various
writers, ancient and modern, chiefly designed to illustrate a
certain view of the history of philosophy. With respect to
these, the Author thinks it proper to add, that he has in no
instance given a citation which he has not believed upon
careful examination to be supported either by parallel
authorities, or by the general tenor of the work from which
it was taken. While he would gladly have avoided the
introduction of anything which might by possibility be con-
strued into the parade of learning, he felt it to be due both
to his subject and to the truth, that the statements in the
text should be accompanied by the means of verification.
And if now, in bringing to its close a labour, of the extent
of which few persons will be able to judge from its apparent
fruits, he may be permitted to speak for a single moment of
the feelings with which he has pursued, and with which he
now lays aside, his task, he would say, that he never doubted
that it was worthy of his best efforts ; that he felt that what-
ever of truth it might bring to light was not a private or
arbitrary thing, not dependent, as to its essence, upon any
human opinion. He was fully aware that learned and able
men maintained opinions upon the subject of [v] Logic
directly opposed to the views upon which the entire argu-
ment and procedure of his work rested. While he believed
those opinions to be erroneous, he was conscious that his
own views might insensibly be warped by an influence of
another kind. He felt in an especial manner the danger
of that intellectual bias which long attention to a particular
aspect of truth tends to produce. But he trusts that out of

BAN eae αν
So ES Le ERS

SP ee OE ee Se Rese ἐς

PREFACE xiii

this conflict of opinions the same truth will but emerge the
more free from any personal admixture ; that its different
parts will be seen in their just proportion ; and that none
of them will eventually be too highly valued or too lightly
regarded because of the prejudices which may attach to the
mere form of its exposition.

To his valued friend, the Rev. George Stephens Dickson,
of Lincoln, the Author desires to record his obligations for
much kind assistance in the revision of this work, and for
some important suggestions.

5, GRENVILLE-PLACE, CORK,
Nov. 30th, 1853.

Sore
ee
oo

aa CONTENTS

CHAPTER =]

PAGE

NATURE AND DESIGN OF THIS Work, . : : ᾿ : Ι
GHAP-ER ΙΙ

SIGNS AND THEIR Laws, : : : : : : 1526

CHAPTER 111
DERIVATION OF THE Laws, . : : Ξ : : ἢ ὩΣ

CHAPTER. Ly.

DIVISION OF PROPOSITIONS, . : : ; : ᾿ gee
CHAR EE Rey,

PRINCIPLES OF SYMBOLICAL REASONING, : ‘ eee
CHAPTER VI

OF INTERPRETATION, . : ey

CHAP LE Rw V.E
Or ELIMINATION, . : : : : : : Ἴ πΘ 7

CHAPTER VIII

Or REDUCTION, . : : ' ᾿ is
CHAPTER. 1x

METHODS OF ABBREVIATION, ; : ᾿ ἱ : 3.5 150

[viii] CHAPTER X

ConpiITIONS OF A PERFECT METHOD, . : : τ 1 50
XV

Xvi LAWS OF THOUGHT

CHAPTER Xl

Or SECONDARY PROPOSITIONS,

CHAPTER ΧΗ

METHODS IN SECONDARY PROPOSITIONS,

CHAPTER XIll

CLARKE AND SPINOZA, .

CPAPTER XEV,

EXAMPLE OF ANALYSIS,

CHAPTER XV

OF THE ARISTOTELIAN LOGIC,

CHAPTER XVI

Or THE THEORY OF PROBABILITIES,

CHAPTER XVII

GENERAL METHOD IN PROBABILITIES, .

CHAP THR XV it

ELEMENTARY ILLUSTRATIONS,

CHAP PER XTX

Or STATISTICAL CONDITIONS,

CHAPTER Xx

PROBLEMS ON CAUSES, .

CHAPTER ΧΧῚ
PROBABILITY OF JUDGMENTS,

CHAPTER XXII
CONSTITUTION OF THE INTELLECT,

PAGE

168

188

197

232

239

257

268

292

ΤΙ

337

393

418

[1] AN INVESTIGATION

OF

Pot PAW s OF THOUGHT

CRATERS |
NATURE AND DESIGN OF THIS WORK

1. THE design of the following treatise is to investigate
the fundamental laws of those operations of the mind by
which reasoning is performed ; to give expression to them
in the symbolical language of a Calculus, and upon this
foundation to establish the science of Logic and construct
its method ; to make that method itself the basis of a general
method for the application of the mathematical doctrine
of Probabilities; and, finally, to collect from the various
elements of truth brought to view in the course of these
inquiries some probable intimations concerning the nature
and constitution of the human mind.

2. That this design is not altogether a novel one it is
almost needless to remark, and it is well known that to its
two main practical divisions of Logic and Probabilities a
very considerable share of the attention of philosophers has
been directed. In its ancient and scholastic form, indeed,
the subject of Logic stands almost exclusively associated
with the great name of Aristotle. As it was presented to
ancient Greece in the partly technical, partly metaphysical
disquisitions of the Organon, such, with scarcely any essential
change, it has continued to the present day. The stream
of origina: inquiry has rather been directed towards questions

of general philosophy, which, though they [2] have arisen
1 I

2 THE LAWS OF THOUGAT

among the disputes of the logicians, have outgrown their
origin, and given to successive ages of speculation their
peculiar bent and character. The eras of Porphyry and
Proclus, of Anselm and Abelard, of Ramus, and of Descartes,
together with the final protests of Bacon and Locke, rise up
before the mind as examples of the remoter influences of the
study upon the course of human thought, partly in suggest-
ing topics fertile of discussion, partly in provoking remon-
strance against its own undue pretensions. The history of
the theory of Probabilities, on the other hand, has presented
far more of that character of steady growth which belongs
to science. In its origin the early genius of Pascal,—in its
maturer stages of development the most recondite of all the
mathematical speculations of Laplace,—were directed to
its improvement ; to omit here the mention of other names
scarcely less distinguished than these. As the study of
Logic has been remarkable for the kindred questions of
Metaphysics to which it has given occasion, so that of
Probabilities also has been remarkable for the impulse which
it has bestowed upon the higher departments of mathematical
science. Each of these subjects has, moreover, been justly
regarded as having relation to a speculative as well as toa
practical end. To enable us to deduce correct inferences
from given premises is not the only object of Logic; nor is
it the sole claim of the theory of Probabilities that it teaches
us how to establish the business or life assurance on a secure
basis ; and how to condense whatever is valuable in the re-
cords of innumerable observations in astronomy, in physics,
or in that field of social inquiry which is fast assuming a
character of great importance. Both these studies have
also an interest of another kind, derived from the light
which they shed upon the intellectual powers. They instruct
us concerning the mode in which language and number serve
as instrumental aids to the processes of reasoning ; they
reveal to us in some degree the connexion between different
powers of our common intellect ; they set before us what,
in the two domains of demonstrative and of probable know-

NATURE AND DESIGN OF THIS WORK — 3

ledge, are the essential standards of truth and correctness,
—standards not derived from without, but deeply founded
in the constitution of the human faculties. These ends of
speculation yield neither in interest nor in dignity, nor yet,
it [3] may be added, in importance, to the practical objects,
with the pursuit of which they have been historically associ-
ated. To unfold the secret laws and relations of those high
faculties of thought by which all beyond the merely per-
ceptive knowledge of the world and of ourselves is attained
or matured, is an object which does not stand in need of
commendation to a rational mind.

3. But although certain parts of the design of this work
have been entertained by others, its general conception, its
method, and, to a considerable extent, its results, are
believed to be original. For this reason 1 shall offer, in the
present chapter, some preparatory statements and explana-
tions, in order that the real aim of this treatise may be
understood, and the treatment of its subject facilitated.

It is designed, in the first place, to investigate the funda-
mental laws of those operations of the mind by which reason-
ing is performed. It is unnecessary to enter here into any
argument to prove that the operations of the mind are in a
certain real sense subject to laws, and that a science of the
mind is therefore possible. If these are questions which
admit of doubt, that doubt is not to be met by an en-
deavour to settle the point of dispute @ przorz, but by direct-
ing the attention of the objector to the evidence of actual
laws, by referring him to an actual science. And thus the
solution of that doubt would belong not to the introduction
to this treatise, but to the treatise itself. Let the assump-
tion be granted, that a science of the intellectual powers is
possible, and let us ‘us for a moment consider how the know-
ledge of it is to be obtained.

4. Like all other sciences, that of the intellectual opera-
tions must primarily rest upon observations, —the subject
of such observation being the very operations and processes
of which we desire to determine the laws. But while the

4 THE LAWS OF THOUGHT

necessity of a foundation in experience is thus a condition
common to all sciences, there are some special differences
between the modes in which this principle becomes available
for the determination of general truths when the subject of
inquiry is the mind, and when the subject is external nature.
To these it is necessary to direct attention.

[4] The general laws of Nature are not, for the most part,
immediate objects of perception. They are either inductive
“inferences from a large body of facts, the common truth in
‘which they express, or, in their origin at least, physical
hypotheses of a causal nature serving to explain phanomena
with undeviating precision, and to enable us to predict new
Combinations of them, ©. Phey, are in all cases, and. in the
strictest sense of the term, probable conclusions, approach-
ing, indeed, ever and ever nearer to certainty, as they
receive more and more of the confirmation of experience.
But of the character of probability, in the strict and proper
sense of that term, they are never wholly divested. On the
other hand, the knowledge of the laws of the mind does not
require as its basis any extensive collection of observations.
The general truth is seen in the particular instance, and it
is not confirmed by the repetition of instances. We may
illustrate this position by an obvious example. It may be
a question whether that formula of reasoning, which is called
the dictum of Aristotle, de omni et nullo, expresses a
primary law of human reasoning or not; but it is no~
question that it expresses a general truth in Logic. Now |
that truth is made manifest in all its generality by reflection
upon a single instance of its application. And this is both
an evidence that the particular principle or formula in
question is founded upon some general law or laws of the
mind, and an illustration of the doctrine that the perception
of such general truths is not derived from an induction
from many instances, but is involved in the clear appre-
hension of a single instance. In connexion with this truth
is seen the not less important one that our knowledge of
the laws upon which the science of the intellectual powers

NATURE AND DESIGN OF THIS. WORK 34

rests, whatever may be its extent or its deficiency, is not
probable knowledge. For we not only see in the particular
example the general truth, but we see it also as a certain
truth,——a truth, our confidence in which will not continue
to increase with increasing experience of its practical
verifications.

5. But if the general truths of Logic are of sucha nature.
that when presented to the mind.they at once command _
assent, wherein consists the difficulty of constructing. the.
Science of Logic? Not, it may be answered, in collecting
the materials of knowledge, but in discriminating _ their
nature, and determining [5] their mutual place and relation.
Alfsciences consist of general truths, but of those truths some
only are primary and fundamental, others are secondary
and derived. The laws of elliptic motion, discovered by
Kepler, are general truths in astronomy, but they are not
its fundamental truths. And it is so also in the purely
mathematical sciences. An almost boundless diversity of
theorems, which are known, and an infinite possibility of
others, as yet unknown, rest together upon the foundation
of a few simple axioms ; and yet these are all genxerad truths.
It may be added, that they are truths which to an intelli-
gence sufficiently refined would shine forth in their own
unborrowed light, without the need of those connecting
links of thought, those steps of wearisome and often painful
deduction, by which the knowledge of them is actually
acquired. Let us define as fundamental those laws. and
principles from which all other general truths of science may
be deduced, and into which they may all be again resolved.
Shall we then err in regarding that as the true science of
Logic which, laying down certain elementary laws, con-
firmed by the very testimony of the mind, permits us thence
to deduce, by uniform processes, the entire chain of its
secondary consequences, and furnishes, for its practical
applications, methods of perfect generality? Let it be
considered whether in any science, viewed either asa system
of truth or as the foundation of a practical art, there can

6 THE LAWS OF THOUGHT

properly be any other test of the completeness and the
fundamental characters of its laws, than the completeness
of its system of derived truths, and the generality of the
methods which it serves to establish. Other questions may
indeed present themselves. Convenience, prescription, in-
dividual preference, may urge their claims and deserve
attention. But as respects the question of what constitutes
science in its abstract integrity, I apprehend that no other
considerations than the above are properly of any value.

6. It is designed, in the next place, to give expression
in this treatise to the fundamental laws of reasoning in the
symbolical language of a Calculus. Upon this head it will
suffice to say, that those laws are such as to suggest this
mode of expression, and to give to it a peculiar and ex-
clusive fitness for the ends in view. [6] There is not only
a close analogy between the operations of the mind in
general reasoning and its operations in the particular science
of Algebra, but there is to a considerable extent an exact
agreement in the laws by which the two classes of operations
are conducted. Of course the laws must in both cases be
determined independently ; any formal agreement between
them can only be established a@ posteriort by actual compari-
son. To borrow the notation of the science of Number,
and then assume that in its new application the laws by
which its use is governed will remain unchanged, would be
mere hypothesis. There exist, indeed, certain general
principles founded in the very nature of language, by which
the use of symbols, which are but the elements of scientific
language, is determined. To a certain extent these
elements are arbitrary. Their interpretation is purely
conventional: we are permitted to employ them in what-
ever sense we please, But this permission is limited by
two indispensable conditions,—first, that from the sense
once conventionally established we never, in the same
process of reasoning, depart ; secondly, that the laws by
which the process is conducted be founded exclusively
_upon the above fixed sense or meaning of the symbols

NATURE AND DESIGN. OF (THIS WORK. °%

employed. In accordance with these principles, any agree-
ment which may be established between the laws of the
symbols of Logic and those of Algebra can but issue in an
agreement of processes. The two provinces of interpreta-
tion remain apart and independent, each subject to its own
laws and conditions.

Now the actual investigations of the following pages
exhibit Logic, in its practical aspect, as a system of pro-
cesses carried on by the aid of symbols having a definite
interpretation, and subject to laws founded upon that inter-
“pretation alone. But at the same time they exhibit those
faws as identical in form with the laws of the general
symbols of Algebra, with this single addition, viz., that the
symbols of Logic are further subject to a special law
(Chap. II.), to which the symbols of quantity, as such, are
not subject. Upon the nature and the evidence of this law
it is not purposed here to dwell. These questions will be
fully discussed in a future page. But as constituting the
essential ground of difference between those forms of infer-
ence with which Logic is [7] conversant, and those which
present themselves in the particular science of Number, the
law i in question is deserving of more than a passing notice. .
it may be said that it lies at the very foundation of general
reasoning, —that it governs those intellectual acts of concep-
tion or of imagination which are preliminary to the processes
of logical deduction, and that it gives to the processes them-
selves much of their actual form and expression. It may
hence be affirmed that this law constitutes the germ or seminal
principle, of which every approximation to a general method
in Logic is the more or less perfect development.

7. The principle has already been laid down (5) that the
sufficiency and truly fundamental character of any assumed
system of laws in the science of Logic must partly be seen
in the perfection of the methods to which they conduct us.
It remains, then, to consider what the requirements of a
general method in Logic are, and how far they are fulfilled
in the system of the present work.

8 THE LAWS OF THOCGHT

Logic is conversant with two kinds of relations, —relations
among things, and relations among facts. But as facts are
-expressed by propositions, the latter species of relation
may, at least for the purposes of Logic, be resolved into a
relation among propositions. The assertion that the fact or
event 4 is an invariable consequent of the fact or event B
may, to this extent at least, be regarded as equivalent to
the assertion, that the truth of the proposition affirming the
occurrence of the event & always implies the truth of the
proposition affirming the occurrence of the event A. Instead,
then, of saying that Logic is conversant with relations
among things and relations among facts, we are permitted
, to say that it is concerned with relations among things and
᾿ relations among propositions. Of the former kind of rela-
tions we have an example in the proposition—‘‘ All men
are mortal” ; of the latter kind in the proposition—‘‘ If the
sun is totally eclipsed, the stars will become visible.” The
one expresses a relation between ‘‘men” and ‘‘ mortal
beings,” the other between the elementary propositions—
‘‘The sun is totally eclipsed”; ‘*The stars will become
visible.” Among such relations I suppose to be included
those which affirm or deny existence with respect to things,
and those which affirm or deny truth with re-[8]spect to
propositions. Now let those things or those propositions
among which relation is expressed be termed the elements
of the propositions by which such relation is expressed.
Proceeding from this definition, we may then say that the
“ premises of any logical argument express given relations
among certain elements, and that the conclusion must ex-
press an zmplied relation among those elements, or among
a part of them, ze. a relation implied by or inferentially
involved in the premises,

8. Now this being premised, the requirements of a general
method in Logic seem to be the following :—

Ist. As the conclusion must express a relation among
the whole or among a part of the elements involved in the
premises, it is requisite that we should possess the means of

NATURE AND DESIGN OF THIS WORK <9

eliminating those elements which we desire not to appear
in the conclusion, and of determining the whole amount of
relation implied by the premises among the elements which
we wish to retain. Those elements which do not present
themselves in the conclusion are, in the language of the
common Logic, called middle terms; and the species of
limination exemplified in treatises on Logic consists in
deducing from two propositions, containing a common
element or middle term, a conclusion connecting the two
remaining terms. But the problem of elimination, as con-
templated in this work, possesses a much wider scope. It
proposes not merely the elimination of one middle term from
two propositions, but the elimination generally of middle
terms from propositions, without regard to the number of
either of them, or to the nature of their connexion. To
this object neither the processes of Logic nor those of
Algebra, in their actual state, present any strict parallel.
In the latter science the problem of elimination is known to
be limited in the following manner :—From two equations
we can eliminate one symbol of quantity; from three
equations two symbols; and, generally, from 2 equations
m—1 symbols. But though this condition, necessary in
Algebra, seems to prevail in the existing Logic also, it has
no essential place in Logic as a science. There, no relation
whatever can be proved to prevail between the number of
terms to be eliminated and the number of propositions
from which the elimination is to be effected. From the
equation representing a single proposition, any num-[9]ber
of symbols representing terms or elements in Logic may be
eliminated ; and from any number of equations represent-
ing propositions, one or any other number of symbols of
this kind may be eliminated in a similar manner. For such
elimination there exists one general process applicable to all
cases. This is one of the many remarkable consequences
of that distinguishing law of the symbols of Logic, to which
attention has been already directed.

2ndly. It should be within the province of a general

10 THE LAWS OF FHOUGAT

method in Logic to express the final relation among the
elements of the conclusion by any admissible £zxd of pro-
position, or in any selected order of terms. Among varieties
of kind we may reckon those which logicians have designated
by the terms categorical, hypothetical, disjunctive, &c. To
a choice or selection in the order of the terms, we may refer
whatsoever is dependent upon the appearance of particular
elements in the subject or in the predicate, in the antecedent
or in the consequent, of that proposition which forms the
‘‘conclusion.” But waiving the language of the schools,
let us consider what really distinct species of problems may
present themselves to our notice. We have seen that the
elements of the final or inferred relation may either be ¢hzngs
or propositions. Suppose the former case ; then it might be
required to deduce from the premises a definition or descrip-
tion of some one thing, or class of things, constituting an
element of the conclusion in terms of the other things in-
volved in it. Or we might form the conception of some
thing or class of things, involving more than one of the
elements of the conclusion, and require its expression in
terms of the other elements. Again, suppose the elements
retained in the conclusion to be propositions, we might
desire to ascertain such points as the following, Viz. ,
Whether, in virtue of the premises, any of those propositions,
taken singly, are true or false?—Whether particular com-
binations of them are true or false ?—Whether, assuming a
particular proposition to be true, any consequences will
follow, and if so, what consequences, with respect to the
other propositions?—Whether any particular condition
being assumed with reference to certain of the propositions,
any consequences, and what consequences, will follow with
respect to the others? and [10] so on. I say that these
are general questions, which it should fall within the scope
or province of a general method in Logic to solve. Perhaps
we might include them all under this one statement of the
final problem of practical Logic. Given a set of premises
expressing relations among certain elements, whether things

NATURE AND DESIGN OF THIS (WORK . 1%

or propositions : required explicitly the whole relation con-
sequent among any of those elements under any proposed
conditions, and in any proposed form. That this problem,
under all its aspects, is resolvable, will hereafter appear.
But it is not for the sake of noticing this fact, that the above
inquiry into the nature and the functions of a general method
in Logic has been introduced. It is necessary that the
reader should apprehend what are the specific ends of the
investigation upon which we are entering, as well as the
principles which are to guide us to the attainment of them.

Ὁ Possibly it may here be said that the Logic. of
Aristotle, in its rules of syllogism and conversion, sets
forth the elementary processes of which all reasoning con-
sists, and that beyond these there is neither scope nor
occasion for a general method. I have no desire to point
out the defects of the common Logic, nor do I wish to refer
to it any further than is necessary, in order to place in its
true light the nature of the present treatise. With this end
alone in view, I would remark :—tIst. That syllogism, con-
version, &c., are not the ultimate processes of Logic. It
will be shown in this treatise that they are founded upon,
and are resolvable into, ulterior and more simple processes
which constitute the real elements of method in Logic. Nor
is it true in fact that all inference is reducible to the par-
ticular forms of syllogism and conversion.—Vzde Chap. XV.
2ndly. If all inference were reducible to these two processes
(and it has been maintained that it is reducible to syllogism
alone), there would still exist the same necessity for a
general method. For it would still be requisite to determine
in what order the processes should succeed each other, as
well as their particular nature, in order that the desired
relation should be obtained. By the desired relation |
mean that full relation which, in virtue of the premises, con-
nects any elements selected out of the premises at will, and
which, moreover, expresses that relation in any desired
form and order. [11] If we may judge from the mathe-
matical sciences, which are the most perfect examples of

12 THE LAWS OF THOUGHT

method known, this drective function of Method constitutes
its chief office and distinction. The fundamental processes
of arithmetic, for instance, are in themselves but the elements
of a possible science. To assign their nature is the first
business of its method, but to arrange their succession is its
subsequent and higher function. In the more complex
examples of logical deduction, and especially in those
which form a basis for the solution of difficult questions in
the theory of Probabilities, the aid of a directive method,
such as a Calculus alone can supply, is indispensable.

10. Whence it is that the ultimate laws of Logic are
mathematical in their form; why they are, except in a
single point, identical with the general laws of Number ;
and why in that particular point they differ ;—are questions
upon which it might not be very remote from presumption
to endeavour to pronounce a positive judgment. Probably
they lie beyond the reach of our limited faculties. It may,
perhaps, be permitted to the mind to attain a knowledge of
the laws to which it is itself subject, without its being also
given to it to understand their ground and origin, or even,
except in a very limited degree, to comprehend their fitness
for their end, as compared with other and conceivable systems
of law. Such knowledge is, indeed, unnecessary for the
ends of science, which properly concerns itself with what is,
and seeks not for grounds of preference or reasons of appoint-
ment, These considerations furnish a sufficient answer to
all protests against the exhibition of Logic in the form of a
Calculus. It is not because we choose to assign to it such
a mode of manifestation, but because the ultimate laws of
thought render that mode possible, and prescribe its character,
and forbid, as it would seem, the perfect manifestation of
the science in any other form, that such a mode demands
adoption. It is to be remembered that it is the business of
science not to create laws, but to discover them. We’do
not originate the constitution of our own minds, greatly as it
may be in our power to modify their character. And as the
laws of the human intellect do not depend upon our will, so

NATURE AND DESIGN OF THIS WORK 13

the forms of the science, of which they constitute the basis,
are in all essential regards independent of individual choice.

[12] 11. Beside the general statement of the principles
of the above method, this treatise will exhibit its applica-
tion to the analysis of a considerable variety of proposi-
tions, and of trains of propositions constituting the premises
of demonstrative arguments. These examples have been
selected from various writers, they differ greatly in com-
plexity, and they embrace a wide range of subjects. Though
in this particular respect it may appear to some that too
great a latitude of choice has been exercised, I do not deem
it necessary to offer any apology upon this account. That
Logic, as a science, is susceptible of very wide applications
is admitted ; but it is equally certain that its ultimate forms
and processes are mathematical. Any objection a priori
which may therefore be supposed to lie against the adoption
of such forms and processes in the discussion of a problem
of morals or of general philosophy must be founded upon
misapprehension or false analogy. It is not of the essence
of mathematics to be conversant with the ideas of number
and quantity. Whether as a general habit of mind it would
be desirable to apply symbolical processes to moral argu-
ment, is another question. Possibly, as I have elsewhere
observed,* the perfection of the method of Logic may be
chiefly valuable as an evidence of the speculative truth of
its principles. To supersede the employment of common
reasoning, or to subject it to the rigour of technical forms,
would be the last desire of one who knows the value of that
intellectual toil and warfare which imparts to the mind an
athletic vigour, and teaches it to contend with difficulties,
and to rely upon itself in emergencies. Nevertheless, cases
may arise in which the value of a scientific procedure, even
in those things which fall confessedly under the ordinary
dominion of the reason, may be felt and acknowledged.
Some examples of this kind will be found in the present

work.

* Mathematical Analysis of Logic. London: G, Bell. 1847.

14 THE PAWS OF THOUGHT

12. The general doctrine and method of Logic above
explained form also the basis of a theory and corresponding
method of Probabilities. Accordingly, the development of
such a theory and method, upon the above principles, will
constitute a distinct object of the present treatise. Of the
nature of this application it may be desirable to give here
some account, more especially as [13] regards the character
of the solutions to which it leads. In connexion with this
object some further detail will be requisite concerning
the forms in which the results of the logical analysis are
presented.

The ground of this necessity of a prior method in Logic,
as the basis of a theory of Probabilities, may be stated in
a few words. Before we can determine the mode in which
the expected frequency of occurrence of a particular event
is dependent upon the known frequency of occurrence of
any other events, we must be acquainted with the mutual
dependence of the events themselves. Speaking technically,
we must be able to express the event whose probability is
sought, as a function of the events whose probabilities are
given. Now this explicit determination belongs in all in-
stances to the department of Logic. Probability, however,
in its mathematical acceptation, admits of numerical measure-
ment. Hence the subject of Probabilities belongs equally
to the science of Number and to that of Logic. In recognis-
ing the co-ordinate existence of both these elements, the
present treatise differs from all previous ones; and as this
difference not only affects the question of the possibility of
the solution of problems in a large number of instances, but
also introduces new and important elements into the solutions
obtained, I deem it necessary to state here, at some length,
the peculiar consequences of the theory developed in the
following pages.

13. The measure of the probability of an event is usually
defined as a fraction, of which the numerator represents the
number of cases favourable to the event, and the denominator
the whole number of cases favourable and unfavourable ; all

NATURE AND DESIGN OF THIS WORK: ἃς

cases being supposed equally likely to happen. That defini-
tion is adopted in the present work. At the same time it
is shown that there is another aspect of the subject (shortly
to be referred to) which might equally be regarded as funda-
mental, and which would actually lead to the same system
of methods and conclusions, It may be added, that so far
as the received conclusions of the theory of Probabilities
extend, and so far as they are consequences of its funda-
mental definitions, they do not differ from the results
(supposed to be equally correct in inference) of the method
of this work.

[14] Again, although questions in the theory of Pro-
babilities present themselves under various aspects, and
may be variously modified by algebraical and other con-
ditions, there seems to be one general type to which all
such questions, or so much of each of them as truly belongs
to the theory of Probabilities, may be referred. Considered
with reference to the data and the gu@sztum, that type may
be described as follows :—1st. The data are the probabilities
of one or more given events, each probability being either
that of the absolute fulfilment of the event to which it re-
lates, or the probability of its fulfilment under given sup-
posed conditions. 2ndly. The gu@siztum, or object sought,
is the probability of the fulfilment, absolutely or condition-
ally, of some other event differing in expression from those
in the data, but more or less involving the same elements.
As concerns the data, they are either causally given,—as
when the probability of a particular throw of a die is
deduced from a knowledge of the constitution of the piece,
—or they are derived from observation of repeated in-
stances of the success or failure of events. In the latter
case the probability of an event may be defined as the limit
toward which the ratio of the favourable to the whole number
of observed cases approaches (the uniformity of nature being
presupposed) as the observations are indefinitely continued.
Lastly, as concerns the nature or relation of the events in
question, an important distinction remains. Those events

16 THE LAWS OF THOUGAT

are either simple or compound. By a compound event is
meant one of which the expression in language, or the con-
ception in thought, depends upon the expression or the con-
ception of other events, which, in relation to it, may be
regarded as simple events. To say ‘‘it. rains,” or to say
‘it thunders,” is to express the occurrence of a simple
event; but to say ‘‘it rains and thunders,” or to say ‘‘it
either rains or thunders,” is to express that of a compound
event. For the expression of that event depends upon the
elementary expressions, ‘‘it rains,” ‘‘it thunders.” The
criterion of simple events is not, therefore, any supposed sim-
plicity in their nature. It is founded solely on the mode of
their expression in language or conception in thought.

14. Now one general problem, which the existing theory
of Probabilities enables us to solve, is the following, viz. :—
Given [15] the probabilities of any simple events : required
the probability of a given compound event, z.e. of an event
compounded in a given manner out of the given simple
events. The problem can also be solved when the compound
event, whose probability is required, is subjected to given
conditions, z.e. to conditions dependent also in a given
manner on the given simple events. Beside this general
problem, there exist also particular problems of which the
principle of solution is known. Various questions relating
to causes and effects can be solved by known methods under
the particular hypothesis that the causes are mutually ex-
clusive, but apparently not otherwise. Beyond this it is
not clear that any advance has been made toward the
solution of what may be regarded as the general problem
of the science, viz. : Given the probabilities of any events,
simple or compound, conditioned or unconditioned : required
the probability of any other event equally arbitrary in ex-
pression and conception. In the statement of this question
it is not even postulated that the events whose probabilities
are given, and the one whose probability is sought, should
involve some common elements, because it is the office of a
method to determine whether the data of a problem are

NATURE AND DESIGN OF THIS WORK 17

sufficient for the end in view, and to indicate, when they
are not so, wherein the deficiency consists.

This problem, in the most unrestricted form of its state-
ment, is resolvable by the method of the present treatise ;
or, to speak more precisely, its theoretical solution is com-
pletely given, and its practical solution is brought to depend
only upon processes purely mathematical, such as the re-
solution and analysis of equations. The order and character
of the general solution may thus be described.

15. In the first place it is always possible, by the pre-
liminary method of the Calculus of Logic, to express the
event whose probability is sought as a logical function of the
events whose probabilities are given. The result is of the
following character : Suppose that Y represents the event
whose probability is sought, A, δ, C, &c. the events whose
probabilities are given, those events being either simple or
compound. Then the whole relation of the event X to the
events, A, B, C, &c. is deduced in the form of what mathe-
maticians term a development, consisting, in [16] the most
’ general case, of four distinct classes of terms. By the first
~class are expressed those combinations of the events A, B, C,
which both necessarily accompany and necessarily indicate
the occurrence of the event X ; by the second class, those
combinations which necessarily accompany, but do not
necessarily imply, the occurrence of the event 1; by the
third class, those combinations whose occurrence in con-
nexion with the event X is impossible, but not otherwise
impossible ; by the fourth class, those combinations whose
occurrence is impossible under any circumstances. I shall
not dwell upon this statement of the result of the logical
analysis of the problem, further than to remark that the
elements which it presents are precisely those by which the
expectation of the event Y, as dependent upon our know-
ledge of the events A, B, C, is, or alone can be, affected.
General reasoning would verify this conclusion ; but general
reasoning would not usually avail to disentangle the com-

plicated web of events and circumstances from which the
2

18 THE LAWS OF THOUGHT

solution above described must be evolved. The attainment
of this object constitutes the first step towards the complete
solution of the question proposed. It is to be noted that
thus far the process of solution is logical, z.e. conducted by
symbols of logical significance, and resulting in an equation
interpretable into a proposttion. Let this result be termed
the final logical equation.

The second step of the process deserves attentive remark.
From the final logical equation to which the previous step
has conducted us, are deduced, by inspection, a series of
algebraic equations implicitly involving the complete solu-
tion of the problem proposed. Of the mode in which this
transition is effected let it suffice to say, that there exists
a definite relation between the laws by which the pro-
babilities of events are expressed as algebraic functions of
the probabilities of other events upon which they depend,
and the laws by which the logical connexion of the events
is itself expressed. This relation, like the other coincidences
of formal law which have been referred to, is not founded
upon hypothesis, but is made known to us by observation
(1. 4), and reflection. If, however, its reality were assumed
a priort as the basis of the very definition of Probability,
strict deduction would thence lead us to the received
numerical definition as a [17] necessary consequence. The
Theory of Probabilities stands, as it has already been
remarked (I. 12), in equally close relation to Logic and to
Arithmetic ; and it is indifferent, so far as results are con-
cerned, whether we regard it as springing out of the latter
of these sciences, or as founded in the mutual relations
which connect the two together. )

16, There are some circumstances, interesting perhaps to
the mathematician, attending the general solutions deduced
by the above method, which it may be desirable to notice.

ist. As the method is independent of the number and the
nature of the data, it continues to be applicable when the
latter are insufficient to render determinate the value sought.
When such is the case, the final expression of the solution

NATURE AND DESIGN OF THIS WORK 19

’ will contain terms with arbitrary constant coefficients. To
such terms there will correspond terms in the final logical
equation (I. 15), the interpretation of which will inform us
what new data are requisite in order to determine the values
of those constants, and thus render the numerical solution
complete. If such data are not to be obtained, we can still,
by giving to the constants their limiting values Ὁ and 1,
determine the limits within which the probability sought
must lie independently of all further experience. When the
event whose probability is sought is guzte independent of
those whose probabilities are given, the limits thus obtained
for its value will be o and 1, as it is evident that they ought
to be, and the interpretation of the constants will only lead
to a re-statement of the original problem.
: 2ndly. The expression of the final solution will in all
cases involve a particular element of quantity, determinable
by the solution of an algebraic equation. Now when that
equation is of an elevated degree, a difficulty may seem to
arise as to the selection of the proper root. There are,
indeed, cases in which both the elements given and the
element sought are so obviously restricted by necessary
conditions that no choice remains. But in complex in-
stances the discovery of such conditions, by unassisted force
of reasoning, would be hopeless. A distinct method is
requisite for this end,—a method which might not inappro-
priately be termed the Calculus of Statistical Conditions.
Into the nature of this method [ shall not here further enter
[18] than to say, that, like the previous method, it is based
upon the employment of the ‘‘ final logical equation,” and
that it definitely assigns, Ist, the conditions which must be
fulfilled among the numerical elements of the data, in order
that the problem may be real, z.e. derived from a fosszble
expertence; 2ndly, the numerical limits, within which the
probability sought must have been confined, if, instead of
being determined by theory, it had been deduced directly
by observation from the same system of phenomena from
which the data were derived. It is clear that these limits

20 THE LAWS OF THOUGAT

will be actual limits of the probability sought. Now, on
supposing the data subject to the conditions above assigned
to them, it appears in every instance which I have examined
that there exists one root, and only one root, of the final
algebraic equation which is subject to the required limita-
tions. Every source of ambiguity is thus removed. It
would even seem that new truths relating to the theory of
algebraic equations are thus incidentally brought to light. It
is remarkable that the special element of quantity, to which
the previous discussion relates, depends only upon the daa,
and not at all upon the guesztum of the problem proposed.
Hence the solution of each particular problem unties the
knot of difficulty for a system of problems, viz., for that
system of problems which is marked by the possession of
common data, independently of the nature of their gu@szta.
This circumstance is important whenever from a particular
system of data it is required to deduce a series of connected
conclusions. And it further gives to the solutions of
particular problems that character of relationship, derived
from their dependence upon a central and fundamental
unity, which not unfrequently marks the application of
general methods.

17. But though the above considerations, with others of
a like nature, justify the assertion that the method of this
treatise, for the solution of questions in the theory of
Probabilities, is a general method, it does not thence follow
that we are relieved in all cases from the necessity of re-
course to hypothetical grounds. It has been observed that
a solution may consist entirely of terms affected by arbitrary
constant coefficients,—may, in fact, be wholly indefinite.
The application of the method of this work to some of the
most important questions within its range would—[19] were
the data of experience alone employed—present results of
this character. ‘To obtain a definite solution it is necessary,
in such cases, to have recourse to hypotheses possessing more
or less of independent probability, but incapable of exact
verification. Generally speaking, such hypotheses will differ

NATURE AND DESIGN OF THIS WORK 21

from the immediate results of experience in partaking of a
logical rather than of a numerical character ; in prescribing
the conditions under which phenomena occur, rather than
assigning the relative frequency of their occurrence. This
circumstance is, however, unimportant. Whatever their
nature may be, the hypotheses assumed must thenceforth
be regarded as belonging to the actual data, although tend-
ing, as is obvious, to give to the solution itself somewhat
of a hypothetical character. With this understanding as
to the possible sources of the data actually employed, the
method is perfectly general, but for the correctness of the
hypothetical elements introduced it is of course no more
responsible than for the correctness of the numerical data
derived from experience.

In illustration of these remarks we may observe that the
theory of the reduction of astronomical observations * rests,
in part, upon hypothetical grounds. It assumes certain
positions as to the nature of error, the equal probabilities of
its occurrence in the form of excess or defect, &c., without
which it would be impossible to obtain any definite con-
clusions from a system of conflicting observations. But
granting such positions as the above, the residue of the
investigation falls strictly within the province of the theory
of Probabilities. Similar observations apply to the important
problem which proposes to deduce from the records of the
“majorities of a deliberative assembly the mean probability
of correct judgment in one of its members. If the method
of this treatise be applied to the mere numerical data, the
solution obtained is of that wholly indefinite kind above
described. And to show in a more eminent degree the
insufficiency of those data by themselves, the interpretation
of the arbitrary constants (I. 16) which appear in the solution,
merely produces [20] a re-statement of the original problem.
Admitting, however, the hypothesis of the independent
formation of opinion in the individual mind, either absolutely,

* The author designs to treat this subject either in a separate work or ina future
Appendix. In the present treatise he avoids the use of the integral calculus,

22 THE LAWS OF THOUGHT

as in the speculations of Laplace and Poisson, or under
limitations imposed by the actual data, as will be seen in
this treatise, Chap. XXI., the problem assumes a far more
definite character. It will be manifest that the ulterior
value of the theory of Probabilities must depend very much
upon the correct formation of such mediate hypotheses,
where the purely experimental data are insufficient for
definite solution, and where that further experience indicated
by the interpretation of the final logical equation is unattain-
able. Upon the other hand, an undue readiness to form
hypotheses in subjects which from their very nature are
placed beyond human ken, must re-act upon the credit of
the theory of Probabilities, and tend to throw doubt in the
general mind over its most legitimate conclusions.

18. It would, perhaps, be premature to speculate here
upon the question whether the methods of abstract science
are likely at any future day to render service in the investi-
gation of social problems at all commensurate with those
which they have rendered in various departments of physical
inquiry. An attempt to resolve this question upon pure
a priori grounds of reasoning would be very likely to mis-
lead us. For example, the consideration of human free-
agency would seem at first sight to preclude the idea that
the movements of the social system should ever manifest
that character of orderly evolution which we are prepared to
expect under the reign of a physical necessity. Yet already
do the researches of the statist reveal to us facts at variance
with such an anticipation. Thus the records of crime and
pauperism present a degree of regularity unknown in regions
in which the disturbing influence of human wants and
passions is unfelt. On the other hand, the distemperature
of seasons, the eruption of volcanoes, the spread of blight
in the vegetable, or of epidemic maladies in the animal
kingdom, things apparently or chiefly the product of natural
causes, refuse to be submitted to regular and apprehensible
laws. ‘‘ Fickle as the wind,” is a proverbial expression.
Reflection upon these points teaches us in some degree to

NATURE AND DESIGN OF THIS WORK δὴ

correct our earlier judgments. We learn that we are not to
[21] expect, under the dominion of necessity, an order per-
ceptible to human observation, unless the play of its produc-
ing causes is sufficiently simple ; nor, on the other hand, to
deem that free agency in the individual is inconsistent with
regularity in the motions of the system of which he forms a
component unit. Human freedom stands out as an apparent
fact of our consciousness, while it is also, 1 conceive, a highly
probable deduction of analogy (Chap. XXII.) from the nature
of that portion of the mind whose scientific constitution we
are able to investigate. But whether accepted as a fact
reposing on consciousness, or as a conclusion sanctioned by
the reason, it must be so interpreted as not to conflict with
an established result of observation, viz. : that phanomena,
in the production of which large masses of men are concerned,
do actually exhibit a very remarkable degree of regularity,
enabling us to collect in each succeeding age the elements
upon which the estimate of its state and progress, so far as
manifested in outward results, must depend. There is thus
no sound objection ὦ priorvt against the possibility of that
species of data which is requisite for the experimental
foundation of a science of social statistics. Again, what-
ever other object this treatise may accomplish, it is presumed
that it will leave no doubt as to the existence of a system
of abstract principles and of methods founded upon those
principles, by which any collective body of social data may
be made to yield, in an explicit form, whatever information
they implicitly involve. There may, where the data are
exceedingly complex, be very great difficulty in obtaining
this information,—difficulty due not to any imperfection of
the theory, but to the laborious character of the analytical
_ processes to which it points. It is quite conceivable that in
many instances that difficulty may be such as only united
effort could overcome. But that we possess theoretically in
all cases, and practically, so far as the requisite labour of
calculation may be supplied, the means of evolving from
statistical records the seeds of general truths which lie buried

24 THE LAWS OF THOUGHT

amid the mass of figures, is a position which may, 1 con-
ceive, with perfect safety be affirmed.

19. But beyond these general positions I do not venture
to speak in terms of assurance. Whether the results which
might be expected from the application of scientific methods
to statis-[22]tical records, over and above those the dis-
covery of which requires no such aid, would so far com-
pensate for the labour involved as to render it worth while
to institute such investigations upon a proper scale of magni-
tude, is a point which could, perhaps, only be determined
by experience. It is to be desired, and it might without
great presumption be expected, that in this, as in other
instances, the abstract doctrines of science should minister
to more than intellectual gratification. Nor, viewing the
apparent order in which the sciences have been evolved, and
have successively contributed their aid to the service of man-
kind, does it seem very improbable that a day may arrive
in which similar aid may accrue from departments of the
field of knowledge yet more intimately allied with the
elements of human welfare. Let the speculations of this
treatise, however, rest at present simply upon their claim
to be regarded as true.

20. I design, in the last place, to endeavour to educe from
the scientific results of the previous inquiries some general
intimations respecting the nature and constitution of the
human mind. Into the grounds of the possibility of this
species of inference it is not necessary to enter here. One
or two general observations may serve to indicate the track
which I shall endeavour to follow. It cannot but be admitted
that our views of the science of Logic must materially in-
fluence, perhaps mainly determine, our opinions upon the
nature of the intellectual faculties. For example, the ques-
tion whether reasoning consists merely in the application
of certain first or necessary truths, with which the mind
has been originally imprinted, or whether the mind is itself
a seat of law, whose operation is as manifest and as con-
clusive in the particular as in the general formula, or whether,

NATURE AND DESIGN OF THIS WORK 25

as some not undistinguished writers seem to maintain, all
reasoning is of particulars ; this question, I say, is one which
not merely affects the science of Logic, but also concerns the
formation of just views of the constitution of the intellectual
faculties. Again, if it is concluded that the mind is by
original constitution a seat of law, the question of the nature
of its subjection to this law,—whether, for instance, it is an
obedience founded upon necessity, like that which sustains
the revolutions of the heavens, and preserves the order of
Nature,—or whether [23] it is a subjection of some quite
distinct kind, is also a matter of deep speculative interest.
Further, if the mind is truly determined to be a subject of
law, and if its laws also are truly assigned, the question of
their probable or necessary influence upon the course of
human thought in different ages is one invested with great
importance, and well deserving a patient investigation, as
matter both of philosophy and of history. These and other
questions I propose, however imperfectly, to discuss in the
concluding portion of the present work. They belong,
_ perhaps, to the domain of probable or conjectural, rather
than to that of positive, knowledge. But it may happen
that where there is not sufficient warrant for the certainties
of science, there may be grounds of analogy adequate for
the suggestion of highly probable opinions. It has seemed
to me better that this discussion should be entirely reserved
for the sequel of the main business of this treatise, —which is
the investigation of scientific truths and laws. Experience
sufficiently instructs us that the proper order of advance-
ment in all inquiries after truth is to proceed from the
known to the unknown. There are parts, even of the
philosophy and constitution of the human mind, which have
been placed fully within the reach of our investigation. To
make a due acquaintance with those portions of our nature
the basis of all endeavours to penetrate amid the shadows
and uncertainties of that conjectural realm which lies beyond
and above them, is the course most accordant with the
limitations of our present condition.

[24] CHAPTER II

OF SIGNS IN GENERAL, AND OF THE SIGNS APPROPRIATE
TO THE SCIENCE OF LOGIC IN PARTICULAR; ALSO
OF THE LAWS TO WHICH THAT CLASS OF SIGNS
ARE SUBJECT

1. THAT Language is-an instrument of human reason, and
not merely a medium for the expression of thought, is a
truth generally admitted. It is proposed in this chapter to
inquire what it is that renders Language thus subservient
to the most important of our intellectual faculties. In the
various steps of this inquiry we shall be led to consider the
constitution of Language, considered as a system adapted
to an end or purpose ; to investigate its elements ; to seek
to determine their mutual relation and dependence ; and to
inquire in what manner they contribute to the attainment of
the end to which, as co-ordinate parts of a system, they
have respect.

In proceeding to these inquiries, it will not be necessary
to enter into the discussion of that famous question of the
schools, whether Language is to be regarded as an essential
instrument of reasoning, or whether, on the other hand, it
is possible for us to reason without its aid. I suppose this
question to be beside the design of the present treatise, for
the following reason, viz., that it is the business of Science
to investigate laws; and that, whether we regard signs as
the representatives of things and of their relations, or as the
representatives of the conceptions and operations of the
human intellect, in studying the laws of signs, we are in
effect studying the manifested laws of reasoning. If there
exists a difference between the two inquiries, it is one which

does not affect the scientific expressions of formal law, which
26

SIGNS AND THEIR LAWS 27

are the object of investigation in the present stage of this
work, but relates only to the mode in which those results
are presented to the mental regard. For though in in-
vestigating the laws of signs, a posterzorz, the immediate
subject of examination is Language, with the rules which
govern its use ; while in making the internal [25] processes

of thought the direct object of inquiry, we appeal in a more

immediate way to our personal consciousness,—it will be
found that in both cases the results obtained are formally
equivalent. Nor could we easily conceive, that the un-
numbered tongues and dialects of the earth should have
preserved through a long succession of ages so much that is
common and universal, were we not assured of the existence
of some deep foundation of their agreement in the laws of
the mind itself.

2. The elements of which all language consists are signs

or symbols. Words are signs. Sometimes they are said

to represent things ; sometimes the operations by which the

mind combines together the simple notions of things into

complex conceptions ; sometimes they express the relations
of action, passion, or mere quality, which we perceive to

exist among the objects of our experience ; sometimes the
emotions of the perceiving mind. But words, although in
this and in other ways they fulfil the office of signs, or re-
presentative symbols, are not the only signs which we are
capable of employing. Arbitrary marks, which speak only

to the eye, and arbitrary sounds or actions, which address

themselves to some other sense, are equally of the nature of

signs, provided that their representative office is defined and

‘understood. Inthe mathematical sciences, letters, and the

symbols +, —, =, &c., are used as signs, although the

_ term ‘‘sign” is applied to the latter class of symbols, which

represent operations or relations, rather than to the former,

which represent the elements of number and quantity. As
the real import of a sign does not in any way depend upon
‘its particular form or expression, so neither do the laws

which determine its use. In the present treatise, however,

28 THE LAWS OF THOUGAT

it is with written signs that we have to do, and it is with
reference to these exclusively that the term ‘‘sign” will be
employed. The essential properties of signs are enumerated
in the following definition.

Definition.—A sign is an arbitrary mark, having a fixed
interpretation, and susceptible of combination with other
signs in subjection to fixed laws dependent upon their
mutual interpretation.

3. Let us consider the particulars involved in the above
definition separately.

[26] (1.) In the first place, a sign is an arbitrary mark.
It is clearly indifferent what particular word or token we
associate with a given idea, provided that the association
once made is permanent. The Romans expressed by the
word ‘‘civitas” what we designate by the word ‘‘ state.”
But both they and we might equally well have employed
any other word to represent the same conception. Nothing,
indeed, in the nature of Language would prevent us from
using a mere letter in the same sense. Were this done, the
laws according to which that letter would require to be used
would be essentially the same with the laws which govern
the use of ‘‘civitas” in the Latin, and of ‘‘state” in the
English language, so far at least as the use of those words .
is regulated by any general principles common to all
languages alike.

(2.) In the second place, it is necessary that each sign
should possess, within the limits of the same discourse or
process of reasoning, a fixed interpretation. The necessity
of this condition is obvious, and seems to be founded in the
very nature of the subject. There exists, however, a dispute
as to the precise nature of the representative office of words
or symbols used as names in the processes of reasoning. By
some it is maintained, that they represent the conceptions
of the mind alone; by others, that they represent things.
The question is not of great importance here, as its decision
cannot affect the laws according to which signs are employed.
I apprehend, however, that the general answer to this and

SIGNS AND THEIR LAWS 29

such like questions is, that in the processes of reasoning,
signs stand in the place and fulfil the office of the conceptions
and operations of the mind ; but that as those conceptions
and operations represent things, and the connexions and
relations of things, so signs represent things with their con-
nexions and relations ; and lastly, that as signs stand in the
place of the conceptions and operations of the mind, they
are subject to the laws of those conceptions and operations.
This view will be more fully elucidated in the next chapter ;
but it here serves to explain the third of those particulars
involved in the definition of a sign, viz., its subjection to
fixed laws of combination depending upon the nature of its
interpretation.

4. The analysis and classification of those signs by which
the [27] operations of reasoning are conducted will Be con-
sidered in the following Proposition :

PROPOSITION I

All the operations of Language, as an instrument of
reasoning, may be conducted by a system of signs composed
of the following elements, viz. :

Ist. Leteral symbols, as x, y, &c., representing things as
subjects of our conceptions.

2nd. Szgus of operation, as +, —, xX, standing for those
operations of the mind by which the conceptions of things are
combined or resolved so as to form new conceptions involving
the same elements.
= 3rd. The sign of identity, =

And these symbols of Logtc are tn their use subject to
definite laws, partly agreeing with and partly differing from
the laws of the corresponding symbols tn the sctence of
Algebra.

Let it be assumed as a criterion of the true elements of
rational discourse, that they should be susceptible of com-
bination in the simplest forms and by the simplest laws,
and thus combining should generate 411 other known and

30 THE LAWS OF THOUGHT

conceivable forms of language ; and adopting this principle,
let the following classification be considered.

CLASS I

5. Appellative or descriptive signs, expressing etther the
name of a thing, or some quality or ctrcumstance belonging
to tt.

To this class we may obviously refer the substantive
proper or common, and the adjective. These may indeed
be regarded as differing only in this respect, that the former
expresses the substantive existence of the individual thing
or things to which it refers; the latter implies that exist-
ence. If we attach to the adjective the universally under-
stood subject ‘‘ being” or ‘‘thing,” it becomes virtually a
substantive, and may for all the essential purposes of reason-
ing be replaced by the substantive. Whether or not, in
every particular of the mental regard, it is the same thing to
say, ‘‘ Water is a fluid thing,” as to say, ‘‘ Water is fluid” ;
it is at least equivalent in the expression of the processes
of reasoning.

[28] It is clear also, that to the above class we must refer
any sign which may conventionally be used to express
some circumstances or relation, the detailed exposition of
which would involve the use of many signs. The epithets
of poetic diction are very frequently of this kind. They
are usually compounded adjectives, singly fulfilling the office
of a many-worded description. Homer’s ‘‘ deep-eddying
ocean” embodies a virtual description in the single word
βαθυδίνης. And conventionally any other description ad-
dressed either to the imagination or to the intellect might
equally be represented by a single sign, the use of which
would in all essential points be subject to the same laws as
the use of the adjective ‘‘ good” or ‘‘ great.” Combined
with the subject ‘“‘thing,” such a sign would virtually —
become a substantive; and by a single substantive the
combined meaning both of thing and quality might ΡῈ
expressed.

SIGNS AND THEIR LAWS 31

6. Now, as it has been defined that a sign is an arbitrary
mark, it is permissible to replace all signs of the species
above described by letters. Let us then agree to represent
the class of individuals to which a particular name or
description is applicable, by a single letter, as x If the
name is ‘‘men,” for instance, let x represent ‘‘ all men,” or
the class ‘‘men.” By a class is usually meant a collection
of individuals, to each of which a particular name or descrip-
tion may be applied ; but in this work the meaning of the
term will be extended so as to include the case in which
but a single individual exists, answering to the required
mame or description, as well as the cases denoted by the
mms “nothing” and “‘universe,” which as “classes”
should be understood to comprise respectively ‘‘no beings,”
‘fall beings.” Again, if an adjective, as ‘‘ good,” is em-
ployed as a term of description, let us represent by a letter,
as y, all things to which the description ‘‘ good” is
applicable, ze, ‘‘all good things,” or the class ‘‘ good
things.” Let it further be agreed, that by the combination
ay shall be represented that class of things to which the
names or descriptions represented by x and y are simultane-
ously applicable. Thus, if x alone stands for ‘‘ white
_ things,” and y for ‘‘ sheep,” let xy stand for ‘‘ white sheep ” ;
and in like manner, if ¢ stand for ‘‘horned things,” and
4 and y retain their previous interpretations, let zy re-
present [29] ‘‘ horned white sheep,” z.e. that collection of
things to which the name ‘‘sheep,” and the descriptions
“white” and ‘‘horned” are together applicable.

Let us now consider the laws to which the symbols 4, y,

: &c., used in the above sense, are subject.
7. First, it is evident, that according to the above com-
_binations, the order in which two symbols are written is
indifferent. The expressions zy and yx equally represent
that class of things to the several members of which the
_fames or descriptions + and y are together applicable.
Hence we have,

XY =X. (1)

32 THE LAWS OF THOUGHT

In the case of x representing white things, and y sheep,
either of the members of this equation will represent the
class of ‘‘white sheep.” There may be a difference as to
the order in which the conception is formed, but there is
none as to the individual things which are comprehended
under it. In like manner, if + represent ‘‘ estuaries,” and
y “rivers,” the expressions xy and yx will indifferently re-
Present | vivers that are estuaries,” or ‘‘ estuaries that are
rivers,” the combination in this case being in ordinary
language that of two substantives, instead of that of a
substantive and an adjective as in the previous instance.
Let there be a third symbol, as 2, representing that class
of things to which the term ‘‘ navigable” is applicable, and
any one of the following expressions,

COV, Bays Χ»5, (ECS

will represent the class of ‘‘navigable rivers that are
estuaries. ”

If one of the descriptive terms should have some implied
reference to another, it is only necessary to include that
reference expressly in its stated meaning, in order to render
the above remarks still applicable. Thus, if x represent
ἐξ wise” and y ‘‘counsellor,” we shall have to define whether
x implies wisdom in the absolute sense, or only the wisdom
of counsel. With such definition the law ry =yx continues
to be valid.

We are permitted, therefore, to employ the symbols x, y, 2,
&c., in the place of the substantives, adjectives, and descriptive
phrases subject to the rule of interpretation, that any expression
in which several of these symbols are written together shall —
vepresent all the objects or tnadt-[30]|viduals to which their
several meanings are together applicable, and to the law that
the order in which the symbols succeed each other ts indifferent.

As the rule of interpretation has been sufficiently ex-
emplified, I shall deem it unnecessary always to express
the subject ‘‘things” in defining the interpretation of a
symbol used for an adjective. When I say, let x represent

SIGNS AND THEIR LAWS 33

‘* good,” it will be understood that x only represents ‘‘ good ”
when a subject for that quality is supplied by another
pool, and that, used alone, its interpretation will be

‘*good things.”

8. Concerning the law above determined, the folate
observations, which will also be more or less appropriate to
certain other laws to be deduced hereafter, may be added.

First, I would remark, that this law is a law of thought,
and not, properly speaking, a law of things. Difference in
the order of the qualities or attributes of an object, apart
from all questions of causation, is a difference in conception
merely. The law (1) expresses as a general truth, that the
same thing may be conceived in different ways, and states
the nature of that difference ; and it does no more than this.

Secondly, As a law of thought, it is actually developed
‘in a law of Language, the product and the instrument of
thought. Though the tendency of prose writing is toward
uniformity, yet even there the order of sequence of adjectives,
absolute in their meaning, and applied to the same subject,
is indifferent, but poetic diction borrows much of its rich
diversity from the extension of the same lawful freedom to
the substantive also. The language of Milton is peculiarly
distinguished by this species of variety. Not only does
the substantive often precede the adjectives by which it is
qualified, but it is frequently placed in their midst. In the
first few lines of the invocation to Light, we meet with such
examples as the following :

“ Offspring of heaven first-born.”
“The rising world of waters dark and deep.”

“Bright effluence of dr7ght essence increate.”

Now these inverted forms are not simply the fruits of
a poetic license. They are the natural expressions of a
freedom sanc-[3I]tioned by the intimate laws of thought,
but for reasons of convenience not exercised in the ordinary
use of language.

Thirdly, The law expressed by (1) may be characterized
3

34 THE LAWS OF THOUGHT

by saying that the literal symbols 7, 7, 5, are commutative,
like the symbols of Algebra. In saying this, it is not affirmed
that the process of multiplication in Algebra, of which the
fundamental law is expressed by the equation

Δ το 8
possesses in itself any analogy with that process of logical
combination which xy has been made to represent above ;
but only that if the arithmetical and the logical process are
expressed in the same manner, their symbolical expressions
will be subject to the same formal law. The evidence of
that subjection is in the two cases quite distinct.

9. As the combination of two literal symbols in the form
ay expresses the whole of that class of objects to which the
names or qualities represented by x and y are together
applicable, it follows that if the two symbols have exactly
the same signification, their combination expresses no more
than either of the symbols taken alone would do. In such
case we should therefore have

Ly =X:

As y is, however, supposed to have the same meaning as
x, we may replace it in the above equation by x, and we
thus get
XX =X.

Now in common Algebra the combination xxv is more briefly
represented by 2%. Let us adopt the same principle of
notation here; for the mode of expressing a particular
succession of mental operations is a thing in itself quite
as arbitrary as the mode of expressing a single idea or
operation (II. 3). In accordance with this notation, then,
the above equation assumes the form

x= x, (2)
and is, in fact, the expression of a second general law of
those symbols by which names, qualities, or descriptions,
are symbolically represented.

[32] The reader must bear in mind that although the

SIGNS AND THEIR LAWS 35

_symbols x and y in the examples previously formed received
significations distinct from each other, nothing prevents us
from attributing to them precisely the same signification,
It is evident that the more nearly their actual significations
approach to each other, the more nearly does the class of
things denoted by the combination xy approach to identity
with the class denoted by 4, as well as with that denoted
by y. The case supposed in the demonstration of the
equation (2) is that of absolute identity of meaning. The
law which it expresses is practically exemplified in language.
To say ‘‘good, good,” in relation to any subject, though
a cumbrous and useless pleonasm, is the same as to say
‘‘sood.” Thus ‘‘ good, good” men, is equivalent to ‘‘ good ”
men. Such repetitions of words are indeed sometimes em-
ployed to heighten a quality or strengthen an affirmation.
But this effect is merely secondary and conventional ; it is
not founded in the intrinsic relations of language and thought.
Most of the operations which we observe in nature, or per-
form ourselves, are of such a kind that their effect is aug-
mented by repetition, and this circumstance prepares us to
expect the same thing in language, and even to use repetition
when we design to speak with emphasis. But neither in
strict reasoning nor in exact discourse is there any just
ground for such a practice.

10. We pass now to the consideration of another class of
the signs of speech, and of the laws connected with their use.

CLASS II

11. Szgus of those mental operations whereby we collect
parts into a whole, or separate a whole into 115 parts.

We are not only capable of entertaining the conceptions
of objects, as characterized by names, qualities, or circum-
stances, applicable to each individual of the group under
consideration, but also of forming the aggregate conception
of a group of objects consisting of partial groups, each of
which is separately named or described. For this purpose
we use the conjunctions ‘‘and,” ‘‘or,” ὅς. ‘‘ Trees and

36 THE LAWS OF THOUGHT

minerals,” ‘‘ barren mountains, or fertile vales,” are examples
of this kind. In strictness, the words [33] ‘‘and,” ‘‘or,”
interposed between the terms descriptive of two or more
classes of objects, imply that those classes are quite distinct,
so that no member of one is found in another. In this and
in all other respects the words ‘‘and” ‘‘or” are analogous
with the sign + in algebra, and their laws are identical.
Thus the expression ‘‘men and women” is, conventional
meanings set aside, equivalent with the expression ‘* women
and men.” Let represent ‘‘men,” y, ‘‘women”; and let
+ stand for ‘‘and” and ‘‘ or,” then we have

X+YH=VH+X, (3)

an equation which would equally hold true if x and y repre-
sented zumbers, and + were the sign of arithmetical addition.

Let the symbol z stand for the adjective ‘‘ European,”
then since it is, in effect, the same thing to say ‘‘ European
men and women,” as to say ‘‘ European men and European

women,” we have
a(x +y) = 2x + 27. (4)

And this equation also would be equally true were 7, y,
and z symbols of number, and were the juxtaposition of
two literal symbols to represent their algebraic product,
just as in the logical signification previously given, it re-
presents the class of objects to which both the epithets
conjoined belong.

The above are the laws which govern the use of the sign
+, here used to denote the positive operation of aggregating
parts into a whole. But the very idea of an operation effect-
ing some positive change seems to suggest to us the idea of
an opposite or negative operation, having the effect of un-
doing what the former one has done. Thus we cannot
conceive it possible to collect parts into a whole, and not
conceive it also possible to separate a part from a whole.
This operation we express in common language by the sign
except, as, ‘‘ All men except Asiatics,” ‘‘ All states except
those which are monarchical.” Here it is implied that the

SIGNS AND THEIR LAWS 37

things excepted form a part of the things from which they
are excepted. As we have expressed the operation of
aggregation by the sign +, so we may express the negative
operation above described by — minus. Thus if + be taken ἡ
to represent men, and y, Asiatics, 2,4. Asiatic men, [34]
then the conception of ‘‘ All men except Asiatics” will be
expressed by x—y. And if we represent by 2, ‘‘ states,”
and by y the descriptive property ‘‘ having a monarchical
form,” then the conception of ‘‘ All states except those
which are monarchical” will be expressed by x—-2y.

As it is indifferent for all the essextzal purposes of reason-
ing whether we express excepted cases first or last in the
order of speech, it is also indifferent in what order we write
any series of terms, some of which are affected by the sign
—. Thus we have, as in the common algebra,

x Y= -Y+x, (5)
Still representing by x the class ‘‘ men,” and by y ‘‘ Asiatics,”
let z represent the adjective ‘‘ white.” Now to apply the
adjective ‘‘ white” to the collection of men expressed by
the phrase ‘‘ Men except Asiatics,” is the same as to say,
‘White men, except white Asiatics.” Hence we have
a(x — y) = 2x -- By. (6)
This is also in accordance with the laws of ordinary algebra.
The equations (4) and (6) may be considered as exempli-
fication of a single general law, which may be stated by
saying, that the literal symbols, x, y, 2, &c. are distributive
in thetr operation, ‘The general fact which that law ex-
presses is this, viz. :—If any quality or circumstance is
ascribed to all the members of a group, formed either by
aggregation or exclusion of partial groups, the resulting
conception is the same as if the quality or circumstance
were first ascribed to each member of the partial groups,
and the aggregation or exclusion effected afterwards. That
which is ascribed to the members of the whole is ascribed
to the members of all its parts, howsoever those parts are
connected together.

38 THE LAWS OF THOUGAT

CLASS ΠῚ

12. Signs by which relation is expressed, and by which we
form propostitons.

Though all verbs may with propriety be referred to this
class, it is sufficient for the purposes of Logic to consider
it as including only the substantive verb zs or ave, since
every other verb [35] may be resolved into this element,
and one of the signs included under Class I. For as those
signs are used to express quality or circumstance of every
kind, they may be employed to express the active or pass-
ive relation of the subject of the verb, considered with refer-
ence either to past, to present, or to future time. Thus the
Proposition, ‘‘Czsar conquered the Gauls,” may be re-
solved into ‘‘Czesar is he who conquered the Gauls.” The
ground of this analysis I conceive to be the following :—
Unless we understand what is meant by having conquered
the Gauls, z.e. by the expression ‘‘ One who conquered the
Gauls,” we cannot understand the sentence in question. It
is, therefore, truly an element of that sentence ; another
element is ‘‘ Cesar,” and there is yet another required, the
copula zs, to show the connexion of these two. I do not,
however, affirm that there is no other mode than the above
of contemplating the relation expressed by the proposition,
‘‘Czesar conquered the Gauls”; but only that the analysis

here given is a correct one for the particular point of view

which has been taken, and that it suffices for the purposes

of logical deduction. It may be remarked that the passive |

and future participles of the Greek language imply the
existence of the principle which has been asserted, viz. :

that the sign zs or ave may be regarded as an element of ἢ

every personal verb.

13. The above sign, zs or ave, may be expressed by the
symbol =. The laws, or as would usually be said, the
axioms which the symbol introduces, are next to be
considered.

Let us take the Proposition, ‘‘ The stars are the suns and

SIGNS AND THEIR LAWS 39

the planets,” and let us represent stars by +, suns by y, and
planets by z; we have then

X=Y+2. (7)

Now if it be true that the stars are the suns and the planets,
it will follow that the stars, except the planets, are suns.
This would give the equation

τ τς (8)

which must therefore be a deduction from (7). Thus a
term z has been removed from one side of an equation to
the other by [36] changing its sign. This is in accordance
with the algebraic rule of transposition.

But instead of dwelling upon particular cases, we may at
once affirm the general axioms :—

ist. If equal things are added to equal things, the wholes
are equal.

2nd. If equal things are taken from equal things, the
remainders are equal.

And it hence appears that we may add or subtract
equations, and employ the rule of transposition above given
just as in common algebra.

Again: If two classes of things, x and y, be identical,
that is, if all the members of the one are members of the
other, then those members of the one class which possess a
given property 2 will be identical with those members of
the other which possess the same property z. Hence if we
have the equation

eee

then whatever class or property # may represent, we have
also
ZX = ZY.

This is formally the same as the algebraic law :—If both
members of an equation are multiplied by the same quantity,
the products are equal.

In like manner it may be shown that if the correspond-

40 THE. LAWS OF THOUGHT

ing members of two equations are multiplied together, the
resulting equation is true.

14. Here, however, the analogy of the present system
with that of algebra, as commonly stated, appears to stop.
Suppose it true that those members of a class x which
possess a certain property 2 are identical with those
members of a class y which possess the same property 2, it
does not follow that the members of the class x universally
are identical with the members of the class y. Hence it
cannot be inferred from the equation

2. ΞΞΖ2..
that the equation

τ πε
is also true. In other words, the axiom of algebraists, that
both [37] sides of an equation may be divided by the same
quantity, has no formal equivalent here. I say no formal
equivalent, because, in accordance with the general spirit of
these inquiries, it is not even sought to determine whether
the mental operation which is represented by removing a
logical symbol, 5, from a combination gx, is in itself
analogous with the operation of division in Arithmetic.
That mental operation is indeed identical with what is
commonly termed Abstraction, and it will hereafter appear
that its laws are dependent upon the laws already deduced
in this chapter. What has now been shown is, that there
does not exist among those laws anything analogous in
form with a commonly received axiom of Algebra.

But a little consideration will show that even in common
algebra that axiom does not possess the generality of those
other axioms which have been considered. The deduction
of the equation r=y from the equation zr=zy is only valid
when it is known that ¢ is not equal to o. If then the
value z=0 is supposed to be admissible in the algebraic
system, the axiom above stated ceases to be applicable, and
the analogy before exemplified remains at least unbroken.

15. However, it is not with the symbols of quantity
generally that it is of any importance, except as a matter of

SIGNS AND THEIR LAWS 41

speculation, to trace such affinities. We have seen (II. 9)
that the symbols of Logic are subject to the special law,

w= x

Now of the symbols of Number there are but two, viz.
o and 1, which are subject to the same formal law. We
know that o?=o, and that 17=1; and the equation τ τ,
considered as algebraic, has no other roots than o and 1.
Hence, instead of determining the measure of formal
agreement of the symbols of Logic with those of Number
generally, it is more immediately suggested to us to com-
pare them with symbols of quantity admitting only of the
values Ὁ and 1. Let us conceive, then, of an Algebra in
which the symbols 2, y, 2, &c. admit indifferently of the
values Ο and 1, and of these values alone. The laws, the
-axioms, and the processes of such an Algebra will be
identical in their whole extent with the laws, the axioms,
and the processes of an Al-[38]gebra of Logic. Difference
of interpretation will alone divide them. Upon this principle
the method of the following work is established.

16. It now remains to show that those constituent parts
of ordinary language which have not been considered in
the previous sections of this chapter are either resolvable
_ into the same elements as those which have been considered,
or are subsidiary to those elements by contributing to their
more precise definition.

The substantive, the adjective, and the verb, together
with the particles and, except, we have already considered.
The pronoun may be regarded as a particular form of the
substantive or the adjective. The adverb modifies the
meaning of the verb, but does not affect its nature. Pre-
positions contribute to the expression of circumstance or
relation, and thus tend to give precision and detail to the
meaning of the literal symbols. The conjunctions 7/,
either, or, are used chiefly in the expression of relation
among propositions, and it will hereafter be shown that the
same relations can be completely expressed by elementary

42 THE LAWS OF THOUGHT

symbols analogous in interpretation, and identical in form
and law with the symbols whose use and meaning have
been explained in this Chapter. As to any remaining
elements of speech, it will, upon examination, be found
that they are used either to give a more definite significance
to the terms of discourse, and thus enter into the inter-
pretation of the literal symbols already considered, or to
express some emotion or state of feeling accompanying the
utterance of a proposition, and thus do not belong to the
province of the understanding, with which alone our present
concern lies. Experience of its use will testify to the
sufficiency of the classification which has been adopted.

[39] CELAL Pik ΠῚ

DERIVATION OF THE LAWS OF THE SYMBOLS OF LOGIC
FROM THE ~*~ LAWS OF THE OPERATIONS OF THE
HUMAN MIND

1. THE object of science, properly so called, is the know-
ledge of laws and relations. To be able to distinguish
what is essential to this end, from what is only accidentally

associated with it, is one of the most important conditions

of scientific progress. I say, to distinguish between these

elements, because a consistent devotion to science does not
require that the attention should be altogether withdrawn
from other speculations, often of a metaphysical nature,

_ with which it is not unfrequently connected. Such ques-
tions, for instance, as the existence of a sustaining ground
— zi .

οὗ phenomena, the reality of cause, the propriety of forms

of speech implying that the successive states of things are

connected by operations, and others of a like nature, may

possess a deep interest and significance in relation to science,
without being essentially scientific. It is indeed scarcely

possible to express the conclusions of natural science without

_ borrowing the language of these conceptions. Nor is there
necessarily any practical inconvenience arising from this
source. They who believe, and they who refuse to believe,

ἘΠῚ
ἑ

that there is more in the relation of cause and effect than

an invariable order of succession, agree in their interpretation
of the conclusions of physical astronomy. But they only

_ agree because they recognise a common element of scientific

truth, which is independent of their particular views of the

ry

- Nature of causation.

2. If this distinction is important in physical science,

much more does it deserve attention in connexion with

43

44 THE TRAWS OF THOCGA SE

the science of the intellectual powers. For the questions
which this science presents become, in expression at least,
almost necessarily mixed up with modes of thought and
language, which betray a metaphysical origin. The idealist
would give to the laws of reasoning [40] one form of ex-
pression; the sceptic, if true to his principles, another.
They who regard the phenomena with which we are con-
cerned in this inquiry as the mere successive states of the
thinking subject devoid of any causal connexion, and they
who refer them to the operations of an active intelligence,
would, if consistent, equally differ in their modes of state-
ment. Like difference would also result from a difference —
of classification of the mental faculties. Now the principle —
which I would here assert, as affording us the only ground |
of confidence and stability amid so much of seeming and of :
real diversity, is the following, viz., that if the laws in ques- ;
tion are really deduced from observation, they have a real F
existence as laws of the human mind, independently of any ~
metaphysical theory which may seem to be involved in the —
mode of their statement. They contain an element of truth —
which no ulterior criticism upon the nature, or even upon —
the reality, of the mind’s operations, can essentially affect. —
Let it even be granted that the mind is but a succession of —
states of consciousness, a series of fleeting impressions un-_
caused from without or from within, emerging out of nothing,
and returning into nothing again,—the last refinement of —
the sceptic intellect,—still, as laws of succession, or at least
of a past succession, the results to which observation had led |
would remain true. They would require to be interpreted
into a language from whose vocabulary all such terms as |
cause and effect, operation and subject, substance and attri- |
bute, had been banished ; but they would still be valid as
scientific truths.

Moreover, as any statement of the laws of thought, —
founded upon actual observation, must thus contain scientific
elements which are independent of metaphysical theories of
the nature of the mind, the practical application of such ele-

DERIVATION OF THE LAWS 45

ments to the construction of a system or method of reason-
ing must also be independent of metaphysical distinctions.
For it is upon the scientific elements involved in the state-
ment of the laws, that any practical application will rest,
just as the practical conclusions of physical astronomy are
independent of any theory of the cause of gravitation, but
rest only on the knowledge of its phenomenal effects. And,
therefore, as respects both the determi-[41]nation of the
laws of thought, and the practical use of them when dis-
covered, we are, for all really scientific ends, unconcerned
with the truth or falsehood of any metaphysical speculations
whatever. .

3. The course which it appears to me to be expedient,

under these circumstances, to adopt, is to avail myself as

far as possible of the language of common discourse, without
regard to any theory of the nature and powers of the mind
which it may be thought to embody. For instance, it is

agreeable to common usage to say that we converse with

each other by the communication of ideas, or conceptions,

such communication being the office of words; and that
‘with reference to any particular ideas or conceptions pre-

sented to it, the mind possesses certain powers or faculties
by which the mental regard may be fixed upon some ideas,
to the exclusion of others, or by which the given conceptions
or ideas may, in various ways, be combined together. To
those faculties or powers different names, as Attention,
Simple Apprehension, Conception or Imagination, Abstrac-
tion, &c., have been given,—names which have not only
furnished the titles of distinct divisions of the philosophy of
the human mind, but passed into the common language of

-men. Whenever, then, occasion shall occur to use these

|

terms, I shall do so without implying thereby that I accept

the theory that the mind possesses such and such powers
and faculties as distinct elements of its activity. Nor is

.

it indeed necessary to inquire whether such powers of the
understanding have a distinct existence or not. We may
merge these different titles under the one generic name of

46 THE LAWS OF THOUGHT

Operations of the human mind, define these operations so far
as is necessary for the purposes of this work, and then seek
to express their ultimate laws. Such will be the general
order of the course which I shall pursue, though reference
will occasionally be made to the names which common
agreement has assigned to the particular states or operations
of the mind which may fall under our notice.

It will be most convenient to distribute the more definite re-
sults of the following investigation into distinct Propositions.

[42] PROPOSITION 1

4. Lo deduce the laws of the symbols of Logic from a con-
sideration of those operations of the mind which are implied
en the strict use of language as an instrument of reasoning.

In every discourse, whether of the mind conversing with
its own thoughts, or of the individual in his intercourse with
others, there is an assumed or expressed limit within which
the subjects of its operation are confined. The most un-
fettered discourse is that in which the words we use are
understood in the widest possible application, and for them
the limits of discourse are co-extensive with those of the
universe itself. But more usually we confine ourselves to a

less spacious field. Sometimes, in discoursing of men we |
imply (without expressing the limitation) that it is of men —
only under certain circumstances and conditions that we ~
speak, as of civilized men, or of men in the vigour of life, —
or of men under some other condition or relation. Now, —
whatever may be the extent of the field within which all the —

objects of our discourse are found, that field may properly
be termed the universe of discourse.
5. Furthermore, this universe of discourse is in the strictest

sense the ultimate swdject of the discourse. The office of any |

name or descriptive term employed under the limitations

supposed is not to raise in the mind the conception of all

the beings or objects to which that name or description is
applicable, but only of those which exist within the supposed

ey Se Le er

Ee he ee ee ee ae ee ee

i i τὸς κου ον

oh tp

τ"

DERIVATION OF THE. EAWS 47

universe of discourse. If that universe of discourse is the

actual universe of things, which it always is when our words
are taken in their real and literal sense, then by men we
mean all men that exist; but if the universe of discourse is
limited by any antecedent implied understanding, then it is
of men under the limitation thus introduced that we speak.
It is in both cases the business of the word mez to direct a
certain operation of the mind, by which, from the proper
universe of discourse, we select or fix upon the individuals
signified.

6. Exactly of the same kind is the mental operation implied
by the use of an adjective. Let, for instance, the universe
of discourse be the actual Universe. Then, as the word men
directs [43] us to select mentally from that Universe all the
beings to which the term ‘‘men” is applicable ; so the ad-
jective ‘‘ good,” in the combination ‘‘ good men,” directs us
still further to select mentally from the class of mez all those
who possess the further quality ‘‘good”; and if another

adjective were prefixed to the combination ‘‘ good men,” it
would direct a further operation of the same nature, having
“reference to that further quality which it might be chosen
_ to express.

It is important to notice carefully the real nature of the
operation here described, for it is conceivable, that it might
have been different from what it is. Were the adjective
simply attributive in its character, it would seem, that when
a particular set of beings is designated by mez, the prefixing
of the adjective good would direct us to attach mentally to
all those beings the quality of goodness. But this is not
the real office of the adjective. The operation which we
really perform is one of selectzon according to a prescribed
principle or cdea. ‘To what faculties of mind such an opera-

_ tion would be referred, according to the received classifica-
tion of its powers, it is not important to inquire, but I
suppose that it would be considered as dependent upon
the two faculties of Conception or Imagination, and Atten-
tion. To the one of these faculties might be referred the

48 THE LAWS. OF THOUGH &

formation of the general conception; to the other the
fixing of the mental regard upon those individuals within
the prescribed universe of discourse which answer to the
conception. If, however, as seems not improbable, the
power of Attention is nothing more than the power of
continuing the exercise of any other faculty of the mind,
we might properly regard the whole of the mental process
above described as referrible to the mental faculty of
Imagination or Conception, the first. step of the process
being the conception of the Universe itself, and each
succeeding step limiting in a definite manner the conception
thus formed. Adopting this view, I shall describe each
such step, or any definite combination of such steps, as
a definite act of conception. And the use of this term I
shall extend so as to include in its meaning not only the

conception of classes of objects represented by particular |

names or simple attributes of quality, but also the combina-
tion of such conceptions in any manner consistent with
the powers and limitations [44] of the human mind;
indeed, any intellectual operation short of that which is
involved in the structure of a sentence or proposition.
The general laws to which such operations of the mind
are subject are now to be considered.

7. Now it will be shown that the laws which in the
preceding chapter have been determined a posterior¢ from
the constitution of language, for the use of the literal
symbols of Logic, are in reality the laws of that definite
mental operation which has just been described. We

commence our discourse with a certain understanding as _
to the limits of its subject, ze. as to the limits of its |
Universe. Every name, every term of description that —
we employ, directs him whom we address to the per-_

formance of a certain mental operation upon that subject.
And thus is thought communicated. But as each name or
descriptive term is in this view but the representative of an

intellectual operation, that operation being also prior in the ©
order of nature, it is clear that the laws of the name or.

ee

a a i

sect tt Stil πῶς Nd 9g a

DERIVATION OF THE LAWS 49

symbol must be of a derivative character,—must, in fact,
originate in those of the operation which they represent.
That the laws of the symbol and of the mental process are
identical in expression will now be shown.

8. Let us then suppose that the universe ot our discourse
is the actual universe, so that words are to be used in the
full extent of their meaning, and let us consider the two
mental operations implied by the words ‘‘ white” and
‘*men.” The word ‘‘ men” implies the operation of select-
ing in thought from its subject, the universe, all men; and
the resulting conception, men, becomes the subject of the
next operation. The operation implied by the word ‘‘ white ”
is that of selecting from its subject, ‘‘men,” all of that class
which are white. The final resulting conception is that of
‘‘white men.” Now it is perfectly apparent that if the
operations above described had been performed in a con-
verse order, the result would have been the same. Whether
we begin by forming the conception of ‘‘ sen,” and then
by a second intellectual act limit that conception to ‘‘ white
men,” or whether we begin by forming the conception of
*< white objects,” and then limit it to such of that class as
are ‘‘men,” is perfectly indifferent so far as the result is
concerned. It is obvious that the order of the mental pro-
cesses would be equally [45] indifferent if for the words
*‘white” and ‘‘men” we substituted any other descriptive
or appellative terms whatever, provided only that their
meaning was fixed and absolute. And thus the indifference
of the order of two successive acts of the faculty of Con-
ception, the one of which furnishes the subject upon which
the other is supposed to operate, is a general condition of
the exercise of that faculty. It isa law of the mind, and
itis the real origin of that law of the literal symbols of Logic
which constitutes its formal expression (1) Chap. II.

g. It is equally clear that the mental operation above
described is of such a nature that its effect is not altered
by repetition. Suppose that by a definite act of conception
the attention has been fixed upon men, and that by another

4

50 THE LAWS OF THOUGHT

exercise of the same faculty we limit it to those of the race ;
who are white. Then any further repetition of the latter :
mental act, by which the attention is limited to white ~
objects, does not in any way modify the conception arrived —
at, viz., that of white men. This is also an example of at
general law of the mind, and it has its formal expression in ἢ
the law ((2) Chap. II.) of the literal symbols. :

10, Again, it is manifest that from the conceptions of two. '
distinct classes of things we can form the conception of παῖ
collection of things which the two classes taken together
compose ; and it is obviously indifferent in what order of
position or of priority those classes are presented to the ᾿
mental view. This is another general law of the mind, δπά
its expression is found in (3) Chap. II. q

11. It is not necessary to pursue this course of inquiry —
and comparison. Sufficient illustration has been given to”
render manifest the two following positions, viz. : :

First, That the operations of the mind, by which, in
the exercise of its power of imagination or conception,
it combines and modifies the simple ideas of things or”
qualities, not less than those operations of the reason which”
are exercised upon truths and propositions, are subject to-
general laws. :

Secondly, That those laws are mathematical in their form, q
and that they are actually developed in the essential laws o ᾿
human language. Wherefore the laws of the symbols of
Logic [46] are deducible from a consideration of the opera-_
tions of the mind in reasoning.

12, The remainder of this chapter will be occupied with
questions relating to that law of thought whose expression —
is #27 =x (II. 9), a law which, as has been implied (IJ. 15}
forms the characteristic distinction of the operations of the
mind in its ordinary discourse and reasoning, as compared ©
with its.operations when occupied with the general algebra
of quantity. An important part of the following inquiry
will consist in proving that the symbols o and 1 occupy 8
place, and are susceptible of an interpretation, among the

DERIVATION OF THE LAWS 51

symbols of Logic; and it may first be necessary to show
how particular symbols, such as the above, may with pro-
priety and advantage be employed in the representation of
distinct systems of thought.

The ground of this propriety cannot consist in any com-
munity of interpretation. For in systems of thought so
truly distinct as those of Logic and Arithmetic (I use the
latter term in its widest sense as the science of Number),
there is, properly speaking, no community of subject. The
one of them is conversant with the very conceptions of
‘things, the other takes account solely of their numerical
relations. But inasmuch as the forms and methods of any
system of reasoning depend immediately upon the laws to
which the symbols are subject, and only mediately, through
the above link of connexion, upon their interpretation, there
may be both propriety and advantage in employing the same
symbols in different systems of thought, provided that such
interpretations can be assigned to them as shall render their
formal laws identical, and their use consistent. The ground
of that employment will not then be community of interpre-
tation, but the community of the formal laws, to which in
their respective systems they are subject. Nor must that
community of formal laws be established upon any other
ground than that of a careful observation and comparison of
those results which are seen to flow independently from the
interpretations of the systems under consideration.

These observations will explain the process of inquiry
adopted in the following Proposition. The literal symbols
of Logic are [47] universally subject to the law whose ex-
pression is #2=2x, Of the symbols of Number there are two
only, o and 1, which satisfy this law. But each of these
symbols is also subject to a law peculiar to itself in the
‘system of numerical magnitude, and this suggests the in-
quiry, what interpretations must be given to the literal
symbols of Logic, in order that the same peculiar and formal
laws may be realized in the logical system also.

52 OTT EAS OF THOUGHT

τ --

PROPOSITION II

13. 70 determine the logical value and significance of the
symbols 0 and 1.

eS ee A

The symbol Ο, as used in Algebra, satisfies the following
formal law,

Ox yO) ΟΣ 6)5Ξ0, )
whatever mumber y may represent. That this formal Lavi
may be obeyed in the system of Logic, we must assign to.
the symbol o such an interpretation that the class repreg
sented by oy may be identical with the class represented by
oO, whatever the class y may be. Ai little consideration will
show that this condition is satisfied if the symbol o reprell
sent Nothing. In accordance with a previous definition, we-
may term Nothing aclass. In fact, Nothing and Universe
are the two limits of class extension, for they are the limits.
of the possible interpretations of general names, none of |
which can relate to fewer individuals than are comprised in|
Nothing, or to more than are comprised in the Universe.
Now whatever the class y may be, the individuals which are
common to it and to the class ‘‘ Nothing” are identical with)
those comprised in the class ‘‘ Nothing,” for they are none.
And thus by assigning to Ο the interpretation Nothing, the:
law (1) is satisfied ; and it is not otherwise satisfied consis- |
tently with the perfectly general character of the class y. :

Secondly, The symbol 1 satisfies in the system of Numba
the following law, viz.

Le ge Onn, γον"

whatever number y may represent. And this formal equa-'
tion being assumed as equally valid in the system of this:
work, in [48] which 1 and y represent classes, it appears:
that the symbol 1 must represent such a class that all the in--
dividuals which are found in any proposed class y are also.
all the individuals Iy that are common to that class y and}
the class represented by 1. A little consideration will here
show that the class represented by 1 must be ‘‘ the Uni-'

PERV ALTON OR Sih L AWS 53

verse,” since this is the only class in which are found αὐ
the individuals that exist in any class. Hence the respective
interpretations of the symbols o and 1 in the system of
Logic are Nothing and Universe.

== 14. As with the idea of any class of objects as ‘‘ men,
there is suggested to the mind the idea of the contrary class
of beings which are not men; and as the whole Universe
is made up of these two classes together, since of every
individual which it comprehends we may affirm either that
it is a man, or that it is not a man, it becomes important

+P

᾿ to inquire how such contrary names are to be expressed.
Such is the object of the following Proposition.

PROPOSITION III

Ὶ Tf x represent any class of objects, then will 1 —%x represent
‘the contrary or supplementary class of objects, 1.6. the class
including all objects which are not comprehended in the
class x.

For greater distinctness of conception let x represent the
class men, and let us express, according to the last Pro-
position, the Universe by 1; now if from the conception

of the Universe, as consisting of ‘‘men” and ‘‘not-men,”
we exclude the conception of ‘‘men,” the resulting con-
‘ception is that of the contrary class, ‘‘not-men.” Hence
the class ‘‘not-men” will be represented by 1—% And,

in general, whatever class of objects is represented i the
symbol a, the contrary class will be expressed by I —

1s. Although the following Proposition belongs in strict-
ness toa future chapter of this work, devoted to the subject
of maxims or necessary truths, yet, on account of the great

importance of that law of thought to which it relates, it
has been thought proper to introduce it here.

[49] PROPOSITION IV

That axiom of metaphysicians which is termed the principle
Of contradiction, and which affirms that it ἐς tmpossible for

$4 THE LAWS OF THOUGHT

any being to possess a quality, and at the same time not to”
possess it, ἐξ a consequence of the fundamental law of thought, ©
whose expression ts =X. :

Let us write this equation in the form

2

x—x*?=0,

whence we have :
x(1-—x)=0; (1)
both these transformations being justified by the axiomatic —
laws of combination and transposition (II. 13). Let us, —
for simplicity of conception, give to the symbol ~# the ᾿
particular interpretation of men, then 1 --α will represent —
the class of ‘‘not-men” (Prop. III.) Now the formal -
product of the expressions of two classes represents that '
class of individuals which is common to them both (II. 6).
Hence 2(1 —%) will represent the class whose members are |
at once ‘‘ men,” and ‘‘ not men,” and the equation (1) thus 3
express the principle, that a class whose members are at q
the same time men and not men does not exist. In other y
words, that 22 7s zmpossible for the same individual to be at
the same time a man and not aman. Now let the meaning ©
of the symbol # be extended from the representing of ‘‘ men,”
to that of any class of beings characterized by the possession 4
of any quality whatever ; and the equation (1) will then j
express that it is impossible for a being to possess a quality .
and not to possess that quality at the same time. But this .
is identically that ‘‘ principle of contradiction” which ,
Aristotle has described as the fundamental axiom of all
philosophy. ““Τ is impossible that the same quality should —
both belong and not belong to the same thing. . . This —
is the most certain of all principles. . . Wherefore they
who demonstrate refer to this as an ultimate opinion. For ἢ
it is by nature the source of all the other axioms.” * ᾿

[50] The above interpretation has been introduced not 7

ns Td yap αὐτὸ ἄμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸν
αὐτὸ. a Αὕτη δὴ πασῶν ἐστὶ βεβαιοτάτη τῶν ἀρχῶν... Διὸ πάντες οἱ ἀποδεικ-
νύντες εἰς ταύτην ἀνάγουσιν ἐσχάτην δόξαν᾽ φύσει γὰρ ἀρχὴ καὶ τῶν ἄλλων ἀξιω- —
μάτων αὕτη mavtwy.—Metaphysica, III, 3. 2

DERIVATION OF THE LAWS Ἢ

on account of its immediate value in the present system,
but as an illustration of a significant fact in the philosophy
of the intellectual powers, viz., that what has been commonly
regarded as the fundamental axiom of metaphysics is but
the consequence of a law of thought, mathematical in its
form. I desire to direct attention also to the circumstance
that the equation (1) in which that fundamental law of
thought is expressed is an equation of the second degree. *
Without speculating at all in this chapter upon the question,
whether that circumstance is necessary in its own nature,
we may venture to assert that if it had not existed, the
whole procedure of the understanding would have been
different from what it is. Thus it is a consequence of the
fact that the fundamental equation of thought is of the
second degree, that we perform the operation of analysis and
classification, by division into pairs of [51] opposites, or,
as it is technically said, by dichotomy. Now ifthe equation
in question had been of the third degree, still admitting of
interpretation as such, the mental division must have been
threefold in character, and we must have proceeded by a

* Should it here be said that the existence of the equation x°= x necessitates
also the existence of the equation «7=., which is of the third degree, and then
inquired whether that equation does not indicate a process of ¢rzchotomy ; the
answer is, that the equation «*=. is not interpretable in the system of logic.
For writing it in either of the forms

x(I-x)\I1+x) =o, (2)
x(1 —*)(-1I-%)=0, (3)

we see that its interpretation, if possible at all, must involve that of the factor
I+, or of the factor -1—x. The former is not interpretable, because we
cannot conceive of the addition of any class x to the universe 1 ; the latter is not
‘interpretable, because the symbol ~—1 is not subject to the law «(1-.~)=0, to
which all class symbols are subject. Hence the equation «*=. admits of no in-
terpretation analogous to that of the equation «2=«. Were the former equation,
however, true independently of the latter, 2,6. were that act of the mind which
is denoted by the symbol .v, such that its second repetition should reproduce the
result of a single operation, but not its first or mere repetition, it is presumable
that we should be able to interpret one of the forms (2), (3), which under the
actual conditions of thought we cannot do. There exist operations, known to
the mathematician, the law of which may be adequately expressed by the equa-
tion «8=x, But they are of a nature altogether foreign to the province of
general reasoning.

In saying that it is conceivable that the law of thought might have been
different from what it is, I mean only that we can frame such an hypothesis, and
study its consequences. The possibility of doing this involves no such doctrine
as that the actual law of human reason is the product either of chance or
of arbitrary will.

56 THEOL AWS, OF TOUGH].

species of ¢vzchotomy, the real nature of which it is impossible —
for us, with our existing faculties, adequately to conceive, |
but the laws of which we might still investigate as δῆ
object of intellectual speculation. 7

16. The law of thought expressed by the equation (1)
will, for reasons which are made apparent by the above
discussion, be occasionally referred to as the ‘‘law of —
duality.”

[52] CHAPTER IV

me ΠΗ PLVISION.. OF PROPOSITIONS: INTO: THE TWO
CRASSES SOF: “SPRIMARY +°AND <o°OSECONDARY >= 7OF
THE CHARACTERISTIC PROPERTIES OF THOSE CLASSES,
AND OF THE LAWS OF THE EXPRESSION OF PRIMARY
PROPOSITIONS

I. THE laws of those mental operations which are con-

cerned in the processes of Conception or Imagination having

been investigated, and the corresponding laws of the symbols

by which they are represented explained, we are led to con-
sider the practical application of the results obtained : first,
in the expression of the complex terms of propositions ;
secondly, in the expression of propositions ; and lastly, in
the construction of a general method of deductive analysis.

In the present chapter we shall be chiefly concerned with

the first of these objects, as an introduction to which it is

necessary to establish the following Proposition :

PROPOSITION I

All logical propositions may be considered as belonging
to one or the other of two great classes, to which the
respective names of ‘‘ Primary” or ‘‘ Concrete Propositions,”
and ‘‘ Secondary” or ‘‘ Abstract Propositions,” may be given.

Every assertion that we make may be referred to one or

the other of the two following kinds. Either it expresses a

relation among ¢hzmgs, or it expresses, or is equivalent to
the expression of, a relation among proposztions. An asser-
tion respecting the properties of things, or the phanomena
which they manifest, or the circumstances in which they are

placed, is, properly speaking, the assertion of a relation

57

58 THE LAWS OF THOUGHT

among things. To say that ‘‘snow is white,” is for the
ends of logic equivalent to saying, that ‘‘snow is a white
thing.” An assertion respecting facts or events, their mutual
connexion and dependence, is, for the same ends, generally
equivalent to the assertion, that such and such propositions
con-[53]cerning those events have a certain relation to each
other as respects their mutual truth or falsehood. The former
class of propositions, relating to ¢izmgs, I call ‘‘ Primary ” ;
the latter class, relating to proposztzons, I call ‘‘ Secondary.”
The distinction is in practice nearly but not quite co-extensive
with the common logical distinction of propositions as cate-
gorical or hypothetical.

For instance, the propositions, ‘‘ The sun shines,” ‘‘ The
earth is warmed,” are primary ; the proposition, ‘‘ If the sun
shines the earth is warmed,” is secondary. To say, ‘‘ The
sun shines,” is to say, ‘‘ The sun is that which shines,” and
it expresses a relation between two classes of things, viz.,
‘‘the sun” and ‘‘things which shine.” The secondary pro-

ee re a ee

position, however, given above, expresses a relation of de- ἢ

pendence between the two primary propositions, ‘‘ The sun
shines,” and ‘‘ The earth is warmed.” I do not hereby affirm

that the relation between these propositions is, like that which
exists between the facts which they express, a relation of

causality, but only that the relation among the propositions |
so implies, and is so implied by, the relation among the ᾿

facts, that it may for the ends of logic be used as ἃ fit
representative of that relation.

2. If instead of the proposition, ‘‘ The sun shines,” νοΐ
say, ‘It is true that the sun shines,” we then speak not _

directly of things, but of a proposition concerning things,
viz., of the proposition, ‘‘ The sun shines.” And, therefore,

the proposition in which we thus speak is a secondary one. |

Every primary proposition may thus give rise toa secondary
proposition, viz., to that secondary proposition which asserts:
its truth, or ee its falsehood.

It will usually happen, that the particles 76 ether, or,
will indicate that a proposition is secondary ; but they do

DIVISTON CF FROPOSIFIONS 59

not necessarily imply that such is the case. The proposition,
‘* Animals are either rational or irrational,” is primary. It
cannot be resolved into ‘‘ Either animals are rational or
animals are irrational,” and it does not therefore express
a relation of dependence between the two propositions con-
nected together in the latter disjunctive sentence. The
particles, ezther, or, are in fact no criterion of the nature
of propositions, although it happens that they are more
frequently found in secondary propositions. Even [54] the
conjunction 27 may be found in primary propositions. ‘‘ Men
are, if wise, then temperate,” is an example of the kind. It
cannot be resolved into ‘‘If all men are wise, then all men
are temperate. ”

3. As it is not my design to discuss the merits or defects
of the ordinary division of propositions, I shall simply remark
here, that the principle upon which the present classification
is founded is clear and definite in its application, that -it
involves a real and fundamental distinction in propositions,
and that it is of essential importance to the development of
ἃ general method of reasoning. Nor does the fact that a
primary proposition may be put into a form in which it
_ becomes secondary at all conflict with the views here main-
tained. For in the case thus supposed, it is not of the things
connected together in the primary proposition that any direct
account is taken, but only of the proposition itself considered
as true or as false.

4. In the expression both of primary and of secondary
propositions, the same symbols, subject, as it will appear,
to the same laws, will be employed in this work. The
difference between the two cases is a difference not of form
but of interpretation. In both cases the actual relation
which it is the object of the proposition to express will be
_ denoted by the sign =. _ In the expression of primary pro-
positions, the members thus connected will usually represent
the ‘‘terms” of a proposition, or, as they are more particu-
larly designated, its subject and predicate.

60 THE LAWS OF THOUGAT

PROPOSITION II

5. To deduce a general method, founded upon the enumera-
tion of possible varieties, for the expression of any class or
collection of things, which may constitute a ‘‘term” of a
Primary Proposttion.

First, If the class or collection of things to be expressed |

is defined only by names or qualities common to all the indi-
viduals of which it consists, its expression will consist of a
single term, in which the symbols expressive of those names
or qualities will be combined without any connecting sign,
as if by the algebraic process of multiplication. Thus, if α'
represents opaque substances, y polished substances, ¢ stones,
we shall have,

[55] «v2 =opaque polished stones ;

xy(1—2)=opaque polished substances which are not ἢ

stones ;
a(1—y)(1 —#%)=opaque substances which are not polished,
and are not stones ;

and so on for any other combination. Let it be observed,

that each of these expressions satisfies the same law οἵ

duality, as the individual symbols which it contains. Thus,

LYS X LYS = LYS ;
χγ(1 --- 2) x xy(1 - 2) =xy(1 - 2);

and soon. Any such term as the above we shall designate ͵
as a ‘‘class term,” because it expresses a class of things ΩΣ
means of the common properties or names of the individual -

members of such class,

Secondly, If we speak of a collection of things, different —

portions of which are defined by different properties, names,

or attributes, the expressions for those different portions must —
be separately formed, and then connected by the sign +. —
But if the collection of which we desire to speak has been ~

formed by excluding from some wider collection a defined
portion of its members, the sign — must be prefixed to

the symbolical expression of the excluded portion. Respect-_

ones i

DIVISION OF PROPOSITIONS 61

ing the use of these symbols some further observations may
be added.

6. Speaking generally, the symbol + is the equivalent of
the conjunctions ‘‘and,” ‘‘or,” and the symbol —, the equi-
valent of the preposition ‘‘except.” Of the conjunctions
‘‘and” and ‘‘or,” the former is usually employed when the
collection to be described forms the subject, the latter when

it forms the predicate, of a proposition. ‘‘ The scholar and
the man of the world desire happiness,” may be taken as an
illustration of one of these cases. ‘‘ Things possessing utility

are ezther productive of pleasure ov preventive of pain,” may

exemplify the other. Now whenever an expression involv-

ing these particles presents itself in a primary proposition, it
becomes very important to know whether the groups or
classes separated in thought by them are intended to be
quite distinct from each other and mutually exclusive, or not.
Does the expression, ‘‘Scholars and men of the world,”

pinclude or exclude those who are both? Does the ex-

ill

[56]pression ‘‘ Either productive of pleasure or preventive

of pain,” include or exclude things which possess both these
qualities ? I apprehend that in strictness of meaning the
conjunctions ‘‘and,” ‘‘or,” do possess the power of separa-
tion or exclusion here referred to; that the formula, ‘‘ All 2’s
are either y’s or 2’s,” rigorously interpreted, means, ‘‘ All 2’s
me either y’s, but not 2's,” or, ‘‘a’s but not y’s.” But it
must at the same time be admitted, that the ‘‘jus et norma
loquendi” seems rather to favour an opposite interpretation.
The expression, ‘‘ Either y’s or 2’s,” would generally be
understood to include things that are y’s and 2’s at the same
time, together with things which come under the one, but
not the other. Remembering, however, that the symbol +
does possess the separating power which has been the sub-
ject of discussion, we must resolve any disjunctive expression
which may come before us into elements really separated in
thought, and then connect their respective expressions by
the symbol +.

And thus, according to the meaning implied, the ex-

62 PE LAWS Crt aCUCEr

pression, ‘‘ Things which are either 1.5 or y’s,” will have two |
different symbolical equivalents. If we mean, ‘‘Things —
which are #’s, but not y’s, or y’s, but not 1.5. the expres-
sion will be

x(t -) +y(1 — x); |
the symbol x standing for #’s, y for y’s. If, however, we
mean, ‘‘ Things which are either 275, or, if notz’s, then y’s,”

the expression will be
x+y(1 -- x).

This expression supposes the admissibility of things which
are both 2.5 and y’s at the same time. It might more fully ©
be expressed in the form

ay + x(t —y) +y(1 - x) ;
but this expression, on addition of the two first terms, only —
reproduces the former one.

Let it be observed that the expressions above given satisfy
the fundamental law of duality (III. 16). Thus we have ἢ

{a(x -9) +9(1- 0) }2=2(1 =) 49(1 τα),

{x+y(1 -—x)}P=x+y(1 - x). :

It will be seen hereafter, that this is but a particular ξ
manifesta-[57]tion of a general law of expressions represent-
ing ‘‘classes or collections of things.” 7
7. The results of these investigations may be embodied ᾿
in the following rule of expression.

RULE.—E£apress simple names or qualities by the symbols 4
4, V, 8, &c., thetr contraries by τ --- α, τ τον, 1-2, ὅς; classes”
of things defined by common names or qualities, by connecting
the corresponding symbols as in multiplication ; collections of
things, consisting of portions different from each other, by
connecting the expressions of those portions by the sign +. )
In particular, let the expression, ‘‘ Either x's or y’s,” be ll
pressed by x1—y)+y(1—x), when the classes denoted by δ᾽
and y are exclusive, by x+y(1 —x) when they are not exclusive, ]
Similarly let the expression, ‘‘ Etther x's, or y's, or 3's,” bem
expressed by x1 -- γ)(1 -- 2)-ΕζΚ -- ῶῦα -- 2) + 2(1 —+\(1-y), 3

DIVISION OF PROPOSITIONS 63

when the classes denoted by x, y, and 2, are designed to be
mutually exclusive, by x+y(1I—*)+2(1 —x)\(1—y), when they
are not meant to be exclustve, and so on.

8. On this rule of expression is founded the converse rule
of interpretation. Both these will be exemplified with, per-
haps, sufficient fulness in the following instances. Omitting
for brevity the universal subject ‘‘ things,” or ‘‘ beings,” let
us assume

tena, y— elastic, ¢= metals -

and we shall have the following results :
““ Non-elastic metals,” will be expressed by 2(1 —y) ;

‘« Elastic substances with non-elastic metals,” by y+ 4(1—y) ;

‘‘ Hard substances, except metals,” by r—2;

‘* Metallic substances, except those which are neither hard nor
elastic,” by —2(1—+)(1 —y), or by 2{1-(1-4)\(1 —y)},
vide (6), Chap. 11.

In the last example, what we had really to express was

' ** Metals, except not hard, not elastic, metals.” Con-

junctions used between adjectives are usually superfluous,
and, therefore, must not be expressed symbolically.

Thus, ‘‘ Metals hard and elastic,” is equivalent to ‘‘ Hard

elastic metals,” and expressed by xyz.

Take next the expression, ‘‘ [lard substances, except those
[58] which are metallic and non-elastic, and those which

are elastic and non-metallic.” Here the word ¢hose means

hard substances, so that the expression really means, Hard
Substances except hard substances, metallic, non-elastic, and
hard substances non-metallic, elastic; the word except ex-
tending to both the classes which follow it. The complete
€xpression is

x — {x2(1 —y)+xy(1 -2)};
or, x -- x2(1 —y) -- γᾷ -- 2).

9. The preceding Proposition, with the different illustra-
tions which have been given of it, is a necessary preliminary
to the following one, which will complete the design of the
present chapter. ,

64 THE LAWS OF ΤΗσΌσ

PROPOSITION III

To deduce from an examination of thetr possible varieties
a general method for the expression of Primary or Concrete
Propositions.

A primary proposition, in the most general sense, consists
of two terms, between which a relation is asserted to exist.
These terms are not necessarily single-worded names, but
may represent any collection of objects, such as we have
been engaged in considering in the previous sections. The
mode of expressing those terms is, therefore, comprehended
in the general precepts above given, and it only remains to —
discover how the relations between the terms are to be ex- —
pressed. This will evidently depend upon the nature of the ἢ
relation, and more particularly upon the question whether, —
in that relation, the terms are understood to be universal or
particular, z.e. whether we speak of the whole of that col-—
lection of objects to which a term refers, or indefinitely of —
the whole or of a part of it, the usual signification of the
prelix some,”

Suppose that we wish to express a relation of identity —
between the two classes, ‘‘ Fixed Stars” and ‘‘ Suns,” 2.4
to express that ‘‘ All fixed stars are suns,” and ‘‘ All sung
are fixed stars.” Here, if x stand for fixed stars, and y for
suns, we shall have

Salas Ss. sani

Baa OK id τεὴν ΛΝ ΤΡ ἡ

ΠΣ pear ΡΣ ἼΤΩ -

oS
for the equation required.

[59] In the proposition, ‘‘ All fixed stars are suns,” the
term ‘‘all fixed stars” would be called the sadject, and—
‘‘suns” the predicate. Suppose that we extend the mean- |
ing of the terms sudject and predicate in the following |
manner. By sudject let us mean the first term of any.
affirmative proposition, z.e. the term which precedes the
copula zs or ave ; and by predicate let us agree to mean the
second term, z.e. the one which follows the copula; and let.
us admit the assumption that either of these may be universal |
or particular, so that, in either case, the whole class may

οὐλιβριος, κασι ee ee

bated) dT
3 ὃ fh

DIVISION OF PROPOSITIONS 65

be implied, or only a part of it. Then we shall have the

following Rule for cases such as the one in the last
example :—

10. RULE.—When both Subject and Predicate of a
Proposition are universal, form the separate expressions for
them, and connect them by the sign =.

This case will usually present itself in the expression of
the definitions of science, or of subjects treated after the
manner of pure science. Mr. Senior’s definition of wealth
affords a good example of this kind, viz.:

‘‘ Wealth consists of things transferable, limited in supply,
and either productive of pleasure or preventive of pain.”

Before proceeding to express this definition symbolically,
it must be remarked that the conjunction and is superfluous.
Wealth is really defined by its possession of three properties

or qualities, not by its composition out of three classes or
collections of objects. Omitting then the conjunction and,

let us make
w = wealth.

< ¢ =things transferable.
s =limited in supply.
2 =productive of pleasure.
7 = preventrye, Ol palit.

Now it is plain from the nature of the subject, that the
expression, ‘‘ Either productive of pleasure or preventive of

pain,” in the above definition, is meant to be equivalent to

« Fither productive of pleasure; or, if not productive of

‘Pleasure, preventive of pain.” Thus the class of things

which the above expression, taken alone, would define,

would consist of all things productive [60] of pleasure, to-

gether with all things not productive of pleasure, but pre-

-ventive of pain, and its symbolical expression would be

p+(1-p)r.
If then we attach to this expression placed in brackets to
denote that both its terms are referred to, the symbols s and
¢ limiting its application to things ‘‘ transferable” and

5

66 THE LAWS OF THOUGHT

‘‘limited in supply,” we obtain the following symbolical
equivalent for the original definition, viz. :

w=st{p+r(1 -- 2)}. (1)
If the expression, ‘‘ Either productive of pleasure or preven-
tive of pain,” were intended to point out merely those things
which are productive of pleasure without being preventive of
pain, (1 —7), or preventive of pain, without being produc- ;
tive of pleasure, r(1—p) (exclusion. being made of those
things which are both productive of pleasure and preventive
of pain), the expression in symbols of the definition would be

w = st(p(1-r)+r(1 ~p)}. (2)

All this agrees with what has before been more generally
stated. ;
The reader may be curious to inquire what effect would be
produced if we literally translated the expression, ‘‘ Things.
productive of pleasure or preventive of pain,” by p+.

i

making the symbolical equation of the definition to be f
w =st(p+r). (3)

The answer is, that this expression would be equivalent to
(2), with the additional implication that the classes of things”
denoted by s¢p and s¢r are quite distinct, so that of things
transferable and limited in supply there exist none in the
universe which are at the same time both productive of
pleasure and preventive of pain. How the full import da
any equation may be determined will be explained hereafter,
What has been said may show that before attempting to
translate our data into the rigorous language of symbols, it
is above all things necessary to ascertain the zztended import
of the words we are using. But this necessity cannot e
regarded as an evil by those who value correctness of [61]
thought, and regard the right employment of language ϑ
both its instrument and its safeguard. 3
11. Let us consider next the case in which the predicate
of the proposition is particular, e.g. ““ All men are mortal. *
In this case it is clear that our meaning is, ‘‘ All mena ¢

ee

.
ἡ

DIVISION OF PROPOSITIONS 67

2)

some mortal beings,” and we must seek the expression of
the predicate, ‘‘some mortal beings.” Represent then by v,
a class indefinite in every respect but this, viz., that some of
its members are mortal beings, and let x stand for ‘‘ mortal
beings,” then will wr represent ‘‘some mortal beings,”
Hence if y represent men, the equation sought will be

y= Ux.

From such considerations we derive the following Rule,
for expressing an affirmative universal Dope ΠΝ whose pre-
dicate is particular :

RULE.—Express as before the subject and the predicate,
attach to the latter the indefinite symbol v, and equate the
expressions.

It is obvious that v is a symbol of the same kind as x, y,
&c., and that it is subject to the general law,

w=v, or v1—v)=0.

Thus, to express the proposition, ‘‘ The planets are either
primary or secondary,” we should, according to the rule, pro-

ceed thus :
Let x represent planets (the subject) ;

y =primary bodies ;
g= secondary bodies ;

then, assuming the conjunction ‘‘or” to separate absolutely
the class of ‘‘ primary” from that of ‘‘secondary ” bodies,
so far as they enter into our consideration in the proposition
given, we find for the equation of the proposition

x =o{y(1— 2) +5(1-9)}- (4)

It may be worth while to notice, that in this case the de¢eral
translation of the premises into the form —

x=0U(y +2) (5)

[62] would be exactly equivalent, v being an indefinite class

symbol. The form (4) is, however, the better, as the ex-
pression

JU. - 2) +2(1—- 9)

68 THE LAWS OF THOUGHT

consists of terms representing classes quite distinct from.
each other, and satisfies the fundamental law of duality.

If we take the proposition, ‘‘ The heavenly bodies are
either suns, or planets or comets,” representing these classes |
of things by w, x, y, 2, respectively, its expression, on the
supposition that none of the heavenly bodies belong at once
to two of the divisions above mentioned, will be

w=v{x(1-y)(1-2)+y1—x)\(1 -2)4+2(1 -- α)α -9}}

If, however, it were meant to be implied that the heavenly
bodies were either suns, or, if not suns, planets, or, if neither.
suns nor planets, fixed stars, a meaning which does not
exclude the supposition of some of them belonging at once
to two or to all three of the divisions of suns, planets, and
fixed stars,—the expression required would be ἢ 4
w= of a+ 3(1— x) +2(4 τ 2)(1 -y)}. (6)

The above examples belong to the class of ΠΝ
not definitions. Indeed the predicates of propositions are
usually particular. When this is not the case, either the
predicate is a singular term, or we employ, instead of the
copula ‘‘is” or ‘‘are,” some form of connexion, which
implies that the predicate is to be taken universally. i

12. Consider next the case of universal negative proposi ie
tions, e.g. ‘‘ No men are perfect beings.” ἕ

Now it is manifest that in this case we do not speak of Ἢ
class termed ‘‘no men,” and assert of this class that all its:
members are ‘‘ perfect beings.” But we virtually make an

Ei
4
ὴ

perfect beings.” Thus the true meaning of the proposition |
is this: [

“.ΑἹ] men (subject) are (copula) not perfect (predicate) ” ;
whence, ify represent ‘‘men,” and x ‘perfect beings,” we
shall have

Y=U1-<x),

[63] and similarly in any other case. Thus we have the
following Rule:

DIVISION OF PROPOSITIONS 69

RULE.—To express any proposition of the form ““ No x’s
are y’s,” convert tt into the form ‘* All xs are not y’s,” and
then proceed as in the previous case.

. 13. Consider, lastly, the case in which the subject of the
proposition is particular, e.g. ‘‘Some men are not wise.”
Here, as has been remarked, the negative zo¢ may properly
be referred, certainly, at least, for the ends of Logic, to the
predicate wzse ; for we do not mean to say that it is not true
that ‘‘ Some men are wise,” but we intend to predicate of
‘some men” a want of wisdom. The requisite form of the
given proposition is, therefore, ‘‘Some men are not-wise.”
Putting, then, y for ‘‘ men,” x for ‘‘ wise,” z.e. ‘‘ wise beings,”
and introducing wv as the symbol of a class indefinite in all
respects but this, that it contains some individuals of the
class to whose expression it is prefixed, we have

vy =U(1 — x).
14. We may comprise all that we have determined in the

following general Rule:

™~

GENERAL RULE FOR THE SYMBOLICAL EXPRESSION OF
PRIMARY PROPOSITIONS

ist. [f the proposition ts affirmative, form the expression
of the subject and that of the predicate. Should etther of them
be particular, attach to it the indefinite symbol v, and then
equate the resulting expressions.
2ndly. If the proposition ts negative, express first its true
meaning by attaching the negative particle to the predicate,
then proceed as above.

One or two additional examples may suffice for illustration.
Ex.—‘‘ No men are placed in exalted stations, and free
from envious regards.”

met y represent ‘‘men,” x, ‘‘ placed in exalted stations,”
8, ‘‘ free from envious regards.”

Now the expression of the class described as ‘‘ placed in
[64] exalted stations,” and ‘‘ free from envious regards,” is

70 LEE LAWS, ΟΕ.

xz. Hence the contrary class, ze. they to whom this de-_
scription does not apply, will be represented by 1—+z, and
to this class all men are referred. Hence we have

y=u(1 -- xz).

If the proposition thus expressed had been placed in ἐπε.
equivalent form, ‘‘ Men in exalted stations are not free from

envious regards,” its expression would have been
yx =U 1 -- 2).
It will hereafter appear that this expression is really equiva- 4
lent to the previous one, 'on the particular hypothesis involved, :
viz., that v is an indefinite class symbol. 3
Ex.—‘‘ No men are heroes but those who unite self-denial —
to courage.” :
Let += ‘‘men,” y= ‘‘heroes,” z=‘‘those who practise ~
self-denial,” w, ‘‘ those who possess courage.” ‘
The assertion really is, that ‘‘men who do not possess q
courage and practise self-denial are not heroes.” ᾿
Hence we have

x(1 —2w)=v(1 -y)

for the equation required. ;

15. In closing this Chapter it may be interesting to com- 2
pare together the great leading types of propositions sym- |
bolically expressed. If we agree to represent by X and Y.
the symbolical expressions of the ‘‘ terms,” or things related, 4
those types will be ᾿

ἋΞ- ὦ
>, G8 G
VX =0Y.

Some minor forms are really included under these. Thus, 4
if Y=o, the second form becomes Ἶ
Doe
and if Y=1 it becomes
ὙΠ Ξ 1:

DIVISION OF PROPOSTIIONS vl

[65] both which forms admit of interpretation. [1 is further
to be noticed, that the expressions X and Y, if founded
upon a sufficiently careful analysis of the meaning of the
‘“‘terms” of the proposition, will satisfy the fundamental
law of duality which requires that we have

‘Xe Xx Ore XC) 556.
Fe Y of 1 — 7) Oo:

[66] CHAPTER V

OF THE FUNDAMENTAL PRINCIPLES OF SYMBOLICAL REASON-
ING, AND OF THE EXPANSION OR DEVELOPMENT OF
EXPRESSIONS INVOLVING LOGICAL SYMBOLS

1. THE previous Chapters of this work have been devoted to”
the investigation of the fundamental laws of the operations”
of the mind in reasoning ; of their development in the laws
of the symbols of Logic ; and of the principles of expression, '
by which that species of propositions called primary may be
represented in the language of symbols. These inquiries -
have been in the strictest sense preliminary. They form an
indispensable introduction to one of the chief objects of this:
treatise—the construction of a system or method of Logic
upon the basis of an exact summary of the fundamental laws
of thought. There are certain considerations touching the
nature of this end, and the means of its attainment, to which
I deem it necessary here to direct attention.

2. I would remark in the first place that the generality of
a method in Logic must very much depend upon the gener-¥
ality of its elementary processes and laws. We have, for
instance, in the previous sections of this work investigated,
among other things, the laws of that logical process of
addition which is symbolized by the sign +. Now thosey
laws have been determined from the study of instances, i
all of which it has been a necessary condition, that the classes
or things added together in thought should be mutually ex-
clusive. The expression ++, seems indeed uninterpretable,
unless it be assumed that the things represented by x and the
things represented by y are entirely separate ; that they em-
brace no individuals in common. And conditions analogous

to this have been involved in those acts of conception from
72 | |

PRINCIPLES OF SYMBOLICAL REASONING 73

the study of which the laws of the other symbolical opera-
tions have been ascertained. Thequestion then arises, whether
[67] it is necessary to restrict the application of these sym-
bolical laws and processes by the same conditions of inter-
pretability under which the knowledge of them was obtained.
If such restriction is necessary, it is manifest that no such
thing as a general method in Logic is possible. On the
other hand, if such restriction is unnecessary, in what light
are we to contemplate processes which appear to be uninter-
pretable in that sphere of thought which they are designed to
aid? These questions do not belong to the science of Logic
alone. They are equally pertinent to every developed form
of human reasoning which is based upon the employment of
a symbolical language.

3. I would observe in the second place, that this apparent
failure of correspondency between process and interpretation
does not manifest itself in the ovdixary applications of human
reason, For no operations are there performed of which the
meaning and the application are not seen ; and to most minds
it does not suffice that merely formal reasoning should con-
nect their premises and their conclusions ; but every step of
the connecting train, every mediate result which is estab-
lished in the course of demonstration, must be intelligible
also. And without doubt, this is both an actual condition
and an important safeguard, in the reasonings and discourses
of common life.

There are perhaps many who would be disposed to extend
the same principle to the general use of symbolical language
as an instrument of reasoning. It might be argued, that as
the laws or axioms which govern the use of symbols are
established upon an investigation of those cases only in
which interpretation is possible, we have no right to extend
their application to other cases in which interpretation is
impossible or doubtful, even though (as should be admitted)
such application is employed in the intermediate steps of
demonstration only. Were this objection conclusive, it must
be acknowledged that slight advantage would accrue from

74 THE LAWS OF THOUGHT

the use of a symbolical method in Logic. Perhaps that ad-—
vantage would be confined to the mechanical gain of em-
ploying short and convenient symbols in the place of more '
cumbrous ones. But the objection itself is fallacious. What- —
ever our ὦ priord anticipations might be, it is an unquestion- |
able fact that the validity of a conclusion arrived at [68] by
any symbolical process of reasoning, does not depend upon
our ability to interpret the formal results which have pre-
sented themselves in the different stages of the investigation.
There exist, in fact, certain general principles relating to the
use of symbolical methods, which, as pertaining to the
particular subject of Logic, I shall first state, and I shall |
then offer some remarks upon the nature and upon the
grounds of their claim to acceptance.

4. The conditions of valid reasoning, by the aidof symbols,
are—

Ist, [hata fixed interpretation be assigned to the symbols ©
employed in the expression of the data; and that the laws —
of the combination of those symbols be correctly determined
from that interpretation.

2nd, That the formal processes of solution or demonstra- 4
tion be conducted throughout in obedience to all the laws _
determined as above, without regard to the question of the
interpretability of the particular results obtained.

3rd, That the final result be interpretable in form, and
that it be actually interpreted in accordance with that system ©
of interpretation which has been employed in the expres- ἢ
sion of the data. Concerning these principles, the following —
observations may be made.

5. The necessity of a fixed interpretation of the symbols —
has already been sufficiently dwelt upon (II. 3). ΤΠ
necessity that the fixed result should be in such a form as %
to admit of that interpretation being applied, is founded on ¥
the obvious principle, that the use of symbols is a means ©
towards an end, that end being the knowledge of some in- |
telligible fact or truth. And that this end may be attained, ~
the final result which expresses the symbolical conclusion 4

PRINCIPLES OF SYMBOLICAL REASONING 75

must be in an interpretable form. It is, however, in con-
nexion with the second of the above general principles or
conditions (V. 4), that the greatest difficulty is likely to
be felt, and upon this point a few additional words are
necessary.

I would then remark, that the principle in question may
be considered as resting upon a general law of the mind,
the knowledge of which is not given to us ὦ priorz, 2. 6.
antecedently to [69] experience, but is derived, like the
knowledge of the other laws of the mind, from the clear
manifestation of the general principle in the particular in-
stance. A single example of reasoning, in which symbols
are employed in obedience to laws founded upon their
interpretation, but without any sustained reference to
that interpretation, the chain of demonstration conducting
us. through intermediate steps which are not interpretable,
to a final result which is interpretable, seems not only to
establish the validity of the particular application, but to

make known to us the general law manifested therein. No
accumulation of instances can properly add weight to such
evidence. It may furnish us with clearer conceptions of
that common element of truth upon which the application
of the principle depends, and so prepare the way for its
reception. It may, where the immediate force of the evidence
is not felt, serve as a verification, a posterzorz, of the practical
validity of the principle in question. But this does not
affect the position affirmed, viz., that the general principle
must be seen in the particular instance,—seen to be general
in application as well as true in the special example. The
employment of the uninterpretable symbol ,/—1, in the
intermediate processes of trigonometry, furnishes an illustra-
tion of what has been said. I apprehend that there is no
mode of explaining that application which does not covertly
assume the very principle in question. But that principle,
though not, as I conceive, warranted by formal reasoning
based upon other grounds, seems to deserve a place among
those axiomatic truths which constitute, in some sense, the

76 IME LAWS OF LHOUGAT

foundation of the possibility of general knowledge, and which
may properly be regarded as expressions of the mind’s own
laws and constitution. ;
6. The following is the mode in which the principle above.
stated will be applied inthe present work. It has been seen,
that any system of propositions may be expressed by equa-
tions involving symbols x, y, z, which, whenever interpreta-
tion is possible, are subject to laws identical in form with the
laws of a system of quantitative symbols, susceptible only
of the values 0 and 1 (II. 15). But as the formal processes
of reasoning depend only upon the laws of the symbols, and
not upon the nature of their interpretation, we are permitted
to treat the above symbols, [70] x, y, 2, as if they were
quantitative symbols of the kind above described. We may
an fact lay aside the logical interpretation of the symbols in
the given equation ; convert them into quantitative symbols,
susceptible only of the values 0 and τ, perform upon them
as such all the requisite processes of solution; and finally
restore to them their logical interpretation. And this is
the mode of procedure which will actually be adopted,
though it will be deemed unnecessary to restate in every
instance the nature of the transformation employed. The
processes to which the symbols x, y, 2, regarded as quanti-
tative and of the species above described, are subject, are.
not limited by those conditions of thought to which they
would, if performed upon purely logical symbols, be subject,
and a freedom of operation is given to us in the use of them,
without which the inquiry after a general method in Logic:
would be a hopeless quest.
Now the above system of processes would conduct us to.
no intelligible result, unless the final equations resulting:
therefrom were in a form which should render their inter-
pretation, after restoring to the symbols their logical signi-
ficance, possible. There exists, however, a general method
of reducing equations to such a form, and the remainder of
this chapter will be devoted to its consideration. I shall say
little concerning the way in which the method renders in-

Me RINCIPLES OF SYMBOLICAL REASONING 77

terpretation possible,—this point being reserved for the next
chapter,—but shall chiefly confine myself here to the mere
process employed, which may be characterized as a process
of ‘‘development.” As introductory to the nature of this
process, it may be proper first to make a few observations.

7. Suppose that we are considering any class of things
with reference to this question, viz., the relation in which
its members stand as to the possession or the want of a
certain property x As every individual in the proposed
class either possesses or does not possess the property in
question, we may divide the class into two portions, the
former consisting of those individuals which possess, the
latter of those which do not possess, the property. This
possibility of dividing in thought the whole class into two
constituent portions, is antecedent to all knowledge of the
constitution of the class derived from any other source ; of
[71] which knowledge the effect can only be to inform us,
more or less precisely, to what further conditions the portions
of the class which possess and which do not possess the given
property are subject. Suppose, then, such knowledge is to
the following effect, viz., that the members of that portion
which possess the property %, possess also a certain property
2, and that these conditions united are a sufficient definition
of them. We may then represent that portion of the original
class by the expression wx (II. 6). If, further, we obtain
information that the members of the original class which do
not possess the property x, are subject to a condition v, and
are thus defined, it is clear, that those members will be re-
presented by the expression v(1—*). Hence the class in
its totality will be represented by

ux +0(1 — x);

which may be considered as a general developed form for
the expression of any class of objects considered with refer-
ence to the possession or the want of a given property +.
The general form thus established upon purely logical
grounds may also be deduced from distinct considerations of

78 LHE LAWS CF THOUGHT

formal law, applicable to the symbols +, y, 2, equally in —
their logical and in their quantitative interpretation already ~
referred.to (V. 6). |

ὃ, Definitton.—Any algebraic expression involving a sym-_
bol x is termed a function of x, and may be represented —
under the abbreviated general form /(v). Any expression
involving two symbols, rand γ, is similarly termed a function |
of x and y, and may be represented under the general form |
_ f(x, v), and so on for any other case. q
Thus the form /(%) would aud represent any of the -

following functions, viz., +, 1 — +, —— =, &e.; ; and /(4, v) would |

equally represent any of the forms ζ.-Ἐ », x—2y, ΣΦ,

On the same principles of notation, if in any function £(2), :
we change # into 1, the result will be expressed by the
form f/(1); if in the same function we change + into ο, the |
result will be expressed oe in form /(0). Thus, if f(x)
ari

represent the [72] function © --

Fe aCe will se ce

and /(0) will represent =

9. Definition.—Any function f(x), in which x is a logical —
symbol, or a symbol of quantity susceptible only of the
values 0 and 1, is said to be developed, when it is reduced —
to the form az+0(1—+), a and ὁ being so determined as to ~
make the result equivalent to the function from which it |
was derived. |

This definition assumes, that it is possible to represent —
any function /(v) in the form supposed. The assumption ~
is vindicated in the following Proposition. |

PROPOSITION [Ι

10. Lo develop any function f(x) in which x ts a logical i
symbol. :

By the principle which has been asserted in this chapter, ᾿

|

|

}
ἢ

;

|
ἢ
4

‘
4

“PRINCIPLES OF SVMBOLICAL REASONING 79

it is lawful to treat x as a quantitative symbol, susceptible
only of the values o and I,
Assume then,
f(x) =ax + b(1 - x),

and making +=1, we have

Wit.
Again, in the same equation making +=0, we have
flo) =0.

Hence the values of ὦ and ὁ are determined, and substi-
tuting them in the first equation, we have

Ax) =f 1) + flor — %) 5 (1)
as the development sought.* The second member of the
equa-['73]tion adequately represents the function /(*),
whatever the form of that function may be. For + regarded

as a quantitative symbol admits only of the values 0 and 1,
and for each of these values the development

7 faye + flo - α),
assumes the same value as the function /(*).

* To some it may be interesting to remark, that the development of /(x)
obtained in this chapter, strictly holds, in the logical system, the place of the
expansion of f(x) in ascending powers of « in the system of ordinary algebra.
Thus it may be obtained by introducing into the expression of Taylor’s well-
known theorem, viz. : : ᾿

.,1.)-- Χο) -)7(ο)α +0) ἘΠ Ὁ ᾿

ἄς. (1)

the condition x(1 -.«)=0, whence we find 27=, «=x, &c., and

But making in (1), x=1, we get

flay=fl0) +7"(0)4 + FOF + 8c.

| whence

γι). LO) + Be. = f(1)- /(0),

᾿ and (2) becomes, on substitution,

S(x*)=S(0) +t SO) - ,(ο) 4 -
=f (1)*+ Χο) -- +),

the form in question. This demonstration in supposing /(*) to be developable
in a series of ascending powers of « is less general than the one in the text.

7

80 THE TAWS OF THOUGHT

As an illustration, let it be required to develop the

ad Here, when +=1, we find Αυτξ, and

function :

I _ a
when +=0, we find TO) == , or I. Hence the expression

required is ἜΣ
AEE =—-X+1-X;
T+2x 5

and this equation is satisfied for each of the values of which
the symbol +x is susceptible. |

PROPOSITION II

To expand or develop a function tnvolving any number of
logical symbols.

Let us begin with the case in which there are two sym-
bole 2 ang -y, and let us represent the ‘function. to be
developed by /(%, 1).

First, considering /(%, vy) as a function οἱ + alone, and
expanding it by the general theorem (1), we have

Λα, Y)=K it, YX +N; YI -- &) 5 (2)

[74] wherein /(1, y) represents what the proposed function —
becomes, when in it for x we write 1, and /(o, y) what |
the said function becomes, when in it for x we write oO. 3
Now, taking the coefficient /(1, v), and regarding it asm

a function of y, and expanding it accordingly,. we have ;

ἢ

Κι, )ξλι, Iy+fA1, o)(1 ἜΝ (3 ᾿

wherein /(1, 1) represents what /(1, γ᾽) becomes when y
is made equal to 1, and /(1, 0) what /(1, v) becomes whe 1
y is made equal to o. 4
In like manner, the coefficient /(0, y) gives by expansion,

KO, γ)-Άο, τὴν +flo, ογ)( -y). (4)
Substitute in (2) for (1, 7), 7χἴο, v), their values given in
(3) and (4), and we have ἢ

Kx; Y) =f(1, τ)ὰν + fl, ο)χᾷι πον ἡ 1)(1 -- “)» ἥ
ἘΛο, χα -- )α -), 6)

πὰς

PRINCIPLES OF SYMBOLICAL REASONING 81

for the expansion required. Here /(1, 1) represents what
ΠῚ v) becomes when we make therein r=1, y=1; /(1, 0)
represents what /(%, vy) becomes when we make therein

“τι, y=0, and so on for the rest.

I—

Thus, if f(x, v) represent the function : τ we find

fa, )=2, flt,0)=2=0, flo, 1)=2, Ao, 0)=1,

whence the expansion of the given function is
ay +on(t —y) + U(r ayy +(1-a)(1 -9).

It will in the next chapter be seen that the forms - and =

the former of which is known to mathematicians as the
symbol of indeterminate quantity, admit, in such ex-
pressions as the above, of a very important logical inter-
pretation.

Suppose, in the next place, that we have three symbols
in the function to be expanded, which we may represent
under the general form /(4, y, 2). Proceeding as before,
we get

[75] A, 9; 2) -- 7, τ, raya t+ f(1, τ, ογαγᾷι - 2) +f/(1, 0, 1)a(r—y)2
+f(1, 0, o)x(1 —y)(1 —2)+f(o, 1, 1)(1 -- x)yz
+/(0, τ, 0 -- αὐγᾷ -- 5) +f/(0, 0, 1)(1 - αλὰ ~y)s
+f(0, 0, ολ(ι -- α)ίι —y)(1 -- δ),
in which /(1, 1, 1) represents what the function /(%, y, 2) be-
comes when we make therein r=1, y=1, 2=1, and so on
for the rest.
_ 11. It is now easy to see the general law which determines
the expansion of any proposed function, and to reduce the
method of effecting the expansion to a rule. But before
proceeding to the expression of such a rule, it will be con-
venient to premise the following observations :—
_ Each form of expansion that we have obtained consists
of certain terms, into which the symbols x, y, ἅς. enter,

‘multiplied by coefficients, into which those symbols do not
6

[

rl

82 THE LAWS OF THOUGAT

enter. Thus the expansion of /() consists of two terms, +
and 1—x, multiplied by the coefficients f(1) and /(0) τες!
spectively. And the expansion of /(@%, 7) consists of the
four terms xy, *(1—y), (1—*)y, and (1—*)(1-y), multi-
plied by the coefficients /(1, 1), (1, ©), F(O, 1), Χο, 0), τοῦ
spectively. The terms x, I—4, in the former case, and the
terms xy, +(1—y), &c., in the latter, we shall call the con-
stituents of the expansion. It is evident that they are in
form independent of the form of the function to be expanded, q
Of the constituent xy, x and y are termed the factors. }

The general rule of development will therefore consist o ᾿
two parts, the first of which will relate to the formation of
the constituents of the expansion, the second to the de-
termination of their respective coefficients. It is as follows

ist. 70 expand any function of the symbols x,y, 2.—Form
a series of constituents in the following manner: Let the:
first constituent be the product of the symbols; change in
this product any symbol z into 1 -- 5, for the second con-
stituent. Then in both these change any other symbol y
into 1—y, for two more constituents. Then in the four
constituents thus obtained change any other symbol » int 0
1 —x, for four new constituents, and so on until the number
of possible changes is exhausted. _ |

andly. To find the coefficient of any constituent.—lf that
con-['76]stituent involves vas a factor, change in the original
function ¢ into 1; but if it involves τ τας a factor, change
in the original function 4 into o. Apply the same rule with
reference to the symbols y, 2, &c.: the final calculated’
value of the function thus transformed will be the coefficie ἰ
sought. | δ

The sum of the constituents, multiplied each by its re-
spective coefficient, will be the expansion required. |

12. It is worthy of observation, that a function may be
developed with reference to symbols which it does not
explicitly contain. Thus if, proceeding according to the
rule, we seek to develop the function 1 -- 2, with referen 8
to the symbols # and y, we have, a

a Ὡς ey

-PRINCIPLES OF SYMBOLICAL REASONING 83

When +=1 and y=I the given function =o.

ἄπ oe ») ) τ Ὁ»
Ὄπ ΘΙ FoF i) 9 τεὸν ἰς
Ma yh ee -- ὸ ” ») τ S

Whence the development is

! I—x=oxy +0x(1 -y) + ( -- α)ν Ἐ{ -- α)ᾷ -- ));
and this is a true development. The addition of the terms
(1 —x)y and (1 —%)(1—y) produces the function 1 -- 1.

The symbol 1 thus developed according to the rule, with
respect to the symbol 4%, gives

X+1-X.
Developed with respect to x and y, it gives
ay κα -- )) + (α —x)y ἘΚ -- x(t -- 3).
Similarly developed with respect to any set of symbols, it
produces a series consisting of all possible constituents of
those symbols.

13. A few additional remarks concerning the nature of the
general expansions may with propriety be added. Let us
take, for illustration, the general theorem (5), which presents
the type of development for functions of two logical symbols.

In the first place, that theorem is perfectly true and in-
telligible when x and y are quantitative symbols of the
species considered in this chapter, whatever algebraic form
may be assigned to the function /(%, y), and it may there-
fore be intelligibly em-['77]ployed in any stage of the pro-
cess of analysis intermediate between the change of inter-
pretation of the symbols from the logical to the quantitative
system above referred to, and the final restoration of the
logical interpretation.

Secondly. The theorem is perfectly true and intelligible
when # and y are logical symbols, provided that the form of
the function χα, v) is such as to represent a class or col-
lection of things, in which case the second member is always
logically interpretable. For instance, if /(4, y) represent the
| function 1 —7+zy, we obtain on applying the theorem

84 THE LAWS OF THOUGHT

1-x+ay=xy+ox(1 —y)+(1-x)y+(1 ~ x1 -y),
=xy+(1-x)y+(1-x)(1-y),

and this result is intelligible and true. .
Thus we may regard the theorem as true and intelligible 1
for quantitative symbols of the species above described, —
always; for logical symbols, always when interpretable,
Whensoever therefore it is employed in this work it must be |
understood that the symbols +, y are quantitative and of the
particular species referred to, if the expansion obtained is

not interpretable.

But though the expansion is not always immediately
interpretable, it always conducts us at once to results which |
are interpretable. Thus the expression r—y gives on de-—
velopment the form

x(t —y) — y(1 4) .
which is not generally interpretable. We cannot take, in
thought, from the class of things which are 1.5 and not y’s,
the class of things which are y’s and not 2#’s, because the
latter class is not contained in the former. But if the form —
4—y presented itself as the first member of an equation, ©
of which the second member was Oo, we should have on
development

x(t -- )) -- (1 -x)=0.,
Now it will be shown in the next chapter that the above
equation, + and y being regarded as quantitative and of the
species described, is resolvable at once into the two equations

x(1-y)=0, y(1-x)=0,

and these equations are directly interpretable in Logic when
lo-[78]gical interpretations are assigned to the symbols 4
and y. And it may be remarked, that though functions do
not necessarily become interpretable upon development, yet
equations are always reducible by this process to inter-~
pretable forms. i
14. The following Proposition establishes some important
properties of constituents. In its enunciation the symbol ¢_
is employed to represent indifferently any constituent of

|

BAIVNCIPLES OF SYMBOLICAL KEASONING 85

an expansion. Thus if the expansion is that of a function
of two symbols x and y, ¢ represents any of the four forms
ay, (1—y), (1—*yy, and (1—*x)(1—y). Where it is neces-
sary to represent the constituents of an expansion by single
symbols, and yet to distinguish them from each other, the
distinction will be marked by suffixes. Thus ¢, might be
employed to represent ἅν, ¢, to represent 4(1 —y), and so on,

PROPOSITION III

Any single constituent ἐ of an expansion satisfies the law
of duality whose expression ts

“#1 -t)=0,

The product of any two distinct constituents of an expansion
ts equal to 0, and the sum of all the constituents ts equal toji.

ist. Consider the particular constituent xy, We have
Ἂν Mey ee

But =, y?=y, by the fundamental law of class symbols ;
hence

xy X xy = XY.
Or representing xy by Ζ,
txt=t,
or “(1 ἢ =0.

Similarly the constituent +(1—¥) satisfies the same law.
For we have
Coe (Uy) εν
{x(1 —y)}2=a2(1-y), or t(1—-Z)=0.

Now every factor of every constituent is either of the form
«or of the form 1—%. Hence the square of each factor is
equal to that [79] factor, and therefore the square of the
product of the factors, z.e. of the constituent, is equal to
the constituent ; wherefore ¢ representing any constituent,
we have

C= or &1—t)=0.

2andly. The product of any two constituents iso. This

|
|
|

86 THE LAWS OF THOUGHT

is evident from the general law of the symbols expressed by ~
the equation #(1—*)=0; for whatever constituents in the 4
same expansion we take, there will be at least one factor ~
x in the one, to which will correspond a factor I1—x in the ¥
other. Ἷ
3rdly. The sum of all the constituents of an expansion is
unity. This is evident from addition of the two constituents }
x and 1—4, or of the four constituents, xy, x(1—y), (1-4), '
(1—2x)(1-y). But it is also, and more generally, proved by 4
expanding 1 in terms of any set of symbols (V. 12). The '
constituents in this case are formed as usual, and all the ᾿
coefficients are unity. j

1s. With the above Proposition we may connect the ~
following.

PROPOSITION IV

If V represent the sum of any sertes of constituents, the 3
separate coefficients of which are τ, then is the condition —
satisfied, !

Κα -- Κ)γΞο.
Let ¢,, 4%. . . ¢, be the constituents in question, then
Vee ee.

Squaring both sides, and observing that 7?=4,, 4,4, =0,|
&c., we have

V+=4,+4, oe pe ἜΖ:
whence

Therefore

[80] CHAPTER VI

OF THE GENERAL INTERPRETATION OF LOGICAL EQUA-
TIONS, AND THE RESULTING ANALYSIS OF PROPOSI-
TIONS. ALSO, OF THE CONDITION OF INTERPRETA-
BILITY OF LOGICAL FUNCTIONS

1. IT has been observed that the complete expansion of any
function by the general rule demonstrated in the last chapter,
involves two distinct sets of elements, viz., the constituents
of the expansion, and their coefficients. I propose in the
present chapter to inquire, first, into the interpretation of
constituents, and afterwards into the mode in which that
interpretation is modified by the coefficients with which they
are connected.

ne: terms. logical equation,” τ logical function,” :&e.,
will be employed generally to denote any equation or function
involving the symbols x, vy, &c., which may present itself
either in the expression of a system of premises, or in the
train of symbolical results which intervenes between the
premises and the conclusion. If that function or equation
is in a form not immediately interpretable in Logic, the
symbols x, vy, &c., must be regarded as quantitative symbols
of the species described in previous chapters (II. 15), (V. 6),
as satisfying the law,

x(I — x) =0.

By the problem, then, of the interpretation of any such
logical function or equation, is meant the reduction of it to
a form in which, when logical values are assigned to the
symbols +, y, &c., it shall become interpretable, together
with the resulting interpretation. These conventional de-

finitions are in accordance with the general principles for the
87

88 THe LAWS OF HOGG.

conducting of the method of this treatise, laid down in the
previous chapter.

[81] PROPOSITION I

2. The constituents of the expansion of any function of the
logical symbols x, y, &c., are interpretable, and represent the
several exclusive divisions of the universe of discourse, formed
by the predication and dental in every possible way of the
gualities denoted by the symbols x, y, &e. |

For greater distinctness of conception, let it be supposed
that the function expanded involves two symbols # and y,
with reference to which the expansion has been effected,
We have then the following constituents, viz. :

xy, x(1-y), (1-x)y, (1-x)(1-y). :

Of these it is evident, that the first ry represents that class
of objects which at the same time possess both the elementary
qualities expressed by x and y, and that the second 4(1 ~yy
represents the class possessing the property x, but not the
property y. In like manner the third constituent represents
the class of objects which possess the property represented
by y, but not that represented by x; and the fourth con-
stituent (I—+)(1—y) represents that class of objects the
members of which possess neither of the qualities in question, |
Thus the constituents in the case just considered represent
all the four classes of objects which can be described by
affirmation and denial of the properties expressed by x and y,
Those classes are distinct from each other. No member of
one is a member of another, for each class possesses some |
property or quality contrary to a property or quality possessed
by any other class. Again, these classes together make up
the universe, for there is no object which may not be de-
scribed by the presence or the absence of a proposed quality,
and thus each individual thing in the universe may be re-
ferred to some one or other of the four classes made by the

possible combination of the two given classes x and y, and
their contraries.

|

|
|
|

OF INTERPRETATION 89

The remarks which have here been made with reference
to the constituents of /(x, v) are perfectly general in char-
acter. The constituents of any expansion represent classes
—those classes [82] are mutually distinct, through the
possession of contrary qualities, and they together make up
the universe of discourse.

3. These properties of constituents have their expression

in the theorems demonstrated in the conclusion of the last
chapter, and might thence have been deduced. From the
fact that every constituent satisfies the fundamental law of
the individual symbols, it might have been conjectured that
each constituent would represent a class, From the fact
that the product of any two constituents of an expansion
vanishes, it might have been concluded that the classes they
represent are mutually exclusive. Lastly, from the fact
that the sum of the constituents of an expansion is unity,
it might have been inferred, that the classes which they
represent, together make up the universe.
4. Upon the laws of constituents and the mode of their
interpretation above determined, are founded the analysis
and the interpretation of logical equations. That all such
equations admit of interpretation by the theorem of develop-
ment has already been stated. I propose here to investigate
the forms of possible solution which thus present themselves
in the conclusion of a train of reasoning, and to show how
those forms arise. Although, properly speaking, they are
but manifestations of a single fundamental type or principle
of expression, it will conduce to clearness of apprehension
if the minor varieties which they exhibit are presented
separately to the mind.

The forms, which are three in number, are as follows:

FORM I

5. The form we shall first consider arises when any logical

equation 7 =0 is developed, and the result, after resolution
into its component equations, is to be interpreted. The

function is. supposed to involve the logical symbols 7, y, &c.,

go THE LAWS OF THOUGHT ἷ

in combinations which are not fractional. Fractional com-_
binations indeed only arise in the class of problems which —
will be considered when we come to speak of the third of
the forms of solution above referred to.

PROPOSITION II |
To interpret the logical equation V =o.

For simplicity let us suppose that V involves but twall
sym-[83]bols, x and y, and let us represent the development
οὔ the given equation by

axy + bx(1—y)+e(1 —x)y+a(1 -x)1-y)=0; Ἵ

a, ὦ, c, and d being definite numerical constants.

Now, suppose that any coefficient, as a, does not vanish, — .
Then multiplying each side of the equation by the constituent
ay, to which that coefficient is attached, we have

axy =0,

whence, as a does not vanish,

Eso). :
and this result is quite independent of the nature of the other
coefficients of the expansion. Its interpretation, on assign-
ing to x and y their logical significance, is ‘‘ No individuals
belonging at once to the class represented by x, and the laa
represented by y, exist.”

But if the coefficient a does vanish, the term ary does rofl
appear in the development (1), and, therefore, the equatio
4+y =O cannot thence be deduced. |

In like manner, if the coefficient ὁ does not vanish, we have.

x(1 -- γ) ΞΞ ο,
which admits of the interpretation, ‘‘ There are no individuals
which at the same time belong to the class x, and do not
belong to the class y.”

Either of the above interpretations may, however, as wil
subsequently be shown, be exhibited in a different form.
The sum of the distinct interpretations thus obtained from

i

OF INTERPRETATION ΟἹ

| the several terms of the expansion whose coefficients do not
vanish, will constitute the complete interpretation of the
equation V=o. The analysis is essentially independent of
| the number of logical symbols involved in the function V,

-and the object of the proposition will, therefore, in all in-
stances, be attained by the following Rule :—

RULE.—Develop the function V, and equate to Ο every con-
stituent whose coefficient does not vanish. The interpretation
of these results collectively will constitute the interpretation of
the given equation.

[84] 6. Let us take as an example the definition of
**clean beasts,” laid down in the Jewish law, viz., ‘‘ Clean
beasts are those which both divide the hoof and chew the
cud,” and let us assume

“= cleat peasts ;
y =beasts dividing the hoof ;
= beasts chewing the cud.

Then the given proposition will be represented by the equation

which we shall reduce to the form
x—-ys=0,

and seek that form of interpretation to which the present
method leads, Fully. developing the first member, we have

oxys + xy(1 - 2) +x«(1 -—y)s+ x(t -- )( - 2)

τᾷ - αὐτο - αν ~2) 4 0(1 - αὐὰ - γ)ε τοί - α)α -χι - 9).
Whence the terms, whose coefficients do not vanish, give
νι: -- το, x2(1-y)=0, x(1-y)(1-2z)=0, (1 -- σ)γετξο.
These equations express a denial of the existence of certain
classes of objects, viz. :

_ Ist. Of beasts which are clean, and divide the hoof, but
do not chew the cud.

2nd. Of beasts which are clean, and chew the cud, but do
not divide the hoof.

: ; x= YZ,

92 THE LAWS OF THOUGHT

3rd. Of beasts which are clean, and neither divide the hoof
nor chew the cud.

Ath. Of beasts which divide the hoof, and chew the cud, |

and are not clean.

Now all these several denials are really involved in the
original proposition. And conversely, if these denials be
granted, the original proposition will follow as a necessary
consequence. They are, in fact, the separate elements of
that proposition. Every primary proposition can thus be
resolved into a series of denials of the existence of certain
defined classes of things, and may, from that system of

denials, be itself reconstructed. It might here be asked, —

how it is possible to make an assertative pro-[85]position

out of a series of denials or negations? From what source
is the positive element derived? I answer, that the mind

assumes the existence of a universe not ὦ 271071 as a fact in-
dependent of experience, but either a posteriori as a deduc-

tion from experience, or hypothetically as a foundation of the

possibility of assertive reasoning. Thus from the Proposition,
“Ὁ There are no men who are not fallible,” which is a negation
or denial of the existence of ‘‘ infallible men,” it may be in-
ferred either hypothetically, ‘‘ All men (if men exist) are
fallible,” or absolutely, (experience having assured us of
the existence of the race), ‘‘ All men are fallible.”

The form in which conclusions are exhibited by the method
of this Proposition may be termed the form of ‘‘ Single or

Conjoint Denial.”
FORM II

7. As the previous form was derived from the development
and interpretation of an equation whose second member is 0,
the present form, which is supplementary to it, will be derived
from the development and interpretation of an equation whose
second member is 1. It is, however, readily suggested by
the analysis of the previous Proposition.

Thus in the example last discussed we deduced from the
equation

Xx -yZ=0

OF INTERPRETATION 93

the conjoint denial of the existence of the classes represented
by the constituents

xy(t—s), x2(1-y), a(1—y)(x—2), (1 -- αὐνε,

whose coefficients were not equal to 0. It follows hence that
the remaining constituents represent classes which make up
the universe. Hence we shall have

ays (1 - αἰνὰ - 2) + (1 - α)α - 7) + (1 -- αὐ --)( 5) =F.

This is equivalent to the affirmation that all existing things
belong to some one or other of the following classes, viz. :

Ist. Clean beasts both dividing the hoof and chewing
the cud.

[86] 2nd. Unclean beasts dividing the hoof, but not chew-
ing the cud.

3rd. Unclean beasts chewing the cud, but not dividing
the hoof.

4th. Things which are neither clean beasts, nor chewers
of the cud, nor dividers of the hoof.

_ This form of conclusion may be termed the form of ‘‘ Single
or Disjunctive Affirmation, ”’—single when but one constituent
appears in the final equation ; disjunctive when, as above,
more constituents than one are there found.

Any equation, VY =o, wherein V satisfies the law of duality,
may also be made to yield this form of interpretation by re-
ducing it to the form 1—V=1, and developing the first
member. The case, however, is really included in the next
general form. Both the previous forms are of slight import-
ance compared with the following one.

FORM III

8. In the two preceding cases the functions to be de-
veloped were equated to oand I respectively. In the present
case I shall suppose the corresponding function equated to
any logical symbol w. We are then to endeavour to inter-
pret the equation V=w, V being a function of the logical
symbols x, v, z, &c. In the first place, however, I deem it

94 THE LAWS OF THOUGHT

a

necessary to show how the equation V=w, or, as it will
usually present itself, w= V, arises.

Let us resume the definition of ‘‘clean beasts,” employed
in the previous examples, viz., ‘‘Clean beasts are those
which both divide the hoof and chew the cud,” and suppose
it required to determine the relation in which ‘‘ beasts chew-
ing the cud” stand to ‘‘clean beasts” and ‘‘ beasts dividing
the hoof.” The equation expressing the given proposition is

aes

and our object will be accomplished if we can determine ¢ as
an interpretable function of x and y. .

Now treating x, y, as symbols of quantity subject to a
peculiar law, we may deduce from the above equation, by
solution,

[87] But this equation is not at present in an interpretable ἷ
form. If we can reduce it to such a form it will furnish the
relation required. |

On developing the second member of the above equation,
we have

a= ay + a(t -y)+o(1 -αὴν τα τὶ x)(1 τ

and it will be shown hereafter (Prop. III.) that this admits
of the following interpretation : |

‘* Beasts which chew the cud consist of all clean beasts
(which also divide the hoof), together with an indefinite re-
mainder (some, none, or all) of unclean beasts which do not
divide the hoof.” ;

9. Now the above is a particular example of a problem of
the utmost generality in Logic, and which may thus be
stated :—‘‘ Given any logical equation connecting the sym-
bols x, y, , w, required an interpretable expression for the
relation of the class represented by w to the classes repre-
sented by the other symbols x, y, z, &c.” ,

The solution of this problem consists in all cases in deter-

Se ee ae eee ἐστὸν

OF INTERPRETATION 95

mining, from the equation gzven, the expression of the above

symbol w, in terms of the other symbols, and rendering that
expression interpretable by development. Now the equation
given is always of the first degree with respect to each of
the symbols involved. The required expression for w can
therefore always be found. In fact, if we develop the given
equation, whatever its form may be with respect to w, we
obtain an equation of the form

Ew+ E'(1 -w)=o9, (1)

ΠΕ and E’ being functions of the remaining symbols. From

the above we have

'—(F' —-E)w.
Therefore Fos δ

να

aE SH @)

and expanding the second member by the rule of develop-

-ment, it will only remain to interpret the result in logic by

the next proposition.

/

[88] If the fraction ἘΠΕ has common factors in its

numerator and denominator, we are not permitted to reject
them, unless they are mere numerical constants. For the
symbols x, vy, &c., regarded as quantitative, may admit of
such values 0 and 1 as to cause the common factors to be-
come equal to 0, in which case the algebraic rule of reduction
fails. This is the case contemplated in our remarks on the
failure of the algebraic axiom of division (II. 14). To ex-
press the solution in the form (2), and without attempting
to perform any unauthorized reductions, to interpret the re-
sult by the theorem of development, is a course strictly in
accordance with the general principles of this treatise.

If the relation of the class expressed by 1—w to the
other classes, x, y, &c. is required, we deduce from (1), in
like manner as above,
es
E-£E”
to the interpretation of which also the method of the follow-
ing Proposition is applicable :

70

96 THE LAWS OF THOUGAT

PROPOSITION III

10. 70 determine the interpretation of any logical equation
of the formw=V, in which τὸ ts a class symbol, and V a
function of other class symbols quite unlimited in rts form.

Let the second member of the above equation be fully
expanded. Each coefficient of the result will belong to
some one of the four classes, which, with their respective
interpretations, we proceed to discuss.

ist. Let the coefficient be 1. As this is the symbol of
the universe, and as the product of any two class symbols
represents those individuals which are found in both classes,
any constituent which has unity for its coefficient must be
interpreted without limitation, z.e. the whole of the class
which it represents is implied.

2nd. Let the coefficient be 0. As in Logic, equally with
Arithmetic, this is the symbol of Nothing, no part of the
class [89] represented by the constituent to which it is pre-
fixed must be taken.

3rd. Let the coefficient be of the form τ Now, as in

: : O : :
Arithmetic, the symbol a represents an zzdefinite number,

except when otherwise determined by some special circum-
stance, analogy would suggest that in the system of this
work the same symbol should represent an zzdefintte class.
That this is its true meaning will be made clear from the
following example :

Let us take the Proposition, ‘‘ Men not mortal do not
exist” ; represent this Proposition by symbols ; and seek, in
obedience to the laws to which those symbols have been
proved to be subject, a reverse definition of ‘‘ mortal beings,”
in terms:of-* men,”

Now if we represent ‘‘men” by y, and ‘‘ mortal beings,”
by x, the Proposition, ‘‘ Men who are not mortals do not
exist,” will be expressed by the equation

WI - x)=0,

OF INTERPRETATION 97

from which we are to seek the value of x Now the above
equation gives
Yryx=O, OT yx=y.

Were this an ordinary algebraic equation, we should, in the
next place, divide both sides of it by y. But it has been
remarked in Chap. II. that the operation of division cannot
be performed with the symbols with which we are now
engaged. Our resource, then, is to express the operation,
and develop the result by the method of the preceding
chapter. We have, then, first,

x=,

hg

and, expanding the second member as directed,
12)
χε AE sy):

This implies that mortals (7) consist of all men (y), together
with such a remainder of beings which are not men (1 -- 7),

as will be indicated by the coefficient 5° Now let us inquire

what [90] remainder of ‘‘not men” is implied by the
premiss. It might happen that the remainder included all
the beings who are not men, or it might include only some
of them, and not others, or it might include none, and any
one of these assumptions wculd be in perfect accordance
with our premiss. In other words, whether those beings
which are not men are aé/, or some, or none, of them mortal,
the truth of the premiss which virtually asserts that all men
are mortal, will be equally unaffected, and therefore the ex-

ἜΣ Ὁ ae
pression 5 here indicates that all, some, or none of the class

to whose expression it is affixed must be taken.
Although the above determination of the significance of

the symbol = is founded only upon the examination of a

particular case, yet the principle involved in the demonstra-
tion is general, and there are no circumstances under which

7

98 THE LAWS OF THOUGHT

the symbol can present itself to which the same mode of | |
"ἢ :
analysis is inapplicable. We may properly term 5 an in~

definite class symbol, and may, if convenience should require, —
replace it by an uncompounded symbol v, subject to the
fundamental law, v(1 —v) =O. :

4th. It may happen that the coefficient of a constituent |
in an expansion does not belong to any of the previous —
cases, To ascertain its true interpretation when this happens, —
it will be necessary to premise the following theorem :

11. THEOREM.—J/f a function V, intended to represent
any class or collection of objects, w, be expanded, and 17 the
numerical coefficient, a, of any constituent in tts development, :

do not satisfy the law,
α(ι —a)=0, ;

then the constituent in question must be made equal to Ο. ᾿
To prove the theorem generally, let us represent the ex=

pansion given, under the form
woah ae cn (x)

in which ¢,, ¢,, ¢;, &c. represent the constituents, and a,, ag)

a3, &c. the coefficients ; let us also suppose that a, and Ay

do not satisfy the law :

αγί(τ -- α]) τε ο, ἀμί! —ay)=0;
[91] but that the other coefficients are subject to the law
in question, so that we have

aes τος --: Re

7 gat
a,"=a,, &C.

Now multiply each side of the equation (1) by itself. The
result will be ,

pit gS ΡΣ

τὺ = A,7t, + apt, + XC (2)

AAiseisrevicent toi the fact that ae inst feprescut the
development of the equation

w= V2,

but it may also be proved by actually squaring (1), an¢
observing that we have

τ, t2=t, %t%44=0, ἄχεα.

a ys 2.

OF INTERPRETATION 99

by the properties of constituents. Now subtracting (2)
from (1), we have
(ay — @y")4 + (ας — ἀφ") = 0.

Or, a,(I — a,)t, +.a,(1 — ay)t, =0.
Multiply the last equation by ¢,; then since ¢,¢,=0, we
have

αι —a,)¢,=0, whence 4, =o.
In like manner multiplying the same equation by 4,, we
have

a1 —@_)t,=0, whence ¢,=0.

Thus it may be shown generally than any constituent
whose coefficient is not subject to the same fundamental
law as the symbols themselves must be separately equated
too. The usual form under which such coefficients occur

is =. This is the algebraic symbol of infinity. Now the

nearer any number approaches to infinity (allowing such an
expression), the more does it depart from the condition of
satisfying the fundamental law above referred to.

The symbol = , whose interpretation was previously dis-

cussed, does not necessarily disobey the law we are here
considering, for it admits of the numerical values Oo and
1 indifferently. Its actual interpretation, however, as an
indefinite class symbol, cannot, I conceive, except upon
‘the ground of analogy, be de-[92]duced from its arith-
metical properties, but must be established experimentally.
' 12, We may now collect the results to which we have
been led, into the following summary :

ist -bhe«symbol 414.45 -the scociicicnt.o1 ἃ terin.in, ‘a
development, indicates that the whole of the class which
that constituent represents, is to be taken.

2nd. The coefficient o indicates that none of the class
are to be taken.

3rd. The symbol indicates that a perfectly zudefinite

\

100 THE LAWS OF THOUGHT

portion of the class, 2.6. some, mone, OF all of its members
are to be taken.

4th. Any other symbol as a coefficient indicates that the
constituent to which it is prefixed must be equated to Ο.

It follows hence that if the solution of a problem, ob-
tained by development, be of the form

SOY ney ste Once 8
Ov 70

that solution may be resolved into the two following equa-
tions, viz., Ὁ
w= A+uC, (3)
DO; (4)

ee ee re one

v being an indefinite class symbol. The interpretation of :

(3) shows what elements enter, or may enter, into the com-
position of w, the class of things whose definition is re-
quired ; and the interpretation of (4) shows what relations
exist among the elements of the original problem, in
perfect independence of w.

Such are the canons of interpretation. It may be added,
that they are universal in their application, and that their
use is always unembarrassed by exception or failure.

13. Corollary. —\f V be an independently interpretable logi-
cal function, it will satisfy the symbolicallaw, V(1— V)=0.

By an independently interpretable logical function, I

mean one which is interpretable, without presupposing any ;
relation among the things represented by the symbols which :

it involves. Thus #(1—y) is independently interpretable,

but x—y is not so. [93] The latter function presupposes, —
as a condition of its interpretation, that the class represented
by y is wholly contained in the class represented by x; the )

former function does not imply any such requirement.

Now if V be independently interpretable, and if w
represent the collection of individuals which it contains, the
equation w= V will hold true without entailing as a con- '
sequence the vanishing of any of the constituents in ἴῃ6
development of V; since such vanishing of constituents :

OF INTERPRETATION IOI

would imply relations among the classes of things denoted
by the symbols in V. Hence the development of V will
be of the form

a,t, + ant, + &c.
the coefficients a,, a, &c. all satisfying the condition

a(i—a@,)=0, a (1-a,)=0, &c.

Hence by the reasoning of Prop. [V., Chap. V., the function
V will be subject to the law

Vii -- V)=0.

This result, though evident @ przorz from the fact that V
is supposed to represent a class or collection of things, is
thus seen to follow also from the properties of the con-
stituents of which it is composed. The condition V(1— V)
=O may be termed ‘‘the condition of interpretability of
logical functions.”

14. The general form of solutions, or logical conclusions
developed in the last Proposition, may be designated as a
Ε΄ Relation between terms.” I use, as before, the word
“*terms” to denote the parts of a proposition, whether
simple or complex, which are connected by the copula
eis” or ‘‘are.” The classes of things represented by the

iadividual symbols may be called the elements of the
proposition.

15. Ex. 1.—Resuming the definition of ‘‘clean beasts,”

(VI. 6), required a description of ‘‘ unclean beasts.”

phere, as belore,.4 standing for “clean beasts, 7 for
‘beasts dividing the hoof,” z for ‘‘ beasts chewing the
cud,” we have
! X= VE 5 (5)
whence I-xX=1-Yyz;

and developing the second member,
[94] rx yr 2) 21 9) Hy) ) 5

which is interpretableinto the following Proposition: Unclean
beasts are all which divide the hoof without chewing the cud,

dese ea aaa

102 THE LAWS OF THOUGHT

all which chew the cud without dividing the hoof, and all
which neither divide the hoof nor chew the cud.

Ex. 2.—The same definition being given, required a
description of beasts which do not divide the hoof.

From the equation + =yz we have

Ζ
therefore ZB-X,

and developing the second member,
I — y= 0005 + —— (1 —z)+(1 —x)e+o(t -- γᾷ - 2).

Here, according to the Rule, the term whose coefficient |
. πὶο'Ἐ
iS eae must be separately equated to 0, whence we have

I-y=(1-x)s+v(1-x)(1—2),
χίι -- 2) το;

whereof the first equation gives by interpretation the Pro-_
position: Beasts which do not divide the hoof consist of all
unclean beasts which chew the cud, and an indefinite γὲ6-
mainder (some, none, or all) of unclean beasts which do not —
chew the cud.

The second equation gives the Proposition: There are
no clean beasts which do not chew the cud. This is one of ©
the independent relations above referred to. We sought the ~
direct relation of ‘‘ Beasts not dividing the hoof,” to ‘‘ Clean
beasts and beasts which chew the cud.” It happens, how- ;
ever, that independently of any relation to beasts not ;
dividing the hoof, there exists, in virtue of the premiss, a Ἷ
separate relation between clean beasts and beasts which
chew the cud. This relation is also necessarily given by —
the process.

Ex. 3.—Let us take the following definition, Viz. 4
‘“ Responsible beings are all rational beings who are either —
free to act, or [95] have voluntarily sacrificed their freedom,” —
and apply to it the preceding analysis,

OF INTERPRETATION 103

Let x stand for responsible beings.
y i rational beings.
Ζ a those who are free to act,
We those who have voluntarily sacrificed their
freedom of action.

In the expression of this definition I shall assume, that
the two alternatives which it presents, viz.: ‘‘ Rational beings
free to act,” and ‘‘ Rational beings whose freedom of action

has been voluntarily sacrificed,” are mutually exclusive, so
_ that no individuals are found at once in both these divisions.
This will permit us to interpret the proposition literally into
the language of symbols, as follows:

x= yZ+yw. (6)

Let us first determine hence the relation of ‘‘ rational
beings” to responsible beings, beings free to act, and beings
whose freedom of action has been voluntarily abjured.
Perhaps this object will be better stated by saying, that we
desire to express the relation among the elements of the pre-
Β miss in such a form as will enable us to determine how far
‘rationality may be inferred from responsibility, freedom of

action, a voluntary sacrifice of freedom, and their contraries.
From (6) we have
x
sw’
and developing the second member, but rejecting terms
whose coefficients are 0,

»-- wesw + απ =) + (τ -- 8)20 + χα - z)(1 — τ)

+2(1-2)(1=3)(1- ἡ,

whence, equating to o the terms whose coefficients are

I I
- and —, we have
2 O

y=x2(1 —w)+xw(1 —2)+0(1-x)(1- 2)(Α -- τ); (7)
oo; (8)
[96] a(1 —2)(1-w)=0; (9)

whence by interpretation—

104 THE LAWS OF THOUGHT

DIRECT CONCLUSION. —Rational beings are all responsible
beings who are either free to act, not having voluntarily
sacrificed thetr freedom, or not free to act, having voluntarily
sacrificed their freedom, together with an indefinite remainder
(some, none, or all) of beings not responstble, not free, and
not having voluntarily sacrificed their freedom.

First INDEPENDENT RELATION.—wVo responsible beings
are at the same time free to act, and tn the condition of
having voluntarily sacrificed their freedom.

SECOND.—Wo responsible beings are not free to act, and

3

2

4
᾿
g

at the same time in the condition of not having sacrificed —

their freedom.

The independent relations above determined may, however, {

be put in another and more convenient form. Thus (8) gives

ο ο
xwW = —=02+4-(1—2), on development ;
Ζ ο

or, xw=v(I - 2); (10) ©

and in like manner (9) gives
12) (9)
Ἰ- — -ὐἷὔς Ὁ ΘΙ - σ᾽)"
m1 τ 2) = 5. 9 σε oft -);
or, wl — 2) oe: (11)
and (10) and (11) interpreted give the following Propositions :
ist. Responsible beings who have voluntarily sacrificed
their freedom are not free.

2nd. Responsible beings who have not voluntarily sacrificed ᾿

their freedom are free.

These, however, are merely different forms of the relations
before determined.

16. In examining these results, the reader must bear in
mind, that the sole province of a method of inference or
analysis, is to determine those relations which are necessitated
by the connexion of the terms in the original proposition.
Accordingly, in estimating the completeness with which this
object is effected, we have nothing whatever to do with those
other relations which [97] may be suggested to our minds

by the meaning of the terms employed, as distinct from their ;

OF INTERPRETATION 105

expressed connexion. Thus it seems obvious to remark,
that ‘‘ They who have voluntarily sacrificed their freedom
are not free,” this being a relation implied in the very mean-
ing of the terms. And hence it might appear, that the first
of the two independent relations assigned by the method is
on the one hand needlessly limited, and on the other hand
superfluous. However, if regard be had merely to the con-
nexion of the terms in the original premiss, it will be seen
that the relation in question is not liable to either of these

charges. The solution, as expressed in the direct conclusion

and the independent relations, conjointly, is perfectly com-
plete, without being in any way superfluous.

If we wish to take into account the implicit relation above
referred to, viz., ‘‘ They who have voluntarily sacrificed their
freedom are not free,” we can do so by making this a distinct
proposition, the proper expression of which would be

w=v(I — 2).

This equation we should have to employ together with

‘that expressive of the original premiss. The mode in which

such an examination must be conducted will appear when
we enter upon the theory of systems of propositions in a
future Chapter. The sole difference of result to which the
analysis leads is, that the first of the independent relations
deduced above is superseded.

17. Ex. 4.—Assuming the same definition as in Ex-
ample 2, let it be required to obtain a description of irrational

persons.
We have
PS Vi aie
Ζ- τ΄
Z+tWw—X
~ g++

I
=p xe + oxs(1 — το) +ox(1 —2)w -- ~ «(1 —z)(1 -- 20) Ὁ (τ - x)zw

+(1-x)2(1-w)+(1-x)(1 — a) +=(1 —x)(1 —2)\(1 -- τυ)

| ec ~ x)ew +(1 - x)2(1 - w) + (1 —x)(1 -- ὅ)0 + σα -- x)(1 -- 2) - το)

= (1 ~x)s+ (1 γα - s)w +o(r —a)(1— 2)(1~ 0),

with xzw=0, x(1-3)(1 -w)=o.

106 THE LAWS OF THOUGHT

[98] The independent relations here given are the same 1

‘
4

as we before arrived at, as they evidently ought to be, since ©

whatever relations prevail independently of the existence of ᾿
a given class of objects y, prevail independently also of ἴπθ

existence of the contrary class I —¥y.

The direct solution afforded by the first equation is :— :
Irrational persons consist of all irresponsible beings who are
either free to act, or have voluntarily sacrificed ther liberty, :
and are not free to act ; together with an indefinite remainder
of irresponsible beings who have not sacrificed their liberty, |

and are not free to act.

18. The propositions analyzed in this Chapter have been .
of that species called definitions. I have discussed none of —
which the second or predicate term is particular, and of which —

the general type is Y=vX, Y and X being functions of the ©
logical symbols x, y, 2, &c., and Ὁ an indefinite class symbol. —
The analysis of such Pct is greatly facilitated (thougill
the step is not an essential one) by the elimination of the |

᾿

symbol v, and this process depends upon the method of thell
next Chapter. I postpone also the consideration of another
important problem necessary to complete the theory ot
single propositions, but of which the analysis really depends
upon the method of the reduction of systems of propositions

to be developed in a future page of this work.

[99] CHAPTER VII

OF ELIMINATION

1. IN the examples discussed in the last Chapter, all the ele-
“ments of the original premiss re-appeared in the conclusion,

only in a different order, and with a different connexion.
But it more usually happens in common reasoning, and
especially when we have more than one premiss, that some
of the elements are required not to appear in the conclusion.
Such elements, or, as they are commonly called, ‘‘ middle
terms,” may be considered as introduced into the original
propositions only for the sake of that connexion which they
assist to establish among the other elements, which are alone
designed to enter into the expression of the conclusion.

2. Respecting such intermediate elements, or middle
terms, some erroneous notions prevail. It is a general
opinion, to which, however, the examples contained in the

Jast Chapter furnish a contradiction, that inference consists

peculiarly in the elimination of such terms, and that the ele-
mentary type of this process is exhibited in the elimination
of one middle term from two premises, so as to produce a

single resulting conclusion into which that term does not

enter. Hence it is commonly held, that syllogism is the
basis, or else the common type, of all inference, which may
thus, however complex its form and structure, be resolved

into a series of syllogisms. The propriety of this view will

be considered in a subsequent Chapter. At present I wish
to direct attention to an important, but hitherto unnoticed,
point of difference between the system of Logic, as expressed
by symbols, and that of common algebra, with reference to
the subject of elimination. In the algebraic system we are able

to eliminate one symbol from two equations, two symbols
107

108 THE LAWS OF, THOUGHT

from three equations, and generally 2—1 symbols from x

equations. There thus exists a definite connexion between ~
the number of independent equations given, [100] and the —

παν a es

number of symbols of quantity which it is possible to eliminate ~
from them. But it is otherwise with the system of Logic, ‘
No fixed connexion there prevails between the number of —
equations given representing propositions or premises, and —
the number of typical symbols of which the elimination can —
be effected. From a single equation an indefinite number —
of such symbols may be eliminated. On the other hand, ;

from an indefinite number of equations, a single class sym-

bol only may be eliminated. We may affirm, that in this —
peculiar system, the problem of elimination is resolvable |
under all circumstances alike. This is a consequence of that |
remarkable law of duality to which the symbols of Logic —
are subject. To the equations furnished by the premises ;
given, there is added another equation or system of equations :
drawn from the fundamental laws of thought itself, and |
supplying the necessary means for the solution of the problem ἱ
in question. Of the many consequences which flow from ἢ
the law of duality, this is perhaps the most deserving of —

attention.

3. As in Algebra it often happens, that the elimination ~
of symbols from a given system of equations conducts to ἃ
mere identity in the form O=0, no independent relations |
connecting the symbols which remain; so in the system of ©
Logic, a like result, admitting of a similar interpretation, :
may present itself. Such a circumstance does not detract %
from the generality of the principle before stated. The |
object of the method upon which we are about to enter is —
to eliminate any number of symbols from any number οὗ

logical equations, and to exhibit in the result the actual

relations which remain. Now it may be, that no such—
residual relations exist. In such a case the truth of the_
method is shown by its leading us to a merely identical

proposition.
4. The notation adopted in the following Propositions is

OF ELIMINATION 109

similar to that of the last chapter. By /(%) is meant any
expression involving the logical symbol x, with or without
other logical symbols. By /(1) is meant what /(7) becomes
when x is therein changed into 1; by /(0) what the same
function becomes when + is changed into o.

[101] PROPOSITION I

5. 77 f(x)=0 be any logical equation tnvolving the class
symbol x, with or without other class symbols, then will the
equatton

F(x) fo) =0

be true, independently of the interpretation of x; and tt will
be the complete result of the elimination of x from the above
equation.

_ In other words, the eliminationof x from any given equatton,
f(a) =0, will be effected by successtvely changing tn that equa-
tion x into τ, and x into Ο, and multiplying the two resulting
equations together.

Similarly the complete result of the elimination of any class
symbols, x,y, &c., from any equation of the form V =o, will
be obtained by completely expanding the first member of that
eguation in constituents of the given symbols, and multiplying
together all the coefficients of those constituents, and equating
the product to Ο.

Developing the first member of the equation ((v)=0, we
have (V. το),
A(r)x +flo)(t — x) =0 5

or, { f(t) —f(0) }# +/(2) = 0. (1)
le
Ho) - fa)"
and © ee ree igs
(0) —/(1)

| Substitute these expressions for # and 1—~# in the funda-

mental equation
x(1 —x)=0,

a

110 Toe LAWS Ole THOCGHT

and there results |
Peg OO. |
iC) AG 9] ce
or (1) f(0) -- ο, (2)
the form required. : ,
6. It is seen in this process, that the elimination is really
effected between the given equation /(7)=o and the uni-
versally true equation 4(1—*)=0, expressing the funda-
mental law of logical symbols, χα logical. There exists,
therefore, no need of more [102] than one premiss or equa-
tion, in order to render possible the elimination of a term,
the necessary law of thought virtually supplying the other
premiss or equation. And though the demonstration of this
conclusion may be exhibited in other forms, yet the same
element furnished by the mind itself will still be virtualli™
present. Thus we might proceed as follows:
Multiply (1) by x, and we have

f(t)x= 0, (3)
and let us seek by the forms of ordinary algebra to eliminate
a from this equation and (1).

Now if we have two algebraic equations of the form

ax+b=0,
ax+b'=0;

it is well known that the result of the elimination of x is
ab’ —ab=o, (4)

But comparing the above pair of equations with (1) and (3) |
respectively, we find

a=f(1)-f(0), b=f(0);
co 01) eo
which, substituted in (4), give
S(1)f(2) = 09,
as before, In this form of the demonstration, the funda-

mental equation #(1—+)=0, makes its appearance in the
derivation of (3) from (1).

OF ELIMINATION Tit

7, 1 shall add yet another form of the demonstration,
partaking of a half logical character, and which may set the
demonstration of this important theorem in a clearer light.

We have as before

S(1)x +f(0)(1 -- x) =0.
Multiply this equation first by x, and secondly by 1 -- ,
we get
fir)x=0, f(o)(1-x)=0.

From these we have by solution and development,

[103]

S(2 -Ξ-τα - x), on development,

The direct interpretation of these equations is—

ist. Whatever individuals are included in the class repre-
sented by /(1), are not 7's.

2nd. Whatever individuals are included in the class repre-
sented by /(0), are vs.

Whence by common logic, there are no individuals at once
in the class f(1) and in the class /(O), z.e. there are no indivi-
duals in the class f(1)/(0). Hence,

f(1)f(0) --ο. (5)

Or it would suffice to multiply together the developed
equations, whence the result would immediately follow.

8. The theorem (5) furnishes us with the following Rule:

TO ELIMINATE ANY SYMBOL FROM A PROPOSED EQUATION

RULE.—The terms of the equation having been brought, by
transposition if necessary, to the first side, give to the symbol
successively the values τ and Ο, and multiply the resulting
equations together.

The first part of the Proposition is now proved.
9. Consider in the next place the general equation

72, Y) = 93

12 LHE LAWS OF THOUGHT

the first member of which represents any function of x, y,

and other symbols.
By what has been shown, the result of the elimination of

y from this equation will be
T(x, 1) f(x, 0) =0;

for such is the form to which we are conducted by successively
changing in the given equation y into 1, and y into 0, and
multiplying the results together.

Again, if in the result obtained we change successively xinto
1, and + into o, and multiply the results together, we have

A(t, DS; ολ)ο, 1)/(0, 0) =0 ; (6)

as the final result of elimination.

[104] But the four factors of the first member of this equa-
tion are the four coefficients of the complete expansion of |
7, y), the first member of the original equation ; whence —
the second part of the Proposition is manifest. q

EX SWIPES
10, Ex, 1.—Having given the Proposition, ‘‘ All men are
mortal,” and its symbolical expression, in the equation,
J) = 0X,
in which y represents ‘‘ men,” and x ‘‘ mortals,” it is required
to eliminate the indefinite class symbol v, and to interpret

the result,
Here bringing the terms to the first side, we have

y —-Ux =o,
When v=1 this becomes
)πτ στο;
and when v=o it becomes
Os

and these two equations multiplied together, give

»-τγατεο,
or (I -- x) =0,

it being observed that y?= y,

OF ELIMINATION 113

The above equation is the required result of elimination,
and its interpretation is, Jen who are not mortal do not exist,
—an obvious conclusion.

If from the equation last obtained we seek a description of

beings who are not mortal, we have

Whence, by expansion, ιττ σεται —y), which interpreted

gives, They who are not mortal are not men. This is an
example of [105] what in the common logic is called con-
version by contraposition, or negative conversion. *

Ex. 2.—Taking the Proposition, ‘‘ No men are perfect,”
as represented by the equation

yout — x),

menerein y represents ‘‘ men,” and x ‘‘ perfect beings,” 1t is re-
quired to eliminate v, and find from the result a description

both of perfect beings and of cmperfect beings. We have

y-Ui-—x)=0.

Whence, by the rule of elimination,

στα - ὦ} xy=0,

or Y-YWI - x)= 0,
or Ἤχ 0.

which is interpreted by the Proposition, Perfect men do not
exist. From the above equation we have

OO
aro) ὃν αν αὶ t;
x ᾿ ὦ y) by developmen
whence, by interpretation, Mo perfect beings are men.
Similarly,
Ὁ, ἢ ο
I-x=1-—--=-+=j)+- ey ἢ ἢ
ΠΣ ay)

* Whately’s ZLogzc, Book II. Chap, ii. sec. 4.

114 THE LAWS. CF THOUGHT

which, on interpretation, gives, /iperfect beings are all men

with an indefinite remainder of beings, which are not men.

11. It will generally be the most convenient course, in the ©
treatment of propositions, to eliminate first the indefinite class _

symbol v, wherever it occurs in the corresponding equations,

This will only modify their form, without impairing their
significance. Let us apply this process to one of the examples }
of Chap. IV. For the Proposition, ‘‘ No men are placed ἸΗ͂Ν
exalted stations and free from envious regards,” we found

the expression
y=U(1 — x2),

and for the equivalent Proposition, ‘‘ Men in exalted stations |

are not free from envious regards,” the expression

yx =0(1 = 2) 5
[106] and it was observed that these equations, v being an
indefinite class symbol, were themselves equivalent. To

prove this, it is only necessary to eliminate from each the

symbol wv. The first equation is

y—v(i -- x2)=0,

whence, first making v=1, and then v=o, and multiplying -

the results, we have

Oe I + x2) =0,
or XS =O.

Now the second of the given equations becomes on trans-

position

yx -Ui-2)=0;
whence (yx -14+2)yx=0,
or YXZ=0,

as before. The reader will easily interpret the result.

12, Ex, 3.—As a subject for the general method of this
chapter, we will resume Mr. Senior’s definition of wealth,

viz. : ‘Wealth consists of things transferable, limited in
supply, and either productive of pleasure or preventive of
pain.” We shall consider this definition, agreeably to a
former remark, as including all things which possess at once

a

OF ELIMINATION 11s

both the qualities expressed in the last part of the definition,
upon which assumption we have, as our representative

equation,
w=st{ pr+p(1—7r)+r(1 -- 2)},
or w=st{p+r(1 —p)},
wherein
w stands for wealth.
ὃ τ things limited in supply.
t - things transferable.
2 a things productive of pleasure.
A a things preventive of pain.

From the above equation we can eliminate any symbols
that we do not desire to take into account, and express the
result by solution and development, according to any pro-
posed arrangement of subject and predicate.

Let us first consider what the expression for w, wealth,
would [107] be if the element 7, referring to prevention
of pain, were eliminated. Now bringing the terms of the
equation to the first side, we get

w—st( pt+r—rp)=o.

Making r=1, the first member becomes w — 57, and making
y=0 it becomes w—stp ; whence we have by the Rule,

(w — st)(w -- sip) =o, (7)
or w—wstp —wst+ sip=o; (8)
whence stp

w=;
st+ stp -τ
the development of the second member of which equation
gives
w = stp Ὁ 5 st(t -- 2). (9)

Whence we have the conclusion,— Wealth consists of all
things limited in supply, transferable, and productive of
pleasure, and an indefinite remainder of things limited in
supply, transferable, and not productive of pleasure. ‘This
is sufficiently obvious.
Let it be remarked that it is not necessary to perform

116 RAE LAWS OF THOUGHT

the multiplication indicated in (7), and reduce that equation
to the form (8), in order to determine the expression of w
in terms of the other symbols. The process of development
may in all cases be made to supersede that of multiplication.
Thus if we develop (7) in terms of w, we find

(1 -- «δ — stp)w + stp(1 -- το) Ξε ο,

whence stp

= stp — (1 - sia — sip)’

and this equation developed will give, as before,

w= stp +— st - 2).

13. Suppose next that we seek a description of things

limited in supply, as dependent upon their relation to ἢ

wealth, transferableness, and tendency to pro duce pleasure,
omitting all reference to the prevention of pain.

[108] From equation (8), which is the result of the ἢ

elimination of ~ from the original equation, we have

w — «(τοί + wip — tp) = ο.
whence a τυ
wet + wip — ip

= wip + wit - 2). w(t 1)p+~ (1-11 - 2)

+ o(1 — w)tp + = (1 — w)t(1 ~pA)+=(1 —w)(1—t)p

Ἐς τα - δῷ -9).

We will first give the direct interpretation of the above

solution, term by term; afterwards we shall offer some ἢ

general remarks which it suggests; and, finally, show how

the expression of the conclusion may be somewhat ab- |

breviated.

First, then, the direct interpretation is, Things limited

in supply consist of AW wealth transferable and productive

of pleasure—all wealth transferable, and not productive of —

pleasure,—an indefinite amount of what ts not wealth, but
as etther transferable, and not productive of pleasure, or in-

OF ELIMINATION 117

transferable and productive of pleasure, or neither trans-
ferable nor productive of pleasure.

I '
To which the terms whose coefficients are ε: permit us

to add the following independent relations, viz. :

Ist. Wealth that ts intransferable, and productive of
pleasure, does not exist.

2ndly. Wealth that ts intransferable, and not productive
of pleasure, does not extst.

14. Respecting this solution I suppose the following re-
marks are likely to be made.

First, it may be said, that in the expression above ob-
tained for ‘‘ things limited in supply,” the term ‘‘ All wealth
transferable,” &c., is in part redundant ; since all wealth is
(as implied in the original proposition, and directly asserted
in the zndependent relations) necessarily transferable.

I answer, that although in ordinary speech we should
not [109] deem it necessary to add to ‘‘ wealth” the epithet
‘‘transferable,” if another part of our reasoning had led us

to express the conclusion, that there is no wealth which is

not transferable, yet it pertains to the perfection of this
method that it in all cases fully defines the objects represented
by each term of the conclusion, by stating the relation they
bear to each quality or element of distinction that we have
chosen to employ. This is necessary in order to keep the
different parts of the solution really distinct and independent,
and actually prevents redundancy. Suppose that the pair
of terms we have been considering had not contained the
word ‘‘ transferable,” and had unitedly been ‘‘ All wealth,”
we could then logically resolve the single term ‘‘ All wealth”

into the two terms ‘‘ All wealth transferable,” and ‘‘ All

wealth intransferable.” But the latter term is shown to
disappear by the ‘‘independent relations.” Hence it forms
no part of the description required, and is therefore re-
dundant. The remaining term agrees with the conclusion
actually obtained.

Solutions in which there cannot, by logical divisions,

118 iii TANS (OL) 100 Gain

be produced any superfluous or redundant terms, may be
termed pure solutions. Such are all the solutions obtained
by the method of development and elimination above ex-
plained. Itis proper to notice, that if the common algebraic
method of elimination were adopted in the cases in which
that method is possible in the present system, we should
not be able to depend upon the purity of the solutions ob-
tained. Its want of generality would not be its only
defect.

15. In the second; place, it will be remarked, that the
conclusion contains two terms, the aggregate significance
of which would be more conveniently expressed by a single
term. Instead of ‘* All wealth productive of pleasure, and
transferable,” and ‘‘ All wealth not productive of pleasure,
and transferable,” we might simply say, ‘‘ All wealth trans-
ferable.” This remark is quite just. But it must be
noticed that whenever any such simplifications are possible,
they are immediately suggested by the form of the equa-
tion we have to interpret ; and if that equation be reduced
to its simplest form, then the interpretation to which it

conducts will be in its simplest form also. Thus in the

original solution the terms wp and w¢(1—p), which have
unity for their [110] coefficient, give, on addition, wz; the

terms w(1—Z)p and w(1—z)(1—f), which have Ξ for their

coefficient give w(1—¢); and the terms (1 —w)1i—t)p and —

- O : ; ,

(I —w)(1—Z)(1—p), which have 5 for their coefficient, give
(I—w)(1—2). Whence the complete solution is

ο ο

Στ 7 — w)(1 eG - w)t(1—/P),

with the independent relation,

w(1—t)=0, or oe

re)

The interpretation would now stand thus :—
Ist. Things limited in supply consist of all wealth trans-
Serable, wtth an endefinite remainder of what ts not wealth

OF ELIMINATION 119

and not transferable, and of transferable articles which are

not wealth, and are not productive of pleasure.

2nd. All wealth ts transferable.

This is the simplest form under which the general con-
clusion, with its attendant condition, can be put.

16. When it is required to eliminate two or more symbols

from a proposed equation we can either employ (6) Prop.

I., or eliminate them in succession, the order of the process
being indifferent. From the equation

w=st(p+r-—fpr),
we have eliminated 7, and found the result,
τὺ —wst — wstp + sip =o.

Suppose that it had been required to eliminate both r and ¢,
then taking the above as the first step of the process, it
remains to eliminate from the last equation 7. Now when
¢t=1 the first member of the equation becomes

WwW — Ws — wsp + 5,

and when ¢=o the same member becomes w. Whence

“we have

zy(w — ws — wsp + 52) =0,
or W — WS =O,

for the required result of elimination.
[1117 If from the last result we determine w, we have

= ἘΣ πος
τσ Ὁ

whence “44 wealth is limited in supply.” As p does not
enter into the equation, it is evident that the above is
true, irrespectively of any relation which the elements of the
conclusion bear to the quality ‘‘ productive of pleasure.”

Resuming the original equation, let it be required to
eliminate s and ΖΦ We have

τὺ -- εί(2 τ 7 -- 27).

Instead, however, of separately eliminating s and ¢ accord-
ing to the Rule, it will suffice to treat s¢ as a single symbol,

oe DTH LAWS OF THOUGHT

seeing that it satisfies the fundamental law of the symbols ;

by the equation
st(1 — st) =0.

Placing, therefore, the given equation under the form
w—st(pt+r—-pr)=o;

and making s¢ successively equal to 1 and to 0, and taking ἡ
the product of the results, we have

(w—-p—r+pr)w=o,
or w—wp—wr+wpr=o,

for the result sought. .
Asa particular illustration, let it be required to deduce Ὁ

an expression for ‘‘things productive of pleasure” (2), in

terms of ‘‘wealth” (w), and ‘‘things preventive of pain” (7). ἢ

We have, on solving the equation,
Ἢ τα τς 7)
“φύ(ι -- 2)

7

Θ ro) Ὁ ᾿
—_wrt+wi-7 I-w ἘΠῚ ιν 7
wr twa =r) 4 2(1 - wr 42 (1 ται -ν)
= w(t ~r)4 creo (1 —w).
ο ο

Whence the following conclusion :—7Things productive of

plea-[112|sure are, all wealth not preventive of pain, an in-

definite amount of wealth that ἐς preventive of pain, and an in- ᾿

definite amount of what ts not wealth.
From the same equation we get

Wi 7) 9

w(t-r) wii—7r)’

I-p=1-—
which developed, gives

zw(1 -p) == wr +2 (1 —w) σε σα —w). (1-7) =Swr+2(1—-w),

Ee ee ee eee MARI ον

Sa
OTN ὙΠ ΗΠ ας

Whence, Things not productive of pleasure are either wealth,
preventive of pain, or what ts not wealth.
Equally easy would be the discussion of any similar case. —
17. Inthe last example of elimination, we have eliminated —
the compound symbol s¢ from the given equation, by treat-

OF ELIMINATION 121

ing it as asingle symbol. The same method is applicable
to any combination of symbols which satisfies the funda-
mental law of individual symbols. Thus the expression
pt+r—~pr will, on being multiplied by itself, reproduce itself,
so that if we represent p+r—pr by a single symbol as y,
we shall have the fundamental law obeyed, the equation

y=y", or γ( - ))Ξ ο,

being satisfied. For the rule of elimination for symbols is
founded upon the supposition that each individual symbol
is subject to that law; and hence the elimination of any
function or combination of such symbols from an equation,
may be effected by a single operation, whenever that law
is satisfied by the function.

Though the forms of interpretation adopted in this and
the previous chapter show, perhaps better than any others,

the direct significance of the symbols 1 and =, modes of

expression more agreeable to those of common discourse
may, with equal truth and propriety, be employed. Thus
the equation (9) may be interpreted in the following manner :
Wealth ts etther limited in supply, transferable, and pro-
ductive of pleasure, or limited in sup-[113|py, transferable,
and not productive of pleasure. And reversely, Whatever
as limited tn supply, transferable, and productive of pleasure,
is wealth, Reverse interpretations, similar to the above,
are always furnished when the final development introduces
terms having unity as a coefficient.

18. NOTE.—The fundamental equation /(1)/(0)=0, ex-
pressing the result of the elimination of the symbol + from
any equation 2 =o, admits of a remarkable interpretation.

It isto be remembered, that by the equation /(7)=0 is
implied some proposition in which the individuals repre-
‘sented by the class x, suppose ‘‘men,” are referred to,
‘together, it may be, with other individuals ; and it is our
object to ascertain whether there is implied in the pro-
position any relation among the other individuals, inde-

122 THE LAWS OF FiOlUGH i

pendently of those found in the class men. Now the equa-

tion /(1)=o expresses what the original proposition would
become if men made up the universe, and the equation

#(0)=0 expresses what that original proposition would ἡ

become if men ceased to exist, wherefore the equation —
(1) f(0) =0 expresses what in virtue of the original proposi- —

tion would be equally true on either assumption, z.e. equally ὶ

)

ite -whether “men ’ were “<‘all things” or. “nothing,

> ὁ

Wherefore the theorem expresses that what zs equally true, |
whether a given class of objects embraces the whole universe

or disappears from existence, ts independent of that class αἰ-
together, and vice versa. Herein we see another example -

of the interpretation of formal results, immediately deduced
from the mathematical laws of thought, into general axioms
of philosophy.

ty
ἘΠ
:
¢

[114] CHAPTER VIII

OF THE REDUCTION OF SYSTEMS OF PROPOSITIONS

1. IN the preceding Chapters we have determined sufficiently

for the most essential purposes the theory of single primary
propositions, or, to speak more accurately, of primary pro-
positions expressed by a single equation. And we have
established upon that theory an adequate method. We have
shown how any element involved in the given system of
equations may be eliminated, and the relation which con-
nects the remaining elements deduced in any proposed form,
whether of denial, of affirmation, or of the more usual relation
of subject and predicate. It remains that we proceed to the
consideration of systems of propositions, and institute with
respect to them a similar series of investigations. We are
to inquire whether it is possible from the equations by which
a system of propositions is expressed to eliminate, ad /zbztum,

“any number of the symbols involved; to deduce by inter-

pretation. of the result the whole of the relations implied
among the remaining symbols ; and to determine in particular
the expression of any single element, or of any interpretable
combination of elements, in terms of the other elements, so
as to present the conclusion in any admissible form that may
be required. These questions will be answered by showing
that it is possible to reduce any system of equations, or any

_of the equations involved in a system, to an equivalent single

equation, to which the methods of the previous chapters may
be immediately applied. It will be seen also, that in this
reduction is involved an important extension of the theory
of single propositions, which in the previous discussion of
the subject we were compelled to forego. This circumstance

is not peculiar in its nature. There are many special depart-
{23

124 CAE LAWS CF ,HOCGHT

ments of science which cannot be completely surveyed from

within, but require to be studied also from an external point —
of view, and to be regarded in connexion with [115] other

ina aenealial

pont

and kindred subjects, in order that their full proportions |

may be understood.
This Chapter will exhibit two distinct modes of reducing

Sante τ

systems of equations to equivalent single equations. The
first of these rests upon the employment of arbitrary constant —
multipliers. It is a method sufficiently simple in theory, but —
it has the inconvenience of rendering the subsequent pro-_
cesses of elimination and development, when they occur, —
somewhat tedious. It was, however, the method of reduction —
first discovered, and partly on this account, and partly on |
account of its simplicity, it has been thought proper to retain —

it. The second method does not require the introduction of .
arbitrary constants, and is in nearly all respects preferable —
to the preceding one. It will, therefore, generally be adopted ©

in the subsequent investigations of this work.
2. We proceed to the consideration of the first method.

PROPOSITION [

Se ee gO Os ee τ ον τ ee ed

Any system of logical equations may be reduced to a single :
equivalent equation, by multiplying each equation after the —

jirst by a distinct arbitrary constant quantity, and adding all”

the results, including the first equation, together.

By Prop. II., Chap. VI., the interpretation of any single —
equation, /(z, vy. .)=0O is obtained by equating to o those Ἷ

constituents of the development of the first member, whose —

coefficients do not vanish. And hence, if there be given two —
equations, χα, y..)=o0, and A(x, y..)=o, their united 7
import will be contained in the system of results formed by —

equating to Ο all those constituents which thus present them- —

selves in both, or in either, of the given equations developed

according to the Rule of Chap. VI. Thus let it be supposed, ~

that we have the two equations

xY —-2X=0, (1) ;
x-Y=0; (2)%

OF REDUCTION 125

The development of the first gives

—xy-2x(1-y)=0;
whence, xy=0, χίι -y)=0. (3)

[116] The development of the second equation gives

x(1 —y)-— (I ὦ)
whence, x(I-y)=0, y(1-x)=0. (4)
The constituents whose coefficients do not vanish in both de-

velopments are xy, x(1—y), and (I1—~*)y, and these would
together give the system

Dao; αἰ) ὁ; (Ue xy eo, (5)
which is equivalent to the two systems given by the develop-
ments separately, seeing that in those systems the equation
ΠῚ -- γ)ΞΞο is repeated. Confining ourselves to the case of
binary systems of equations, it remains then to determine a
single equation, which on development shall yield the same
constituents with coefficients which do not vanish, as the
given equations produce.

Now if we represent by
: V,=0, V,=0,

the given equations, Κ΄, and V, being functions of the logical
symbols x, y, 2, &c. ; then the single equation

V,+cV,=0, (6)
¢ being an arbitrary constant quantity, will accomplish the
‘required object. For let Af represent any term in the full
development V, wherein Ζ is a constituent and 4 its numerical
coefficient, and let 282 represent the corresponding term in
the full development of V,, then will the corresponding term
in the development of (6) be
| (A+cB)t.
‘The coefficient of ¢ vanishes if A and B both vanish, but not
otherwise. For if we assume that A and B do not both
vanish, and at the same time make
, A+cB=0, 7)
the following cases alone can present themselves.

126 PHE LAWS OF PhO Gia

ist. That A vanishes and B does not vanish. In this case
the above equation becomes
2.9. -Ὁ.
[117] and requires thatc=o. But this contradicts the hypo-
thesis that c is an arbztrary constant.
and. That B vanishes and A does not vanish. This assump-
tion reduces (7) to
AO.
by which the assumption is itself violated.
3rd. That neither A nor B vanishes. The equation (7)

then gives
—A

B 2

which is a definite value, and, therefore, conflicts with the

a

hypothesis that ¢ is arbitrary.

Hence the coefficient 4+ cB vanishes when A and & both
vanish, but not otherwise. Therefore, the same constituents
will appear in the development of (6), with coefficients which
do not vanish, as in the equations V,=0, V,=0, singly or
together. And the equation V,+cV,=0, will be equivalent
to the system V,=0, V,=0.

By similar reasoning it appears, that the general system
of equations

Kio πὸ a0, we.
may be replaced by the single equation
Vi +¢cV,+¢V,+ &c. =0,
c, ¢, &c., being arbitrary constants. The equation thus
formed may be treated in all respects as the ordinary logical
equations of the previous chapters. The arbitrary constants
C1) Cg, &c., are not logical symbols. They do not satisfy
the law,
GA iC) 0; (1-6) — Ὁ.

But their introduction is justified by that general principle
which has been stated in (11. 15) and (V. 6), and exemplified
in nearly all our subsequent investigations, viz., that equa-
tions involving the symbols of Logic may be treated in all

OF REDUCTION 127

respects as if those symbols were symbols of quantity, subject
to the special law 2(1—2)=0, until in the final stage of
solution they assume a form interpretable in that system of
thought with which Logic is conversant.

[118] 3. The following example will serve to illustrate the
above method.

Ex. 1.—Suppose that an analysis of the properties of a
particular class of substances has led to the following general
conclusions, viz. :

ist. That wherever the properties A and & are combined,

“either the property C, or the property D, is present also ; but
they are not jointly present.

2nd. That wherever the properties B and C are combined,
the properties A and D are cither both present with them,
or both absent.

3rd. That wherever the properties A and ὦ are both
absent, the properties C and D are both absent also; and
vice versd, where the properties C and D are both absent,
A and B# are both absent also.

Let it then be required from the above to determine what
may be concluded in any particular instance from the presence
of the property A with respect to the presence or absence of

the properties B and C, paying no regard to the property D.

Represent the property 4 by 7;
ne the property B by y ;
᾿ the property C by 2;
| τ the property D by w.
_ Then the symbolical expression of the premises will be
| xy =v{w(t -- 5) -Ἐ 2(1 -- 20}} ;
γε-ε υχτυ Ὁ (τ -- α)(1 --- 29}} ;
ἀντ τ τε 2)(1 - ®).
From the first two of these equations, separately eliminating
| the indefinite class symbol v, we have

xy{1 -- τυ(τ -- 2) -- ζ(1 -- 20}} =0;
y2{t -- χτῦ -- (- -- α)( -- τ9}} -- ο.

Now if we observe that by development

128 PAE LAWs Of THOUGHT,

1 --w(1 -2)—2(1 -- 10) Ξε τῦς -Ἑ (τ -- w)(1 -- 2),
and 1—xw—(1—x)(1 -- w)=x(1 -- 29) τσ -- α);

[119] and in these expressions replace, for simplicity,

1-xbyx, 1-ybyy, &c,

we shall have from the three last equations,

xy(w2z + WZ) =0; (x)
y2(xt + XW) = ; (2)}
ny 02: (3)a

and from this system we must eliminate zw.
Multiplying the second of the above equations by ς, and '
the third by c, and adding the results to the first, we have

xy(we + 2) + ἐντ(ατο + Xw) + ¢(LY — WZ) -- ο.
When w is made equal to 1, and therefore @ to o, the first —
member of the above equation becomes

Lys + XYZ HORS. |
And when in the same member w is made 0 and τὺ =1, it.
becomes

xyz +exys + xy — ἐξ.
Hence the result of the elimination of w may be expressed
in the form ᾿
(xyz + cays + ἐχ))(αγξ + crys + ἐὰν -- CZ) =0; (4)
and from this equation x is to be determined. |
Were we now to proceed as in former instances, we .
should multiply together the factors in the first member οὗ.
the above equation ; but it may be well to show that such '
a course is not at all necessary. Let us develop the first
member of (4) with reference to x, the symbol whose ex-_
pression is sought, we find ’
yal p2 toys — e2)w + (cyz + ἐγ) - ἐγ -- x) =O;
or cyax + (cyz + y)(c'y — ἐξ)γᾳ — x) =0;
whence we find,
lost em(ep— 23)
(cya + ἐγ)(εν -- 5) — cys’

and developing the second member with respect to y and 2,

OF REDUCTION 129

ὦ»
Po! w= OY +O y8 + IE + OIE;
Ο A Ο
τὰ στοᾷ -)6 —y(1- 2) 4=(1 -)α - 2);
or,

= (1 —y)B +2 (1-5);

the interpretation of which is, Wherever the property A ἐς
present, there etther C ts present and B absent, or C ts absent.
And inversely, Wherever the property C ts present, and the
property B absent, there the property A ts present.

These results may be much more readily obtained by the
method next to be explained. It is, however, satisfactory
to possess different modes, serving for mutual verification,
of arriving at the same conclusion.

4. We proceed to the second method.

PROPOSITION II

Tf any equations, V,=0, V,=0, &c., are such that the
developments of their first members consist only of constituents
with positive coefficients, those equations may be combined
together into a single equivalent equation by addition.

For, as before, let A¢ represent any term in the develop-
ment of the function V,, δὲ the corresponding term in the
development of Y,, and soon. Then will the correspond-
ing term in the development of the equation

V,+ V+ &c.=0, (1)
formed by the addition of the several given equations, be
(4+ B+&c.)é.

But as by hypothesis the coefficients A, 8, &c. are none of
them negative, the aggregate coefficient A+B, &c. in the
derived equation will only vanish when the separate co-
efficients A, B, ἄς. vanish together. Hence the same con-
stituents will appear in the development of the equation (1)
as in the several equations V,=0, V,=0, &c. of the original
system taken collectively, and therefore the interpretation
of the equation (1) will be equiva-[121]lent to the collective
ϑ

130 RHE LAWS OF THOUGH?

interpretations of the several equations from which it is
derived.

Proposition III

s. If V,=0, V,=0, Se. represent any system of equations,

the terms of which have by transposition been brought to the
first side, then the combined interpretation of the system will
be involved in the single equation,

Vit Vit Se. =0,
formed by adding together the squares of the given equations.

For let any equation of the system, as V,=0, produce
on development an equation
at, + at, + &C. Ξε ο,

in which ¢,, Z, &c. are constituents, and a, a, &c. their
corresponding coefficients. Then the equation V,?=o will
produce on development an equation

a,*t, + a,7t, + &c. = 9,
as may be proved either from the law of the development
or by squaring the function a,¢,+a@,¢,, &c. in subjection to

the conditions
teat, t2=ty tl=0,

assigned in Prop. III., Chap. V. Hence the constituents
which appear in the expansion of the equation V,?=o0, are
the same as those which appear in the expansion of the
equation /7,=0, and they have positive coefficients. And
the same remark applies to the equations V,=0, &q¢&
Whence, by the last Proposition, the equation

Vi2+ V2 + &c.=0
will be equivalent in interpretation to the system of equations
V,=0, V,=0, ὅσ,

Corollary.—Any equation, VY =o, of which the first mem-
ber already satisfies the condition

i= Vor Vir VY) —o,

[122] does not need (as it would remain unaffected by) the

]

OF REDUCTION 131

process of squaring. Such equations are, indeed, immedi-

ately developable into a series of constituents, with co-

efficients equal to 1, Chap. V. Prop. IV.

PROPOSITION IV

6. Whenever the equations of a system have by the above
process of squaring, or by any other process, been reduced to
a form such that all the constituents exhibited in thetr de-
velopment have positive coeffictents, any derived equations
obtained by elimination will possess the same character, and
may be combined with the other equations by addition.

Suppose that we have to eliminate a symbol + from any
equation / =o, which is such that none of the constituents,
in the full development of its first member, have negative
coefficients. That expansion may be written in the form

Vix +V,(1 - x) =09,
V, and V, being each of the form
Ab, + Agly . ΤΟΤΕ

in which ¢,, ¢,, . . ¢, are constituents of the other symbols,
and ay, a, - . a, in each case positive or vanishing quantities.
The result of elimination is

ΚΤ ΞΟ:

and as the coefficients in VY, and V, are none of them
negative, there can be no negative coefficients in the product
VV. Hence the equation V,V,=0 may be added to any
other equation, the coefficients of whose constituents are
positive, and the resulting equation will combine the full
significance of those from which it was obtained.

PROPOSITION V

7. To deduce from the previous Proposttions a practical
rule or method for the reduction of systems of equations ex-
pressing propositions in Logtc.

We have by the previous investigations established the
following points, viz. :

132 THE LAWS OF THOUGAT

“a

[123] 1st. That any equations which are of the form q
V =o, V satisfying the fundamental law of duality Κα -- Κ)
--ο, may be combined together by simple addition. ;

andly. That any other equations of the form V =o may be :
reduced, by the process of squaring, to a form in which the ;
same principle of combination by mere addition is applicable. —

It remains then only to determine what equations in the :
actual expression of propositions belong to the former, and ᾿
what to the latter, class.

Now the general types of propositions have been set forth
in the conclusion of Chap. IV. The division of propositions q
which they represent is as follows : .

Ist. Propositions, of which the subject is universal, and ,
the predicate particular.

The symbolical type (IV. 15) is

IG 51)

X and Y satisfying the law of duality. Eliminating v, we j

have ‘
X(1 -- Y)=0, (1) q

and this will be found also to satisfy the same law. No ;
further reduction by the process of squaring is needed. ‘

2nd. Propositions of which both terms are universal, and ;
of which the symbolical type is :

2G)

X and Y separately satisfying the law of duality. Writing :

the equation in the form XY — Y=o, and squaring, we have
2X Vi Vo:

or X(1- Y)+ Κα -- Χ)γτο. (2)

The first member of this equation satisfies the law of duality, ὦ

as is evident from its very form.

We may arrive at the same equation in a different manner. —

The equation
1G %

is equivalent to the two equations

A=DY, Vox,

OF REDUCTION 138

; [124] (for to affirm that X’s are identical with Y’s is to
_ affirm both that All X’s are Y’s, and that All Y’s are X’s),
Now these equations give, on elimination of v,

Χίι -- Y)=0, YV(r-—X)=0,
which added, produce (2).

3rd. Propositions of which both terms are particular. The
form of such propositions is

7 ΞΖ

but v is not quite arbitrary, and therefore must not be
eliminated. For v is the representative of some, which,
though it may include in its meaning αὐ, does not include
none. We must therefore transpose the second member to
the first side, and square the resulting equation according to
the rule.
The result will obviously be

vX(1- Y)+0V1-X)=0.,

The above conclusions it may be convenient to embody in
a Rule, which will serve for constant future direction.

ὃ. RULE.—The equations being so expressed as that the
terms X and Υ in the following typical forms obey the law
of duality, change the equations

X=vVinto X(1 - Y)=0,
X= VYinto X(1- VY)+ V(1-X)=0.
vX =vY into vX(1 -- Y)+uV(1-X)=0.

Any equation whitch ts given in the form X =o will not need
transformation, and any equation which presents itself in the

form X =1 may be replaced by 1 —X =0, as appears from the
second of the above transformations.

When the equations of the system have thus been reduced,

_ any of them, as well as any equations derived from them by
the process of elimination, may be combined by addition.

9. Norr.—It has been seen in Chapter IV. that in literally

translating the terms of a proposition, without attending to

its real meaning, into the language of symbols, we may pro-

134 THE LAWS OF THOUGHT

duce equations in which the terms XY and Y do not obey
the law of duality. The equation w=st(p+7), given in (3)
Prop. III. of [125] the Chapter referred to, is of this kind.
Such equations, however, as it has been seen, have a mean-
ing. Should it, for curiosity, or for any other motive, be
determined to employ them, it will be best to reduce them
by the Rule (VI. 5). |
10. Ex. 2.—Let us take the following Propositions of E
Elementary Geometry : :
ist. Similar figures consist of all whose corresponding ~
angles are equal, and whose corresponding sides are pro- 3
portional. 7
2nd. Triangles whose corresponding angles are equal have :
their corresponding sides proportional, and vzce versd. .
To represent these premises, let us make

ὅτε oiinilar.

¢= triangles.

g =having corresponding angles equal.
y=having corresponding sides proportional.

Then the premises are expressed by the following equations : 4

S= 272, (1) 5
tq = tr. (2) Γ
Reducing by the Rule, or, which amounts to the same thing, ;
bringing the terms of these equations to the first side, squar- —
ing each equation, and then adding, we have

S+qr—29grs+igt+tr—2tgr=o. (3) 4

Let it be required to deduce a description of dissimilar figures q
formed out of the elements expressed by the terms, ér7- ~
angles, having corresponding angles equal, having correspond- ~
ing sides proportional.
We have from (3),
ee Ue ian, 2tgr

297 --ἶ '

tie Was Ch ASI Ξ
297 --τ

bas

(4)

OF REDUCTION 135

And fully developing the second member, we find

I—s=olgr Ὁ 2tg(1 -- 7) 2¢r(1 —g) +41 -- φ)ᾷ -- 2) Ἐ ο(ι - Agr
+(1 = λίτ )} (1 -A)r(1—g) + (1 - 21-947). (5)

[126] In the above development two of the terms have the
coefficient 2, these must be equated to o by the Rule, then
those terms whose coefficients are o being rejected, we have

1-s=A1-g)(1—7)+(1-Ag(1 -7) + (1 -διίε -9)
(7 ΗΠ ΠΥ ~~ (6)
tg(1—7)=0; (7)
tr(1—g)=0; (8)

the direct interpretation of which is

Ist. Dessimilar figures consist of all triangles which have
not thetr corresponding angles equal and sides proportional,
and of all figures not being triangles which have etther thetr
angles equal, and sides not proportional, or thetr corresponding
sides proportional, and angles not equal, or netther thetr corre-
sponding angles equal nor corresponding sides proportional.

2nd, There are no triangles whose corresponding angles
are equal, and sides not proportional.

3rd. There are no triangles whose corresponding stdes are
proportional and angles not equal.

11. Such are the immediate interpretations of the final
equation. It is seen, in accordance with the general theory,
that in deducing a description of a particular class of objects,
viz., dissimilar figures, in terms of certain other elements of
the original premises, we obtain also the independent rela-
tions which exist among those elements in virtue of the same
premises. And that this is not superfluous information, even
as respects the immediate object of inquiry, may easily be
shown. For example, the independent relations may always
be made use of to reduce, if it be thought desirable, to a
briefer form, the expression of that relation which is directly
sought. Thus if we write (7) in the form

o=“g(1 -- 7),

136 THE LAWS OF THOUGHT

and add it to (6), we get, since
H1-g)(i—-7)+M(i—r)=H(1 —7),

1—s=H1—7)+(1- Ag —7) + (1 — Ar (1-9) + (1-1 - g)(1—-1),
[127] which, on interpretation, would give for the first term
of the description of dissimilar figures, ‘‘ Triangles whose
corresponding sides are not proportional,” instead of the
fuller description originally obtained. A regard to con-
venience must always determine the propriety of such
reduction.

12, A reduction which is always advantageous (VII. 15)
consists in collecting the terms of the immediate description
sought, as of the second member of (5) or (6), into as few
groups as possible. Thus the third and fourth terms of the
second member of (6) produce by addition the single term
( -- δά --φ). If this reduction be combined with the last,
we have

1-s=1-v)+(1-Ag(i-7)+(1 -- δὲ - 9),
the interpretation of which is

Dissimtlar figures consist of all triangles whose corresponda-
ing sides are not proportional, and all figures not being trt-
angles which have etther their corresponding angles unequal,
or thetr corresponding angles equal, but sides not proportional.

The fulness of the general solution is therefore not a super-
fluity. While it gives us all the information that we seek, it
provides us also with the means of expressing that informa-
tion in the mode that is most advantageous.

13. Another observation, illustrative of a principle which
has already been stated, remains to be made. Two of the
terms in the full development of 1—s in (5) have 2 for their

I she
coefficients, instead of a It will hereafter be shown that

this circumstance indicates that the two premises were not
independent. To verify this, let us resume the equations of
the premises in their reduced forms, viz.,

s(1 — gr) +97(1 - 5) =0,
ig(1 -r)+tr(1-¢9)=0.

eee

7 ἘΠῚ

OF REDUCTION 137

Now if the first members of these equations have any
common constituents, they will appear on multiplying the
equations together. If we do this we obtain

stg(1 -- 7) + str(1 -- φ) =0.
[128] Whence there will result
stg(1 -r)=0, str(1 -- 7) =9,
these being equations which are deducible from either of the
primitive ones. Their interpretations are—

Similar triangles which have thetr corresponding angles
equal have their corresponding sides proportional.

Similar triangles which have thetr corresponding sides
proportional have their corresponding angles equal.

_ And these conclusions are equally deducible from either
premiss sézgly. In this respect, according to the definitions
laid down, the premises are not independent.

14. Let us, in conclusion, resume the problem discussed
in illustration of the first method of this Chapter, and
endeavour to ascertain, by the present method, what may
be concluded from the presence of the property C, with
reference to the properties A and Ρ.

We found on eliminating the symbols v the following
equations, viz. :

xy(ws + WZ) =o, (1)
y2(xi + xw) =o, (2)
ap = wi (3)

From these we are to eliminate w and determine z. Now
(1) and (2) already satisfy the condition V(1— V)=0. The
third equation gives, on. bringing the terms to the first side,
and squaring

ἀγα -- WZ) + WH -- KV) =0. (4)

Adding (1) (2) and (4) together, we have
ay(we + WZ) -- γε(ατ + Xw) + ἄγ — WZ) + WAI ~ XY) =O.
Eliminating w, we get

(xyz +y2e + £9) { cys + you + ἄγε + 2(1 -- Χ))} =0.

138 LHe bPAWS. Of THOOGH YT
Now, on multiplying the terms in the second factor by those
in the first successively, observing that
XX=0, YY=O, 2ξ-Ξ-ο,
[129] nearly all disappear, and we have only left
XYZ + XYZ=O; (5)

whence oa
KY + XY

One ΘΟ. ae
= OXY + — XY 4 — KY + OXY
©) [9]
oat eat χα
τ τῶν τ 4),
Ο [9]

furnishing the interpretation. Wherever the property C ts
found, either the property A or the property B will be found
with it, but not both of them together.

From the equation (5) we may readily deduce the result ὦ
arrived at in the previous investigation by the method of
arbitrary constant multipliers, as well as any other proposed
forms of the relation between x, y, and z; e.g. [f the property
B ts absent, either A and C will be jotntly present, or C will
be absent. And conversely, [f A and C are jointly present,
B will be absent. The converse part of this conclusion is
founded on the presence of a term xz with unity for its co-
efficient in the developed value of 7.

[130] ΠΕ ΕΝ ΙΧ
OF CERTAIN METHODS OF ABBREVIATION

1. THOUGH the three fundamental methods of development,
elimination, and reduction, established and illustrated in the
previous Chapters, are sufficient for all the practical ends of
Logic, yet there are certain cases in which they admit, and
especially the method of elimination, of being simplified in an
important degree ; and to these 1 wish to direct attention in
the present Chapter. I shall first demonstrate some proposi-
tions in which the principles of the above methods of abbrevia-
tion are contained, and 1 shall afterwards apply them to
particular examples. τς

Let us designate as class terms any terms which satisfy
the fundamental law V(1— V)=0. Such terms will individu-
ally be constituents ; but, when occurring together, will not,
as do the terms of a development, necessarily involve the same
symbols in each. Thus ax+6xy+cyg may be described as
an expression consisting of three class terms, 7, zy, and yz,
multiplied by the coefficients a, 0, c respectively. The prin-
ciple applied in the two following Propositions, and which,
_ in some instances, greatly abbreviates the process of elimina-
tion, is that of the resectzon of superfluous class terms ; those
being regarded as superfluous which do not add to the
constituents of the final result.

PROPOSITION I
2. From any equation, V=o, in whitch V consists of a
sertes of class terms having positive coeffictents, we are per-
mitted to reject any term which contains another term as a
factor, and to change every posttive coeffictent to unity.

For the significance of this series of positive terms depends
139

140 THE LAWS*OP 1 OCG FE

only upon the number and nature of the constituents of its —
final expansion, z.e. of its expansion with reference to all the :

symbols [131] which it involves, and not at all upon the
actual values of the coefficients (VI. 5). Now let x be any
term of the series, and xy any other term having + as a factor.
The expansion of x with reference to the symbols + and y
will be
xy + «(1 -- 4),
and the expansion of the sum of the terms x and xy will be
2xy + x(1 —y).
But by what has been said, these expressions occurring in
the first member of an equation, of which the second member
is Ο, and of which all the coefficients of the first member are

positive, are equivalent ; since there must exist simply the —

two constituents vy and x#(1—y) in the final expansion,

whence will simply arise the resulting equations
or AT =) ὃ;

And, therefore, the aggregate of terms ++zy may be re-

placed by the single term ~.

The same reasoning applies to all the cases contemplated
in the Proposition. Thus if the term x is repeated, the
aggregate 2% may be replaced by x, because under the circum-
stances the equation r=0 must appear in the final reduction.

PROPOSITION [1

3. Whenever in the process of elimination we have to
mulliply together two factors, each consisting solely of positive
terms, satisfying the fundamental law of logical symbols, it
ts permitted to reject from both factors any common term, or
Jrom either factor any term whitch ἐς divisible by a term in
the other factor ; provided always, that the rejected term be
added to the product of the resulting factors.

In the enunciation of this Proposition, the word ‘‘ divisible ”
is a term of convenience, used in the algebraic sense, in
which xy and χ( -- )}) are said to be divisible by *.

a ee a a iss ‘he -.

METHODS OF ABBREVIATION [41

To render more clear the import of this Proposition, let it
be supposed that the factors to be multiplied together are
aty+eand%+yw+é. It is then asserted, that from these
two factors we may reject the term x, and that from the
second factor we may reject the term yw, provided that
these terms be transferred [132] to the final product. Thus,
the resulting factors being y+<s and ¢, if to their product
yt+aet we add the terms x and yw, we have

x+ywt+yt + 22,

as an expression equivalent to the product of the given
factors ++y+e2 and ++yw+¢; equivalent namely in the
process of elimination.

Let us consider, first, the case in which the two factors
have a common term 4%, and let us represent the factors by
the expressions ++ δ, ++ Q, supposing P in the one case
and Q in the other to be the sum of the positive terms addi-
tional to %.

Now,

(c+ P)(x+ Q)=x+xP4+x4Q0+ PQ. (1)
But the process of elimination consists in multiplying certain
factors together, and equating the result to o. Either
then the second member of the above equation is to be
equated too, or it is a factor of some expression which is
to be equated to Ο.

If the former alternative be taken, then, by the last Pro-
position, we are permitted to reject the terms #P and σῷ,
inasmuch as they are positive terms having another term +
as a factor. The resulting expression is

x+PQ,

which is what we should obtain by rejecting x from both
factors, and adding it to the product of the factors which
remain.

Taking the second alternative, the only mode in which
the second member of (1) can affect the final result of
elimination must depend upon the number and nature of its
constituents, both which elements are unaffected by the

142 THE LAWS Of THOGCGHT

rejection of the terms xP and χῷ. For that development

of x includes all possible constituents of which z is a factor.
Consider finally the case in which one of the factors con-

tains a term, as xy, divisible by a term, x, in the other factor.
Let ++ P and xy+ @ be the factors. Now

(c+ P)(xy+ Q)=xy+xQ4+xyP+ PQ.

But by the reasoning of the last Proposition, the term ryP
may be rejected as containing another positive term xy as a
factor, whence we have
[133] αν +xQ+ PQ

=xy+(x+P)Q.
But this expresses the rejection of the term xy from the
second factor, and its transference to the final product.
Wherefore the Proposition is manifest.

PROPOSITION III

4. If t be any symbol which ts retained in the final result
of the elimination of any other symbols from any system of
equations, the result of such elimination may be expressed 771
the form

ἘΠ o
an which E ts formed by making in the proposed system t=1,
and eliminating the same other symbols ; and Κ΄ by making
in the proposed system t=0, and eliminating the same other
symbols.

For let #(¢)=0 represent the final result of elimination.
Expanding this equation, we have

b(1)t+ φί(ο)( -- 4) =0.

Now by whatever process we deduce the function ¢(¢) from
the proposed system of equations, by the same process
should we deduce ¢(1), if in those equations ¢ were changed
into I; and by the same process should we deduce p(O), if
in the same equations ¢ were changed into 0. Whence the
truth of the proposition is manifest.

METHODS OF ABBREVIATION 143

5. Of the three propositions last proved, it may be re-
marked, that though quite unessential to the strict develop-
ment or application of the general theory, they yet ac-
complish important ends of a practical nature. By Prop. I.
we can simplify the results of addition ; by Prop. II. we can
simplify those of multiplication ; and by Prop. III. we can
break up any tedious process of elimination into two distinct
processes, which will in general be of a much less complex
character. This method will be very frequently adopted,
when the final object of inquiry is the determination of the
value of ¢, in terms of the other symbols which remain after
elimination is performed.

6. Ex. 1.—Aristotle, in the Vzcomachean Ethics, Book II.
cap. 3, having determined that actions are virtuous, not as
possessing in themselves a certain character, but as implying
a cer-[134]tain condition of mind in him who performs them,
viz., that he perform them knowingly, and with deliberate
preference, and for their own sakes, and upon fixed principles
of conduct, proceeds in the two following chapters to con-
sider the question, whether virtue is to be referred to the
genus of Passions, or Faculties, or Habits, together with
some other connected points. He grounds his investigation
upon the following premises, from which, also, he deduces
the general doctrine and definition of moral virtue, of which
the remainder of the treatise forms an exposition.

PREMISES

1. Virtue is either a passion (πάθος), or a faculty (δύναμις),
or a habit (ἕξις).

2. Passions are not things according to which we are
praised or blamed, or in which we exercise deliberate
preference.

3. Faculties are not things according to which we are
praised or blamed, and which are accompanied by deliberate
preference.

4. Virtue is something according to which we are praised
or blamed, and which is accompanied by deliberate preference.

2

144 THE LAWS OF THOUGHT

5. Whatever art or science makes its works to be in a
good state avoids extremes, and keeps the mean in view
relative to human nature (τὸ μέσον . . . πρὸς ἡμᾶς).

6. Virtue is more exact and excellent than any art or
science.

This is an argument ἃ fortzore. If science and true art
shun defect and extravagance alike, much more does virtue
pursue the undeviating line of moderation. If ¢hey cause
their work to be in a good state, much more reason have
we to say that Virtue causeth her peculiar work to be ‘‘in
a good state.” Let the final premiss be thus interpreted.
Let us also pretermit all reference to praise or blame, since
the mention of these in the premises accompanies only the
mention of deliberate preference, and this is an element
which we purpose to retain. We may then assume as our
representative symbols—

v =virtue.

2 = passions.

7=taculties.

h= habits.

d=things accompanied by deliberate preference.

[135] g=things causing their work to be in a good state.

m= things keeping the mean in view relative to human

nature.

Using, then, g as an indefinite class symbol, our premises
will be expressed by the following equations :

v=g{o(1 Αι - 2) + f(t — pa — 2) + A(t - p(t -f)}.

2-- 4(1 -- 4).
f= -- @).
ΦΕΞ σα,
:ΞΞ 972.
ΞΕ’:

And separately eliminating from these the symbols g,
vit — p(x ἐγ. 2) -flr—p)(1 — 2) - A(x -p)r—f)} =o. (1)

pad=o. (2)
Nake (3)

:
|
Ἢ

METHODS OF ABBREVIATION 145

o(1-d)=0. (4)
g(t -m)=0. (5)
721 —g) ΞΟ: (6)

We shall first eliminate from (2), (3), and (4) the symbol
d, and then determine vin relation to 2, f, and ἡ. Now
the addition of (2), (3), and (4) gives
(p+f)d+v(1 -- 4) -Ξ ο.
From which, eliminating d in the ordinary way, we find
(P+f)v=0. (7)
Adding this to (1), and determining v, we find

OS pf 1 PE --6 --ἢ fa — pyle ~ ἢ — Mx -ῆᾷ - ἢ)
Whence by development, ©

v= Lit - f(x 2),

The interpretation of this equation is: Virtue ἐς a hadzt,
and not a faculty or a passton.

[136] Next, we will eliminate /, f, and g from the original
system of equations, and then determine v in relation to
h, d, and m. We will in this case eliminate 2 and 7 to-
gether. On addition of (1), (2), and (3), we get
o{r—plr—f)(1 — 4) f(z — pa - 2) - Hr -p)\(r-f)} + pat fa=o.
Developing this with reference to p and f, we have
(9 Ὁ 2d\pf+ (vh+d)p(x -- 7) + (wh + a(x — p)f

+u(1 —A)(1 -- 2) -- 33 =0.
Whence the result of elimination will be
(v+2d)(vh+d)(vh+ d)u(1 — h)=0.
Now v+2d=v+d+d, which by Prop. I. is reducible to
v+d. The product of this and the second factor is

(v+d)(vh +4),
which by Prop. II. reduces to

d+u(vh) or vh+d.
10

146 THE LAWS Of THOUGHT

In like manner, this result, multiplied by the third factor,
gives simply vk+d. Lastly, this multiplied by the fourth
factor, v(1—/), gives, as the final equation,

vd(x — hk) =o. (8)

It remains to eliminate g from (5) and (6). The result is
v(I —m)=0. (9)
Finally, the equations (4), (8), and (9) give on addition
v(1 —d) + vd(1 -- Δ) + σ( -- m) =0,

from which we have
[ΦῚ

Ἴρις

And the development of this result gives
ἘΣ ham,
12)

of which the interpretation is, —V7rtue ἐς a habit accompanied
by [137] deliberate preference, and eid in view the
mean relative to human nature.

Properly speaking, this is not a definition, but a descrip-
tion of virtue. It is αὐ, however, that can be correctly
inferred from the premises. Aristotle specially connects
with it the necessity of prudence, to determine the safe and
middle line of action ; and there is no doubt that the ancient
theories of virtue generally partook more of an intellectual
character than those (the theory of utility excepted) which
have most prevailed in modern days. Virtue was regarded
as consisting in the right state and habit of the whole
mind, rather than in the single supremacy of conscience or
the moral faculty. And to some extent those theories were
undoubtedly right. For though unqualified obedience to the
dictates of conscience is an essential element of virtuous
conduct, yet the conformity of those dictates with those un-
changing principles of rectitude (αἰώνια δίκαια) which are —
founded in, or which rather are themselves the foundation of
the constitution of things, isanotherelement. And generally
this conformity, in any high degree at least, is inconsistent

METHODS OF ABBREVIATION 147

with a state of ignorance and mental hebetude. Reverting to
the particular theory of Aristotle, it will probably appear
to most that it is of too negative a character, and that the
shunning of extremes does not afford a sufficient scope for
the expenditure of the nobler energies of our being. Aris-
totle seems to have been imperfectly conscious of this de-
fect of his system, when in the opening of his seventh book
he spoke of an ‘‘ heroic virtue” * rising above the measure
of human nature.

7. | have already remarked (VIII. 1) that the theory of
single equations or propositions comprehends questions which
cannot be fully answered, except in connexion with the
theory of systems of equations. This remark is exemplified
when it is proposed to determine from a given single equa-
tion the relation, not of some single elementary class, but
of some compound class, involving in its expression more
than one element, in terms of the remaining elements. The
following particular example, and the succeeding general
problem, are of this nature.

_ [138] Ex. 2.—Letus resume the symbolical expression
of the definition of wealth employed in Chap. VII., viz.,
w=st{ptr(1-p)},

wherein, as before,

w = wealth,

s=things limited in supply,

¢=things transferable,

p=things productive of pleasure ;

y=things preventive of pain.

and suppose it required to determine hence the relation of
things transferable and productive of pleasure, to the other
elements of the definition, viz., wealth, things limited in
supply, and things preventive of pain.

The expression for things transferable and productive of
pleasure is ¢f. Let us represent this by a new symbol ».
We have then the equations

* τὴν ὑπὲρ ἡμᾶς ἀρετὴν ἡρωϊκήν τίνα καὶ θείαν. --- ες. Eth. Book VII.

148 LAEOLAWS Of, PHCUGH

w=stip+r(t -)},
Da;

from which, if we eliminate ¢ and f, we may determine y

as a function of w,s, and 7 The result interpreted will
give the relation sought.

Bringing the terms of these equations to the first side, we
have

w — stp — str(1 -- 23) =0.

Gee (3)

And adding the squares of these equations together,

w+ stp + str(1 — 3) -- 2wstp -- 2wstr(1-—p)+y+tp—2ytp=o. (4)

ΤῸ nee ek

Developing the first member with respect to ¢ and 2, in —

order to eliminate those symbols, we have

(w+s—2ws+1—y)ip+(wt+sr—2wsrt+y)t(1 -- 2)
τς 2 οὐ τ ἡ 5. Ὁ (5)

and the result of the elimination οἵ ¢ and 2. will be obtained
by equating to 0 the product of the four coefficients of

Lp, 1 —p); (1 -t)p, and (1 = ἢ) (α = 2):

[139] Or, by Prop. III., the result of the elimination of ¢
and 2. from the above equation will be of the form

Ey+ E'(1-y),

wherein & is the result obtained by changing in the given

equation y into I, and then eliminating ¢ and 2; and Ε΄ the ,

result obtained by changing in the same equation y into Ο,
and then eliminating ¢ and 2. And the mode in each case

of eliminating ¢ and 2) is to multiply together the coefficients —

of the four constituents 7p, Ζ(1 --- 2), &c.
If we make y=1, the coefficients become—
Ist. w(I—s)+s(1—w).
2nd. 1+w(I—sr)+s(1—w)r, equivalent to 1 by Prop, I.
3rd and 4th. 1+w, equivalent to 1 by Prop. I.
Hence the value of Z will be

w(I—s)+s(1-w).

ΝΣ a ae a ed ee λοι

METHODS OF ABBREVIATION 149

Again, in (5) making y=0, we have for the coefficients—
Ist. I1+w(1—s)+s5(1—w), equivalent to 1.

2nd. w(t —sr)+sr(1 —w).

3rd and qth. w.

The product of these coefficients gives

2 =w(i - sr).
The equation from which y is to be determined, there-
ΤΟΙ, ἴ9
{w(1 —s)+s(1 —w)}y+w(1 -- s7)(1 -- )) Ξξ ο,

5 w(1 -- 57) ᾿
~ w(t -- sr) -- w(t -- 5) -- ((( -- το)

a
and expanding the second member,
y=" wsr + ws(1 —r) +2 w(1 —s)r+ w(t —s)(1 -- 2)
re) οΟ Oo

4+ 0(1 —7w)sr +0(1 — w)s(1 -- 7) ἐξα —w)(1—s)r

εθα - α)α -ὴα - 7);
whence reducing
[140] y=ws(t — 7) + —wsr $2 (1 -w)(1 -- 5), (6)
with w(I —S)=0. (7)
The interpretation of which is—

ist. Things transferable and productive of pleasure consist
of all wealth (limited in supply and) not preventive of pain,
an indefinite amount of wealth (limited in supply and) pre-
ventive of pain, and an indefinite amount of what ts not
wealth and not limited tn supply.

2nd. All wealth is limited tn supply.

I have in the above solution written in parentheses that
part of the full description which is implied by the accom-
-panying independent relation (7).

8. The following problem is of a more general nature, and
will furnish an easy practical rule for problems such as the last.

150 Ripe AVS Orn ΕΓ

GENERAL PROBLEM

Given any equation connecting the symbols x,y... τὺ, δ...

Required to determine the logical expression of any class
expressed in any way by the symbols x, y . . in terms of the
remaining symbols, W, 5, &C.

Let us confine ourselves to the case in which there are but
two symbols, x, 7, and two symbols, w, 2, a case sufficient
to determine the general Rule.

Let V =o be the given equation, and let (4, y) represent
the class whose expression is to be determined.

Assume ¢=¢(%, 7), then, from the above two equations,
x and y are to be eliminated.

Now the equation Κ᾽ =o may be expanded in the form

Axy + Bx(1 —y) + (( - x)y+ Δ -- x)(1 -- γ) =09, (πὶ

A, B, C, and D being functions of the symbols w and 2.

Again, as ¢(%, 7) represents a class or collection of things,
it τα Consist Gi a-constituent, Of series Of constituents,
whose coefficients are I.

[141] Wherefore if the fudZ development of ¢(%, y) be re-
presented in the form

axy + ὀχ(ι -- )) Ὁ «(τ -- Δ)» Ὁ 4{ -- α)ζι -- 7),

the ceetheienis ὦ 0,-¢,a must each be1. or ©:

Now reducing the equation ¢= (4, v) by transposition and
squaring, to the form

t{1 - (x, ))} + O(% 91-2) = 0;
and expanding with reference to x and y, we get
{{ -—a)+a(1-7)}xyt+ {{τ -- ὁΚ ἘΔ(ἃ - δ) }.χ -- ))

+ {1 -- ) Ἐπ -- 2} -- “)ν
+ {{(-- 4) -Ἐ 4Ἃ(τ -- 4) (α -- χα --))Ξξο;

whence, adding this to (1), we have

{A4+H1-a)+a(1—7#)}xy
+{B+¢t(1-6)4+6(1 - 2)}.χα -- )) + &c. =0.

Let the result of the elimination of a and y be of the form
BEE.

METHODS OF ABBREVIATION 151

then £ will, by what has been said, be the reduced product
of what the coefficients of the above expansion become when
ἔτει, and &’ the product of the same factors similarly reduced
by the condition ¢=o0.

Hence £& will be the reduced product
(4+1-a)(B+1-6)(C+1-c(D+1-2a).
Considering any factor of this expression, as d+1—a, we
see that when a=I it becomes A, and when a=0 it becomes
1+ A, which reduces by Prop. I. ἴοι. Hence we may infer
that & will be the product of the coefficients of those con-
stituents in the development of V whose coefficients in the

development of #(%, γ) are I.
Moreover £’ will be the reduced product

(4 +a)(B+0)(C+c)(D+d).

Considering any one of these factors, as A+a, we see that
this becomes A when ὦ ΞΟ, and reduces to I whena=1;
and so on for the others, Hence £’ will be the product of
the coefficients [142] of those constituents in the develop-
ment of y, whose coefficients in the development (4, y) are
o. Viewing these cases together, we may establish the
following Rule:

9. To deduce from a logical equation the relation of any
class expressed by a given combination of the symbols x, 7, &¢.,
to the classes represented by any other symbols tnvolved tn the
given equation.

RULE.—Expand the given equation with reference to the
symbols x, y. Then form the equation
Et+£E'(1-t)=09,
in which E ἐς the product of the coefficients of all those con-
stituents in the above development, whose coefficients in the
expression of the given class are τ, and δ΄ the product of the
coefficients of those constituents of the development whose co-
efficients in the expression of the given class areo. The value
of t deduced from the above equation by solution and tnterpre-
tation will be the expression required.

152 THE LAWS OF THOUGHT

NotTre.—Although in the demonstration of this Rule V ts
supposed to consist solely of positive terms, it may easily be
shown that this condition ts unnecessary, and the Rule general,
and that no preparation of the gtven equation ts really requtred.

10. Ex. 3.—The same definition of wealth being given as
in Example 2, required an expression for ¢hings transferable,
but not productive of pleasure, 1(1 -- 3), in terms of the other
elements represented by w, 5, and 7.

The equation

w — stp - str(1 -- 2) =0,
gives, when squared,
w+ stp + str(1 -- 2) — 2wstp — 2wstr(1 -- p)=0;
and developing the first member with respect to ¢ and 2,
(τυ +5 -- 22s)tp + (w+ sr - 2zsr)t(1 -- 2) -Ἐ τοι -- 2)2
+z(1 -- 2) -f)=0.
The coefficients of which it is best to exhibit as in the follow-
ing equation :
{ze(1 -- 5) Ἐ σα —w)} tp + {τὐ(α -- sr) +57r(1 -- Ζ0)}} 1 -- fp) + w(1 -- 2)2
Ἔτυ(ι -- (αι - 23) =o.
[143] Let the function Ζ(1 -- 2) to be determined, be repre-
sented by z; then the full development of 2 in respect of ¢
and 2 is
2=of+t1—-f)+0(1 -f4)Pt+o(1-7)(1 — 2).
Hence, by the last problem, we have
Ez+ E(1-2)=0;
where E=w(t -sr)+sr(1 -- το) ;
2’ = {w(t -- 5) -- «(1 -- ΟἹ} xwxw=w(i-s);
{w(1 -- 52) +s7(1 -- 29}} -Ἑ (1 —5)(1 —2)=0.
Hence,

ee z(I -- 5)
21057 — WS -- 57
et Sd :

SG ας ΣῈ τι] -)γα + w(t —s)(1 -- 3),
ο

Ἔσ(ι —w)sr $2 (1 ~w)s(1 -- 7) +2 (1—w)(1 —s)r
ο

ἐξ Sw) - ογίτ - 2).

METHODS OF ABBREVIATION 183
Or,

ο ο ο
Zr σα — w)s(1 sid ic: —w)(I — 5),
with w(I —s)=0,

Hence, Things transferable and not productive of pleasure
ave either wealth (limited in supply and preventive of pain) ;
or things which are not wealth, but limited in supply and not
preventive of pain; or things which are not wealth, and are
unlimited in supply.

The following results, deduced in a similar manner, will be
easily verified :

Things limited in supply and productive of pleasure which
are not wealth,—are tntransferable.

Wealth that ts not productive of pleasure ts transferable,
limited in supply, and preventive of pain.

Things limited in supply which are etther wealth, or are
productive of pleasure, but not both,—are etther transferable
and preventive of pain, or intransferable.

11. From the domain of natural history a large number of
curious examples might be selected. I do not, however, con-
[144]ceive that such applications would possess any inde-
pendent value. They would, for instance, throw no light upon
the true principles of classification in the science of zoology.
For the discovery of these, some basis of positive knowledge
is requisite, —some acquaintance with organic structure, with
teleological adaptation ; and this is a species of knowledge
which can only be derived from the use of external means of
observation and analysis. Taking, however, any collection of
propositions in natural history, a great number of logical
problems present themselves, without regard to the system
of classification adopted. Perhaps in forming such examples,
it is better to avoid, as superfluous, the mention of that
property of a class or species which is immediately suggested
by its name, e.g. the ring-structure in the annelida, a class
of animals including the earth-worm and the leech.

Ex. 4.—1. The annelida are soft-bodied, and either naked
or enclosed in a tube.

154 tHE LAWS OF THOUGHT

2. The annelida consist of all invertebrate animals having |
red blood in a double system of circulating vessels.

Assume a=annelida ;
s =soft-bodied animals ;
m= naked ;

¢=enclosed in a tube;
z=invertebrate ;
v=having red blood, ὅς:
Then the propositions given will be expressed by the equations
a=vs{n(1—-f)+H1-2)}; (1)
air. (2)
to which we may add the implied condition,

ao: (3)
On eliminating v, and reducing the system to a single equa-
tion, we have
α{τ —sn(1—7¢)—st(1-n)}+a(1 -- 2) γᾷ -- α) Ἐ πέτεο. (4)
Suppose that we wish to obtain the relation in which soft-
bodied animals enclosed in tubes are placed (by virtue of the
[145] premises) with respect to the following elements, viz.,
the possession of red blood, of an external covering, and of
a vertebral column.
We must first eliminate a The’result is
ir{1—sn(1 -- ἢ) —st(1 —2)}+nt=0.
Then (IX. 9) developing with respect to sand ¢, and reducing
the first coefficient by Prop. 1, we have
mst +ir(1 —n)s(1 —2)+(tr+n)(1—s)¢+ir(1—s)(1-Z)=0. (5)
Hence, if st=w, we find
NW + ir(I —n) x (ir +2) x ir(1 -- w)=0;
or, nw + ir(I —2)1{ -w)=o0;
_ γ1τ - πὴ
17γ(1 -- 2) -- 2

= orn + 77(1 -- 2) +001 -- 2)» +2 i(1 —r)(1 - 2)
+ 0o(1 - i)rn + (τ -2)7(1-2)+0(1 -2)(1 -- γ)5

+2 (1-A(r-r)(1-2);

METHODS OF ABBREVIATION Iss

se w= ir(r—n) +2 (a r)(x- 2) +o (4 τὴ —n).

Hence, soft-bodied animals enclosed in tubes consist of all
invertebrate animals having red blood and not naked, and an
indefinite remainder of invertebrate animals not having red
blood and not naked, and of vertebrate animals which are
not naked.

And in an exactly similar manner, the following reduced
equations, the interpretation of which is left to the reader,
have been deduced from the development (5).

s(1 - i) =irn~ ils ~n)+o(1 - ἢ
(1 - πετᾷ ~i)r(1 —n)+2(1 ~r)(1—2)
(1-s)(1 -ἢ- πήττα -- ἢ).

[146] In none of the above examples has it been my object
to exhibit in any special manner the power of the method.
That, I conceive, can only be fully displayed in connexion
with the mathematical theory of probabilities. I would,
however, suggest to any who may be desirous of forming a
correct opinion upon this point, that they examine by the
rules of ordinary logic the following problem, Jefore inspect-
τ ing its solution ; remembering at the same time, that what-
ever complexity it possesses might be multiplied indefinitely,
with no other effect than to render its solution by the method
of this work more operose, but not less certainly attainable.

Ex. 5. Let the observation of a class of natural productions
be supposed to have led to the following general results.

Ist, That in whichsoever of these productions the pro-
perties A and Care missing, the property Z is found, together
with one of the properties B and D, but not with both.

and, That wherever the properties A and D are found
while Z is missing, the properties B and C will either both be
found, or both be missing.

3rd, That wherever the property A is found in conjunction
with either B or Z, or both of them, there either the property

156 THE ΤΙΣ ΟΣ TAHCCOGHT

C or the property D will be found, but not both of them.
And conversely, wherever the property C or D is found singly,
there the property A will be found in conjunction with either
B or £, or both of them.

Let it then be required to ascertain, first, what in any
particular instance may be concluded from the ascertained
presence of the property A, with reference to the properties
B, C, and D; also whether any relations exist independently
among the properties 4, C, and D. Secondly, what may be
concluded in like manner respecting the property 4, and the
properties. A. C, and 2).

It will be observed, that in each of the three data, the in-
formation conveyed respecting the properties A, B, C, and
D, is complicated with another element, 35, about which we
desire to say nothing in our conclusion. It will hence be
requisite to eliminate the symbol representing the property
f from the system of equations, by which the given pro-
positions will be expressed.

[147] Let us represent the property A by x, B by y, C
by z, Dby w, HE by v. The data are

XE = φυί yw + wy) ; (1)
ὅατο = 9( 92 Ὁ 75); (2)
XY + XVY = WE + ZW; (2)

# standing for 1-2, &c., and g being an indefinite class
symbol. Eliminating g separately from the first and second
equations, and adding the results to the third equation re-
duced by (5), Chap. VIII., we get

ἀξᾷι — σντῦ — UWP) + Dxw( yz + Zh) + (xy + xv7)(we + το)
+ (wz + 27)(1 — xy — xvy) =o. (4)
From this equation v must be eliminated, and the value of
᾿ 4, determined from the result. For effecting this object, it
will be convenient to employ the method of Prop. III. of
the present Chapter.
Let then the result of elimination be represented by the

equation
ἔχ (ι -x)=0.

q
;
ῃ

METHODS OF ABBREVIATION 187

To find Z make x=1 in the first member of (4), we find
ou γ8 + 27) + (y + wP)(we + WZ) + (ws + zw) Ody.
Eliminating v, we have
(we + 22){w( yz + 27) + (we + Wz) + (ws + 3w)} ;
which, on actual multiplication, in accordance with the con-
ditions ww=0, z2=0, &c., gives
EL = we + yw.
Next, to find &’ make +=0 in (4), we have
2(1 — vyw — VywW) + τοῦ + 2D,

whence, eliminating v, and reducing the result by Propositions
I. and II., we find

4 =wit+2sw+ yw ;
and, therefore, finally we have

(we -- γεῦξ)α + (we + 2 + jw2)xX =0 ; (5)
from which
[148] ao we + ai + ye ae
WS + ZW + YWS — WZ — PWS

wherefore, by development,
X= OYZW + YZW + YSW + OYSW
+ OYZW + YZW + YZW + SW ;
or, collecting the terms in vertical columns,
x= 2W + Zw + YZW ; (6)
the interpretation of which is—

In whatever substances the property A ts found, there will
also be found either the property C or the property D, but not
both, or else the properties B, C, and D, will all be wanting.
And conversely, where either the property C or the property

D ts found singly, or the properties B, C, and D, are together
missing, there the property A will be found.

It also appears that there is no independent relation among
the properties B, C, and D.

158 THE LAWS OF THOUGHT

Secondly, we are to find y. Now developing (5) with
respect to that symbol,

(xs + XZ + XWE + X2W)y + (KS + XWE + X2W + XZW) =O;

whence, proceeding as before,

y =H02 +2 (Swe + xwi + x2w), (7)
ο

XZ =0; (8)

XZW =O; (9)

X2 =0 ; (10)

From (10) reduced by solution to the form
= Oo
XZ=-W;
ce)

we have the independent relation,—// the property A ἐς absent
and C present, Dts present. Again, by addition and solution

(8) and (9) give

Whence we have for the general solution and the remaining
independent relation :

ist. [149] 17 the property B be present in one of the produc-
tions, etther the properties A, C, and D, are all absent, or some
one alone of them ts absent. And conversely, ¢f they are all
absent ἐξ may be concluded that the property A ts present (7).
2nd. 27 A and C are both present or both absent, D will
be absent, quite independently of the presence or absence of B

(8) and (9).

I have not attempted to verify these conclusions.

4

[150] CHAPTER X
OF THE CONDITIONS OF A PERFECT METHOD

1. THE subject of Primary Propositions has been discussed
at length, and we are about to enter upon the consideration
of Secondary Propositions. The interval of transition be-
tween these two great divisions of the science of Logic may
afford a fit occasion for us to pause, and while reviewing
some of the past steps of our progress, to inquire what it
is that in a subject like that with which we have been occupied
constitutes perfection of method. Ido not here speak of
that perfection only which consists in power, but of that
also which is founded in the conception of what is fit and
beautiful. It is probable that a careful analysis of this
question would conduct us to some such conclusion as the
following, viz., that a perfect method should not only be
an efficient one, as respects the accomplishment of the ob-
jects for which it is designed, but should in all its parts and
processes manifest a certain unity and harmony. This con-
ception would be most fully realized if even the very forms
of the method were suggestive of the fundamental principles,
and if possible of the oze fundamental principle, upon which
they are founded. In applying these considerations to the
science of Reasoning, it may be well to extend our view
beyond the mere analytical processes and to inquire what is
best as respects not only the mode or form of deduction,
but also the system of data or premises from which the
deduction is to be made.

2. As respects mere power, there is no doubt that the
first of the methods developed in Chapter VIII. is, within
its proper sphere, a perfect one. The introduction of arbi-

trary constants makes us independent of the forms of the

159

160 ΤΉ ΣΤ Ol THOUGH

premises, as well as of any conditions among the equations
by which they are represented. But it seems to introduce
a foreign element, and while it is a more laborious, it is
also a less elegant form of solution than the second method
of reduction demonstrated in the same [151] Chapter. There
are, however, conditions under which the latter method as-
sumes a more perfect form than it otherwise bears. To
make the one fundamental condition expressed by the
equation
x(1 -x)=0,

the universal type of form, would give a unity of character
to both processes and results, which would not else be at-
tainable. Were brevity or convenience the only valuable
quality of a method, no advantage would flow from the
adoption of such a principle. For to impose upon every
step of a solution the character above described, would in-
volve in some instances no slight labour of preliminary re-
duction. Bue itis still interesting to know that-this can
be done, and it is even of some importance to be acquainted
with the conditions under which such a form of solution
would spontaneously present itself. Some of these points
will be considered in the present chapter.

PROPOSITION I

3. To reduce any equation among logical symbols to the
form V=0, in which V satisfies the law of duality,

Via yo.

It is shown in Chap. V. Prop. IV., that the above con-
dition is satisfied whenever V is the sum of a series of
constituents. And it 1s evident from Prop..1l. Chap, V1.;
that all equations are equivalent which, when reduced by
transposition to the form V=o, produce, by development
of the first member, the same series of constituents with
coefficients which do not vanish; the particular numerical
values of those coefficients being immaterial.

CONDITIONS OF A PERFECT METHOD 161

Hence the object of this Proposition may always be accom-
plished by bringing all the terms of an equation to the first
side, fully expanding that member, and changing in the result
all the coefficients which do not vanish into unity, except such
as have already that value.

But as the development of functions containing many
symbols conducts us to expressions inconvenient from their
great [152] length, it is desirable to show how, in the only
cases which do practically offer themselves to our notice,
this source of complexity may be avoided.

The great primary forms of equations have already been
discussed in Chapter VIII. They are—

ΕΞ a
f= en
vX=0Y.
Whenever the conditions X¥(1-—X)=0, Y(1— Y)=o0, are
satisfied, we have seen that the two first of the above equa-
tions conduct us to the forms
X(1 - Y)=09, (1)
X(1- Y)+ Vau-X)=0; (2)
and under the same circumstances it may be shown that
the last of them gives 7

o{X(1- V)+ VG-X)} =o; (3)

all which results obviously satisfy, in their first members, the
condition

V(i1-V)=o.
Now as the above are the forms and conditions under which
the equations of a logical system properly expressed do
actually present themselves, it is always possible to reduce
them by the above method into subjection to the law re-
quired. Though, however, the separate equations may
thus satisfy the law, their equivalent sum (VIII. 4) may
not do so, and it remains to show how upon it also the
requisite condition may be imposed.

Let us then represent the equation formed by adding the
II

162 THE LAWS OF ΓΕΘ ΕΣ

several reduced equations of the system together, in the

form
V+u+v', &c.=0, (4)

this equation being singly equivalent to the system from —
which it was obtained. We suppose v, wv’, ν΄, &c. to be®
class terms (IX. 1) satisfying the conditions

w1i-v)=0, v(rI-v)=o0, &c.

Now the full interpretation of (4) would be found by deve-
[153]loping the first member with respect to all the ele-
mentary symbols x, vy, &c. which it contains, and equating
to Ο all the constituents whose coefficients do not vanish ; in
other words, all the constituents which are found in either
Ὁ ὦ &c. But those constituents consist of—tIst, such
as are found in vw; 2nd, such as are not found in v, but are
found in v’; 3rd, such as are neither found in wv nor v’, but
are found in wv’, and so on. Hence they will be such as are
found in the expression

v+(1-v)v'+(1-v)(1-v)0" +&c, (5)

an expression in which no constituents are repeated, and
which obviously satisfies the law V(1— V)=o,
Thus if we had the expression

(1-7) +04+(1-2)+ew,

in which the terms I —Z, 1 —2 are bracketed to indicate that
they are to be taken as single class terms, we should, in
accordance with (5), reduce it to an expression satisfying the
condition V(1— V)=o0, by multiplying all the terms after
the first by ¢, then all after the second by 1—v; lastly, the
term which remains after the third by z; the result being

1-¢+f+¢(1-—v)(1-2)+e(1 —v) sw. (6)

4. All logical equations then are reducible to the form
V=o, V satisfying the law of duality. . But it would ob-
viously be a higher degree of perfection if equations always
presented themselves in such a form, without preparation of
any kind, and not only exhibited this form in their original

ἊΣ

CONDITIONS OF A PERFECT METHOD 163

statement, but retained it unimpaired after those additions
which are necessary in order to reduce systems of equations
to single equivalent forms. That they do not spontaneously
present this feature is not properly attributable to defect of
method, but is a consequence of the fact that our premises
are not always complete, and accurate, and independent.
They are not complete when they involve material (as dis-
tinguished from formal) relations, which are not expressed.
They are not accurate when they imply relations which are
not intended. But setting aside these points, with which,
in the present instance, we are less concerned, let it be con-
sidered in what sense they may fail of being independent.

[154] 5. A system of propositions may be termed inde-
pendent, when it is not possible to deduce from any portion
of the system a conclusion deducible from any other portion
of it. Supposing the equations representing those proposi-
tions all reduced to the form

Yo
then the above condition implies that no constituent which
can be made to appear in the development of a particular

function V of the system, can be made to appear in the de-
velopment of any other function V’ of the same system.

When this condition is not satisfied, the equations of the

system are not independent. This may happen in various
cases. Let all the equations satisfy in their first members
the law of duality, then if there appears a positive term # in
the expansion of one equation, and a term xy in that of
another, the equations are not independent, for the term x
is further developable into ry+24(1—y), and the equation

xy=0o
is thus involved in both the equations of the system. Again,
let a term xy appear in one equation, and a term +z in another.

- Both these may be developed so as to give the common con-

stituent xyz. And other cases may easily be imagined in
which premises which appear at first sight to be quite inde-
pendent are not really so. Whenever equations of the form

|

V=o are thus not truly independent, though individually |
they may satisfy the law of duality, Ἷ
V(1 -- V)=0,
the equivalent equation obtained by adding them together :
will not satisfy that condition, unless sufficient reductions by ©
the method of the present chapter have been performed. —
When, on the other hand, the equations of a system both .
satisfy the above law, and are independent of each other, ©
their sum will also satisfy the same law. I have dwelt upon
these points at greater length than would otherwise have been —
necessary, because it appears to me to be important to en- |
deavour to form to ourselves, and to keep before us in all our
investigations, the pattern of an ideal perfection, —the object
and the guide of future efforts. In [155] the present class
of inquiries the chief aim of improvement of method should be —
to facilitate, as far as is consistent with brevity, the trans- ἢ
formation of equations, so as to make the fundamental con- |
dition above adverted to universal.
In connexion with this subject the following Propositions

164 (HE LAWS OF THOUGHT.

are deserving of attention.

PROPOSITION II

[f the first member of any equation V=o satisfy the con-
dition Κ7(1 --- V)=0, and tf the expression of any symbol t of
that equation be determined as a developed function of the other
symbols, the coefficients of the expansion can only assume the

I
"ὋΣ

ot ee

forms 1, O,

For if the equation be expanded with reference to ¢, we

obtain as the result,
Et+E (1-2), (1)

E and &’ being what V becomes when ¢ is successively
changed therein into 1 and 0. Hence & and £’ will them-
selves satisfy the conditions

#Ha-£)=0, £(Uu-E)=0. (2)

CONDITIONS OF A PERFECT METHOD 165

Now (1) gives rth Ff

| E-E
the second member of which is to be expanded as a function
of the remaining symbols. It is evident that the only
numerical values which # and #’ can receive in the calcula-
tion of the coefficients will be 1 and o. The following cases
alone can therefore arise :

Ist. E'=1, E=1, then BE e
2nd. E'=1, E=o0, then —— I.
3rd. ΞΟ ae Meee ae tae) —
4th. E'=0, EL=o0, then oe

Whence the truth of the Proposition is manifest.
[156] 6. It may be remarked that the forms 1, ὁ, and ©

appear in the solution of equations independently of any
reference to the condition V(1-— V)=o0. But itis not so with

᾿ I : ; ᾿ ᾿
the coefficient 5° The terms to which this coefficient is

attached when the above condition is satisfied may receive

O
any other value except the three values 1, 0, and = when

that condition is not satisfied. It is permitted, and it would
conduce to uniformity, to change any coefficient of a develop-
ment not presenting itself in any of the four forms referred

to in this Proposition into regarding this as the symbol

proper to indicate that the coefficient to which it is attached
- should be equated too. This course I shall frequently adopt.

PROPOSITION III

7. The result of the elimination of any symbols x, y, &¢.
from an equation V =o, of which the first member tdentically
satisfies the law of duality,

Vii- V)=09,

166 THE PAWS CP THOUGHT.

may be obtained by developing the given equation with reference |
to the other symbols, and equating to 0 the sum of those con-

stituents whose coefficients in the expansion are equal to untty,

Suppose that the given equation V =o involves but three
symbols, +, y, and ¢, of which x and y are to be eliminated. ἢ
Let the development of the equation, with respect to Ζ, be

At+ δὰ -t)=0, (1)
A and B being free from the symbol ¢.

By Chap. IX. Prop. III., the result of the elimination off

x and y from the given equation will be of the form
Et+ E'(1 -t)=0, (2)
in which & is the result obtained by eliminating the symbols
xand y from the equation 4A=o, £’ the result obtained by

eliminating from the equation B=o.
[157] Now 4 and 4 must satisfy the condition
A(1-A)=0, B(1-B)=0.
Hence A (confining ourselves for the present to this co-
efficient) will either be Oo or 1, or a constituent, or the sum
of a partofthe constituents which involve the symbols # and γ.
If A —o it ts evident that ΞΟ if A isa single constituent,
or the sum of a part of the constituents involving # and y,
E will beo. For the με development of A, with respect to x
and y, will contain terms with vanishing coefficients, and £ is
the product of αὐ the coefficients. Hence when A=1, £ is
equal to A, but in other cases £ is equal too. Similarly,
when £=1, £’ is equal to 8, but in other cases #’ vanishes.
Hence the expression (2) will consist of that part, if any there
be, of (1) in which the coefficients A, Bare unity. And this
reasoning is general. Suppose, for instance, that V involved
the symbols x, y, 5, ¢, and that it were required to eliminate
aandy. Then if the development of V, with reference to
z and ¢, were
at+x2(1 -¢)+y(1 —2)f+(1 -- 2) -- ἢ),
the result sought would be
5.-Ὲ (1 -- )(1 -- ὃ. Ξ ο,

CONDITIONS OF A PERFECT METHOD 167

this being that portion of the development of which the co-
efficients are unity.

Hence, if from any system of equations we deduce a single
equivalent equation V=o, V satisfying the condition

V(1 - V)=0,
the ordinary processes of elimination may be entirely dis-
pensed with, and the single process of development made to
supply their place.

8, It may be that there is no practical advantage in the
method thus pointed out, but it possesses a theoretical unity
and completeness which render it deserving of regard, and I
shall accordingly devote a future Chapter (XIV.) to its illus-
tration. The progress of applied mathematics has presented
other and signal examples of the reduction of systems of pro-
blems or equations to the dominion of some central but per-
vading law.

[158] 9. It is seen from what precedes that there is one
class of propositions to which all the special appliances of the
above methods of preparation are unnecessary. It is that
which is characterized by the following conditions :

First, That the propositions are of the ordinary kind, im-
plied by the use of the copula zs or ave, the predicates being
particular.

Secondly, That the terms of the proposition are intelligible
without the supposition of any understood relation among
the elements which enter into the expression of those terms.

Thirdly, That the propositions are independent.

We may, if such speculation is not altogether vain, permit
ourselves to conjecture that these are the conditions which
would be obeyed in the employment of language as an in-
strument of expression and of thought, by unerring beings,
declaring simply what they mean, without suppression on
the one hand, and without repetition on the other. Con-
sidered both in their relation to the idea of a perfect language,
and in their relation to the processes of an exact method, these
conditions are equally worthy of the attention of the student.

[159] CHEAP TIC XxX)

OF SECONDARY PROPOSITIONS, AND OF THE PRINCIPLES
OF THEIR SYMBOLICAL EXPRESSION

1. THE doctrine has already been established in Chap. IV.,
that every logical proposition may be referred to one or the

4

other of two great classes, viz., Primary Propositions and ἢ

Secondary Propositions. The former of these classes has
been discussed in the preceding chapters of this work, and
we are now led to the consideration of Secondary Proposi-
tions, z.e. of Propositions concerning, or relating to, other
propositions regarded as true or false. The investigation
upon which we are entering will, in its general order and
progress, resemble that which we have already conducted.
The two inquiries differ as to the subjects of thought which
they recognise, not as to the formal and scientific laws which
they reveal, or the methods or processes which are founded
upon those laws. Probability would in some measure favour
the expectation of such a result. It consists with all that
we know of the uniformity of Nature, and all that we believe
of the immutable constancy of the Author of Nature, to
suppose, that in the mind, which has been endowed with
such high capabilities, not only for converse with surround-
ing scenes, but for the knowledge of itself, and for reflection
upon the laws of its own constitution, there should exist a
harmony and uniformity not less real than that which the
study of the physical sciences makes known to us. Anticipa-
tions such as this are never to be made the primary rule of
our inquiries, nor are they in any degree to divert us from
those labours of patient research by which we ascertain what
is the actual constitution of things within the particular pro-

vince submitted to investigation. But when the grounds of
168

ee a ὙΡΎΝΌα

OF SECONDARY PROPOSITIONS 169

resemblance have been properly and independently deter-
mined, it is not inconsistent, even with purely scientific ends,
to make that resemblance a subject of meditation, to trace its
extent, and to receive the intimations of truth, yet undis-
covered, which it may [160] seem to us to convey. ‘The
necessity of a final appeal to fact is not thus set aside, nor is
the use of analogy extended beyond its proper sphere,—the
suggestion of relations which independent inquiry must either
verify or cause to be rejected.

2. Secondary Propositions are those which concern or relate
to Propositions considered as true or false. ‘The relations
of things we express by primary propositions, But we are
able to make Propositions themselves also the subject of
thought, and to express our judgments concerning them.
The expression of any such judgment constitutes a secondary
proposition. There exists no proposition whatever of which
a competent degree of knowledge would not enable us to make
one or the other of these two assertions, viz., either that
the proposition is true, or that it is false ; and each of these
assertions is a secondary proposition. ‘‘It is true that the
sun shines”; ‘‘It is not true that the planets shine by their
own light”; are examples of this kind. In the former ex-
ample the Proposition ‘‘ The sun shines,” is asserted to be
true. In the latter, the Proposition, ‘‘ The planets shine by
their own light,” is asserted to be false. Secondary proposi-
tions also include all judgments by which we express a relation
-or dependence among propositions. To this class or division
we may refer conditional propositions, as, ‘‘ If the sun shine
the day will be fair.” Also most disjunctive propositions, as,
‘¢Kither the sun will shine, or the enterprise will be post-
poned.” In the former example we express the dependence
of the truth of the Proposition, ‘‘ The day will be fair,” upon
the truth of the Proposition, ‘‘ The sun will shine.” In the
latter we express a relation between the two Propositions,
‘©The sun will shine,” ‘‘ The enterprise will be postponed,”
implying that the truth of the one excludes the truth of the
other. To the same class of secondary propositions we must

170 THE LAWS. OF THOUGHT

also refer all those propositions which assert the simultaneous

truth or falsehood of propositions, as, ‘‘It is not true both
that ‘the sun will shine’ and that ‘the journey will be post-
poned.’” The elements of distinction which we have noticed
may even be blended together in the same secondary pro-
position. It may involve both the disjunctive element ex-
pression by ezther, or, and the conditional element expressed
by zf, in addition to which, the connected propositions may
themselves be of a compound [161] character. /f ‘‘ the sun
shine,” avd ‘‘ leisure permit,” then ezther ‘‘the enterprise shall
be commenced,” 97. ‘‘ some preliminary step shall be taken.”
In this example a number of propositions are connected
together, not arbitrarily and unmeaningly, but in such a
manner as to express a definite connexion between them,—
a connexion having reference to their respective truth or
falsehood. This combination, therefore, according to our
definition, forms a Secondary Proposition.

The theory of Secondary Propositions is deserving of at-
tentive study, as well on account of its varied applications,
as for that close and harmonious analogy, already referred to,
which it sustains with the theory of Primary Propositions.
Upon each of these points I desire to offer a few further
observations.

3. I would in the first place remark, that it is in the form
of secondary propositions, at least as often as in that of
primary propositions, that the reasonings of ordinary life are
exhibited. The discourses, too, of the moralist and the meta-
physician are perhaps less often concerning things and their
qualities, than concerning principles and hypotheses, con-
cerning truths and the mutual connexion and relation of
truths. The conclusions which our narrow experience sug-
gests in relation to the great questions of morals and society
yet unsolved, manifest, in more ways than one, the limita-
tions of their human origin; and though the existence of
universal principles is not to be questioned, the partial formulz
which comprise our knowledge of their application are subject
to conditions, and exceptions, and failure. Thus, in those

|
3

Ι

OF SECONDARY PROPOSITIONS L741

- departments of inquiry which, from the nature of their subject-
matter, should be the most interesting of all, much of our
actual knowledge is hypothetical. That there has been a
strong tendency to the adoption of the same forms of thought
in writers on speculative philosophy, will hereafter appear.
Hence the introduction of a general method for the discussion
of hypothetical and the other varieties of secondary proposi-
tions, will open to us a more interesting field of applications
than we have before met with.

4. The discussion of the theory of Secondary Propositions
is in the next place interesting, from the close and remarkable
analogy which it bears with the theory of Primary Proposi-
tions. It [162] will ‘appear, that the formal laws to which
the operations of the mind are subject, are identical in ex-
pression in both cases. The mathematical processes which
are founded on those laws are, therefore, identical also.
Thus the methods which have been investigated in the former
portion of this work will continue to be available in the new
applications to which we are about to proceed. But while
the laws and processes of the method remain unchanged, the
rule of interpretation must be adapted to new conditions.
Instead of classes of things, we shall have to substitute pro-
positions, and for the relations of classes and individuals, we
shall have to consider the connexions of propositions or of
events. Still, between the two systems, however differing
in purport and interpretation, there will be seen to exist a
pervading harmonious relation, an analogy which, while it
serves to facilitate the conquest of every yet remaining diffi-
culty, is of itself an interesting subject of study, and a con-
clusive proof of that unity of character which marks the
constitution of the human faculties.

PROPOSITION I

5. To investigate the nature of the connexion of Secondary
Propositions with the tdea of Time.

It is necessary, in entering upon this inquiry, to state

172 THE LAWS. CF THOCGAT

clearly the nature of the analogy which connects Secondary
with Primary Propositions.

ia ae
SS

Primary Propositions express relations among things, —

viewed as component parts of a universe within the limits of
which, whether coextensive with the limits of the actual uni-
verse or not, the matter of our discourse is confined. The
relations expressed are essentially swbstantzve. Some, or all,
or none, of the members of a given class, ave also members
of another class. The subjects to which primary proposi-
tions refer—the relations amongst those subjects which they
express—are all of the above character.

But in treating of secondary propositions, we find ourselves
concerned with another class both of subjects and relations.
For the subjects with which we have to do are themselves
propositions, so that the question may be asked,—Can we
regard these subjects [163] also as ¢hzngs, and refer them,
by analogy with the previous case, to a universe of their own ?
Again, the relations among these subject propositions are
relations of coexistent truth or falsehood, not of substantive
equivalence. Wedo not say, when expressing the connexion
of two distinct propositions, that the one zs the other, but
use some such forms of speech as the following, according to

the meaning which we desire to convey: ‘‘ Ezther the pro-
position X is true, ov the proposition Y is true”; ‘‘If the
proposition Y is true, the proposition Y is true”; ‘‘ The

propositions X and Y are jointly true”; and so on.

Now, in considering any such relations as the above, we
are not called upon to inquire into the whole extent of their
possible meaning (for this might involve us in metaphysical
questions of causation, which are beyond the proper limits
of science); but it suffices to ascertain some meaning
which they undoubtedly possess, and which is adequate
for the purposes of logical deduction. Let us take, as an
instance for examination, the conditional proposition, ‘‘ If
the proposition XY is true, the proposition Y is true.” An
undoubted meaning of this proposition is, that the time
in which the proposition X is true, is ¢¢me in which the

OF SECONDARY PROPOSITIONS 173

proposition Y is true. This indeed is only a relation of
coexistence, and may or may not exhaust the meaning of
the proposition, but it is a relation really involved in the
statement of the proposition, and further, it suffices for all
the purposes of logical inference.

The language of common life sanctions this view of the
essential connexion of secondary propositions with the notion
of time. Thus we limit the application of a primary pro-
position by the word ‘‘some,” but that of a secondary pro-
position by the word ‘‘sometimes.” To say, ‘‘ sometimes
injustice triumphs,” is equivalent to asserting that there
are times in which the proposition ‘‘ Injustice now triumphs,”
is a true proposition. There are indeed propositions, the
truth of which is not thus limited to particular periods or
conjunctures ; propositions which are true throughout all
time, and have received the appellation of ‘‘ eternal truths.”
The distinction must be familiar to every reader of Plato
and Aristotle, by the latter of whom, especially, it is em-
ployed to denote the contrast between the abstract verities
of science, such as the propositions of geometry which are
always [164] true, and those contingent or phanomenal
relations of things which are sometimes true and sometimes
false. But the forms of language in which both kinds of
propositions are expressed manifest a common dependence
upon the idea of time ; in the one case as limited to some
finite duration, in the other as stretched out to eternity.

6. It may indeed be said, that in ordinary reasoning we
are often quite unconscious of this notion of time involved
in the very language we are using. But the remark, how-
ever just, only serves to show that we commonly reason by
the aid of words and the forms of a well-constructed language,
without attending to the ulterior grounds upon which those
very forms have been established. The course of the present
investigation will afford an illustration of the very same
principle. I shall avail myself of the notion of time in
order to determine the laws of the expression of secondary
propositions, as well as the laws of combination of the sym-

174 THE LAWS OF THOUGHT

bols by which they are expressed. But when those laws and —
those forms are once determined, this notion of time (essen- —
tial, as I believe it to be, to the above end) may practically

be dispensed with. We may then pass from the forms of ὦ
common language to the closely analogous forms of the |
symbolical instrument of thought here developed, and use ἢ
its processes, and interpret its results, without any conscious ~
recognition of the idea of time whatever.

PROPOSITION II
7. To establish a system of notation for the expression of —
Secondary Propositions, and to show that the symbols which
tt involves are subject to the same laws of combination as
the corresponding symbols employed in the expression of
Primary Propositions.
Let us employ the capital letters Y, Y, Z, to denote the
elementary propositions concerning which we desire to make
some assertion touching their truth or falsehood, or among
which we seek to express some relation in the form of a
secondary proposition. And let us employ the corresponding
small letters x, v, 5, considered as expressive of mental opera-
tions, in the following [165] sense, viz. : Let + represent an Ὁ
act of the mind by which we fix our regard upon that portion —
of time for which the proposition X is true; and let this
meaning be understood when it is asserted that x denotes the
time for which the proposition X is true. Let us further
employ the connecting signs +, —, =, &c., in the follow-
ing sense, viz.: Let x+y denote the aggregate of those
portions of time for which the propositions X and Y are
respectively true, those times being entirely separated from
each other. Similarly let +—y denote that remainder of
time which is left when we take away from the portion of
time for which XY is true, that (by supposition) included
portion for which Y is true. Also, let r=y denote that
the time for which the proposition XX is true, is identical
with the time for which the proposition Y is true. We shall
term x the representative symbol of the proposition X, &c.

ον ee ee ΤᾺ

ee ee ΜΡ ΡΨ ee

OF SECONDAKY PROPOSITIONS 175

From the above definitions it will follow, that we shall

always have
XA+AVYVHVH+X,
for either member will denote the same aggregate of time.

Let us further represent by xy the performance in suc-
cession of the two operations represented by y and 4%, z.e.
the whole mental operation which consists of the following
elements, viz., Ist, The mental selection of that portion of
time for which the proposition Y is true. 2ndly, The men-
tal selection, out of that portion of time, of such portion as
it contains of the time in which the proposition X is true,—
the result of these successive processes being the fixing of
the mental regard upon the whole of that portion of time
for which the propositions X and Y are both true.

From this definition it will follow, that we shall always have

XY =X. (1)
For whether we select mentally, first that portion of time
for which the proposition Y is true, then out of the result
that contained portion for which X is true; or first, that
portion of time for which the proposition X is true, then
out of the result that contained portion of it for which the
proposition Y is true; we shall arrive at the same final
result, viz., that portion of time for which the propositions
X and Y are both true.

[166] By continuing this method of reasoning it may be
established, that the laws of combination of the symbols 4,
y, 2, &c., in the species of interpretation here assigned to
them, are identical in expression with the laws of combina-
tion of the same symbols, in the interpretation assigned to
them in the first part of this treatise. The reason of this
final identity is apparent. For in both cases it is the same
faculty, or the same combination of faculties, of which we
study the operation ; operations, the essential character of
which is unaffected, whether we suppose them to be en-
gaged upon that universe of things in which all existence is
contained, or upon that whole of time in which all events

176 TAR LAWS Ok THOUGH &

are realized, and to some part, at least, of which all asser-
tions, truths, and propositions, refer.

Thus, in addition to the laws above stated, we shall have
by (4), Chap. IJ., the law whose expression is

u(y +2) = xy + x2 ; (2)
and more particularly the fundamental law of duality (2)
Chap. II., whose expression is

oe, Ole xl τ ΞΘ: (3)

a law, which while it serves to distinguish the system of
thought in Logic from the system of thought in the science
of quantity, gives to the processes of the former a com-
pleteness and a generality which they could not otherwise
possess,

8. Again, as the law (3) (as well as the other laws) is
satisfied by the symbols 0 and 1, we are led, as before, to
inquire whether those symbols do not admit of interpretation
in the present system of thought. The same course of reason-
ing which we before pursued shows that they do, and war-
rants us in the two following positions, viz. :

ist, That in the expression of secondary propositions, o
represents nothing in reference to the element of time.

2nd, That in the same system 1 represents the universe,
or whole of time, to which the discourse is supposed in any
manner to relate.

As in primary propositions the universe of discourse is
sometimes limited to a small portion of the actual universe
of things, and is sometimes co-extensive with that universe ;
so in secon-[167]dary propositions, the universe of discourse
may be limited to a single day or to the passing moment,
or it may comprise the whole duration of time. It may,
in the most literal sense, be ‘‘ eternal.” Indeed, unless there
is some limitation expressed or implied in the nature of the
discourse, the proper interpretation of the symbol 1 in second-
ary propositions is ‘‘eternity”; even as its proper inter-
pretation in the primary system is the actually existent
universe,

OF SECONDARY PROPOSITIONS 177

9. Instead of appropriating the symbols z, Ps Z, to the re-
presentation of the truths of propositions, we might with
equal propriety apply them to represent the occurrence of
events. In fact, the occurrence of an event both implies,
and is implied by, the truth of a proposition, viz., of the
proposition which asserts the occurrence of the event. The
one signification of the symbol # necessarily involves the
other. It will greatly conduce to convenience to be able to
employ our symbols in either of these really equivalent inter-
pretations which the circumstances of a problem may suggest
to us as most desirable ; and of this liberty I shall avail my-
self whenever occasion requires. In problems of pure Logic
I shall consider the symbols 4, y, &c. as representing ele-
mentary propositions, among which relation is expressed in
the premises. In the mathematical theory of probabilities,
which, as before intimated (I. 12), rests upon a basis of
Logic, and which it is designed to treat in a subsequent
portion of this work, I shall employ the same symbols to
denote the simple events, whose implied or required frequency
of occurrence it counts among its elements.

PROPOSITION III

10. 70 deduce general Rules for the expression of Secondary
Propositions.

In the various inquiries arising out of this Proposition,
fulness of demonstration will be the less necessary, because
of the exact analogy which they bear with similar inquiries
already completed with reference to primary propositions.
We shall first consider the expression of terms; secondly,
that of the propositions by which they are connected.

[168] As 1 denotes the whole duration of time, and x
that portion of it for which the proposition X is true, 1 --ὰα
will denote that portion of time for which the proposition Χ
is false.

Again, as xy denotes that portion of time for which the

propositions X and Y are both true, we shall, by combining
12

178 THE LAWS OF THOUGHT

this and the previous observation, be led to the following
interpretations, viz. : Ἷ
The expression +#(1—y) will represent the time ἀυγίησ,
which the proposition YY is true, and the proposition Y false.
The expression (1 -- .)(1 --- 7) will represent the time during
which the propositions XY and Y are simultaneously false.
The expression #(1-—y)+y(1—*%) will express the time

during which either αὶ is true or Y true, but not both ; for
that time is the sum of the times in which they are singly —
and exclusively true. The expression ry +(1 —%)(1 —y) will -
express the time during which X and Y are either both true

or both false.

If another symbol z presents itself, the same principles —
remain applicable. Thus zyz denotes the time in which —
the propositions X, Y, and Z are simultaneously true Βα
(1 —*)(1 —y)(1 — 2) the time in which they are simultaneously —
false; and the sum of these expressions would denote the >

time in which they are either true or false together.

The general principles of interpretation involved in the ~

above examples do not need any further illustrations or more
explicit statement.

11. The laws of the expression of propositions may now
be exhibited and studied in the distinct cases in which they

present themselves. There is, however, one principle of!
fundamental importance to which I wish in the first place to—

direct attention. Although the principles of expression

Pe 4. ἢ

which have been laid down are perfectly general, and enable —
us to limit our assertions of the truth or falsehood of pro- ἢ
positions to any particular portions of that whole of time —

(whether it be an unlimited eternity, or a period whose
beginning and whose end are definitely fixed, or the passing

moment) which constitutes the universe of our discourse, |

yet, in the actual procedure of human reasoning, such limita-

tion is not commonly employed. When we assert that a_
proposition is true, we generally mean that it is true through-
out the whole [169] duration of the time to which our |

discourse refers; and when different assertions of the

OF SECONDARY PROPOSITIONS © 179

i unconditional truth or falsehood of propositions are jointly
_ made as the premises of a logical demonstration, it is to the
same universe of time that those assertions are referred, and

not to particular and limited parts of it. In that necessary
matter which is the object or field of the exact sciences every
assertion of a truth may be the assertion of an ‘‘ eternal
truth.” In reasoning upon transient phenomena (as of some
social conjuncture) each assertion may be qualified by an
immediate reference to the present time, ‘‘ Now.” But in
both cases, unless there is a distinct expression to the con-
trary, it is to the same period of duration that each separate
proposition relates. The cases which then arise for our con-
sideration are the following :

ist. Zo express the Proposition, ‘‘ The proposition X ts
true.”

Weare here required to express that within those limits of

time to which the matter of our discourse is confined the pro-

position XY is true. Now the time for which the proposition
X is true is denoted by x, and the extent of time to which
our discourse refers is represented by 1. Hence we have

ατεὶ (4)

as the expression required.

2nd. 70 express the Proposttion, ‘‘ The proposition X ts
false.”

We are here to express that within the limits of time to
which our discourse relates, the proposition YX is false ; or
that within those limits there is no portion of time for which
it is true. Now the portion of time for which it is true is 4.
Hence the required equation will be

x= 0. (5)

This result might also be obtained by equating to the whole
duration of time 1, the expression for the time during which
the proposition X is false, viz, 1-#. This gives

I-x=I,

whence x= 0.

180 THE LAWS: OF ΤΗΝ

3rd. 70 express the disjunctive Proposttion, ‘‘ Either the
pro-[170] position X ἐς true or the proposition Υ ts true”; wt ©

being thereby implied that the said propositions are mutually
exclusive, that ts to say, that one only of them ts true.

The time for which either the proposition X is true or the

proposition Y is true, but not both, is represented by the

expression 4(1—y)+y(1—*). Hence we have

x(1 —y) + y(t - x)= 1, (6)

for the equation required.
If in the above Proposition the particles ezther, or, are
supposed not to possess an absolutely disjunctive power, so
that the possibility of the simultaneous truth of the proposi-
tions X and Y is not excluded, we must add to the first
member of the above equations the term xy. We shall thus
have
xy χᾷι -- )) Ὁ (α -- α)ν Ξε,
or x+(1-x)yy=t.

(7)

Ath. To express the conditional Proposition, ‘‘ If the pro-
position Y ts true, the proposttion X ts true.”

Since whenever the proposition Y is true, the proposition
X is true, it is necessary and sufficient here to express, that
the time in which the proposition Y is true is time in which
the proposition XY is true ; that is to say, that it is some in-
definite portion of the whole time in which the proposition
X is true. Now the time in which the proposition Y is true
is y, and the whole time in which the proposition YX is true
is x Let v be a symbol of time indefinite, then will vx re-
present an indefinite portion of the whole time x Accord-

ingly, we shall have
Y=Ux

as the expression of the proposition given.

12. When v is thus regarded as a symbol of time indefinite,
vx may be understood to represent the whole, or an indefinite
part, or no part, of the whole time +; for any one of these
meanings may be realized by a particular determination of

i

P

OF SECONDARY PROPOSITIONS 181

the arbitrary symbol v. Thus, if v be determined to repre-
sent a time in which the whole time + is included, vx will
represent the whole time x. If v be determined to represent
a time, some part of which is in-[1’71]cluded in the time 2,
but which does not fill up the measure of that time, vx will
represent a part of the time x If, lastly, v is determined
to represent a time, of which no part is common with any
part of the time +, vx will assume the value o, and will be
equivalent to ‘‘no time,” or ‘‘ never.”
Now it is to be observed that the proposition, ‘‘lf Y is
true, X is true,” contains no assertion of the truth of either
of the propositions X and Y. It may equally consist with
the supposition that the truth of the proposition Y is a con-
dition indispensable to the truth of the proposition α΄, in
which case we shall have v=1 ; or with the supposition that
although Y expresses a condition which, when realized,
assures us of the truth of Y, yet X may be true without
implying the fulfilment of that condition, in which case v
denotes a time, some part of which is contained in the whole
time x; or, lastly, with the supposition that the proposition
Y is not true at all, in which case v represents some time,
no part of which is common with any part of the time Ὁ
All these cases are involved in the general supposition that
Εν is a symbol of time indefinite.

5th. 70 express a proposition in which the condztional and
the disjunctive characters both exist.

The general form of a conditional proposition is, ‘‘If Y is
true, X is true,” and its expression is, by the last section,
y=vx, We may properly, in analogy with the usage which
has been established in primary propositions, designate Y and
X as the terms of the conditional proposition into which they
enter ; and we may further adopt the language of the ordinary
Logic, which designates the term Y, to which the particle
if is attached, the ‘‘ antecedent” of the proposition, and the
term X the ‘‘ consequent.”

Now instead of the terms, as in the above case, being simple

182 THE LAWS OF THOUGHAT

propositions, let each or either of them be a disjunctive pro- |
position involving different terms connected by the particles _
either, or, as in the following illustrative examples, in which —

X, Y, Z, &c. denote simple propositions.

Ist. If either XY is true or Y is true, then Z is true.
2nd. If X is true, then either Y is true or Z true.

[172] 3rd. If either XY is true or Y is true, then either Z —

and W are both true, or they are both false.

It is evident that in the above cases the relation of the —
antecedent to the consequent is not affected by the circum-_
stance that one of those terms or both are of a disjunctive »
character. Accordingly it is only necessary to obtain, in con-_

formity with the principles already established, the proper

expressions for the antecedent and the consequent, to affect —
the latter with the indefinite symbol Ὁ, and to equate the —
results. Thus for the propositions above stated we shall have ©

the respective equations,

Ist. x(1 -- γ) - (1 - x)y=02.
2nd. x=o{ y(1-2)+2(1-y)}.
3rd, x(1 —y) +y(1 -- x) = σευ 4+ (1 -2){1 -- w)}.

The rule here exemplified is of general application.

Cases in which the disjunctive and the conditional elements |
enter ina manner different from the above into the expression —

of a compound proposition are conceivable, but I am not

aware that they are ever presented to us by the natural ©
exigencies of human reason, and 1 shall therefore refrain —

from any discussion of them. No serious difficulty will arise
from this omission, as the general principles which have
formed the basis of the above applications are perfectly
general, and a slight effort of thought will adapt them to any
imaginable case. ,

13. In the laws of expression above stated those of inter-
pretation are implicitly involved. The equation

x=I

= “

OF SECONDARY PROPOSITIONS 183

must be understood to express that the proposition X is true ;
the equation
a -0 Φ

that the proposition X is false. The equation
Gy a

will express that the propositions Y and Y are both true
together ; and the equation

xy=0

that they are not both together true.
[173] In like manner the equations

x(t -y) +y(t-@)=1,
x(t —y) + H(t — 4) =0,

will respectively assert the truth and the falsehood of the
disjunctive Proposition, ‘‘ Either X is true or Y is true.”
The equations

y=UX

y= (1 -- x)
will respectively express the Propositions, ‘‘If the proposi-
tion Y is true, the proposition X is true.” ‘‘If the proposi-
tion Y is true, the’ proposition 1 is false.”

Examples will frequently present themselves, in the suc-
ceeding chapters of this work, of a case in which some terms
of a particular member of an equation are affected by the in-
definite symbol v, and others not so affected. The following
instance will serve for illustration. Suppose that we have

Y =XZ+UxK(I — 2).

Here it is implied that the time for which the proposition VG
is true consists of all the time for which X and Z are together
true, together with an indefinite portion of the time for which
X is true and Z false. From this it may be seen, 1st, That
if Vis true, either Y and Z are together true, or X is true
and Z false ; 2ndly, If X and Z are together true, Y is true.
The latter of these may be called the reverse interpretation,
and it consists in taking the antecedent out of the second

184 THE LAWS, Of ΟΥΑΙ.

member, and the consequent from the first member of the —

equation. The existence of a term in the second member,

whose coéfficient is unity, renders this latter mode of in- |

terpretation possible. The general principle which it involves
may be thus stated:

14. PRINCIPLE.—Axny constituent term or terms in a par-
ticular member of an equation which have for their coeffictent
unity, may be taken as the antecedent of a proposition, of which
all the terms tn the other member form the consequent.

Thus the equation
YHHZ+OK(T = 2) + (1 $%)(1 -- 2)
would have the following interpretations :

[174] DrrECT INTERPRETATION. —/J/ the proposttion Y is
true, then either X and Z are true, or X ts trueand Z false,
or X and Z are both false.

REVERSE INTERPRETATION. —/f ezther X and Z are true,
or X and Z are false, ΤΥ ts true,

The aggregate of these partial interpretations will express
the whole significance of the equation given.

15. We may here call attention again to the remark, that
although the idea of time appears to be an essential element
in the theory of the interpretation of Secondary Propositions,
it may practically be neglected as soon as the laws of ex-
pression and of interpretation are definitely established. The
forms to which those laws give rise seem, indeed, to corre-
spond with the forms of a perfect language. Let us imagine
any known or existing languages freed from idioms and
divested of superfluity, and let us express in that language
any given proposition in a manner the most simple and
literal, —the most in accordance with those principles of pure
and universal thought upon which all languages are founded,
of which all bear the manifestation, but from which all have
more or less departed. The transition from such a language
to the notation of analysis would consist of no more than the
substitution of one set of signs for another, without essential

Pe ee ee

al i ta

bas Sant”

OF SECONDARY PROPOSITIONS 185

change eitherof form or character. For the elements, whether
things or propositions, among which relation is expressed, we
should substitute letters ; for the disjunctive conjunction we
should write +; for the connecting copula or sign of relation,
we should write =. This analogy I need not pursue. Its
reality and completeness will be made more apparent from
the study of those forms of expression which will present
themselves in subsequent applications of the present theory,
viewed in more immediate comparison with that imperfect
yet noble instrument of thought-—the English language.

16. Upon the general analogy between the theory of Pri-
mary and that of Secondary Propositions, I am desirous of
adding a few remarks before dismissing the subject of the
present chapter.

We might undoubtedly have established the theory of Pri-
mary Propositions upon the simple notion of space,in the same
[175] way as that of Secondary Propositions has been estab-
lished upon the notion of time. Perhaps, had this been done,
the analogy which we are contemplating would have been
in somewhat closer accordance with the view of those who
regard space and time as merely ‘‘ forms of the human under-
standing,” conditions of knowledge imposed by the very
constitution of the mind upon all that is submitted to its
apprehension. But this view, while on the one hand it is
incapable of demonstration, on the other hand ties us down
to the recognition of ‘‘ place,” τὸ ποῦ, as an essential cate-
gory of existence. The question, indeed, whether it is so
or not, lies, I apprehend, beyond the reach of our faculties ;
but it may be, and I conceive has been, established, that
the formal processes of reasoning in primary propositions do
not require, as an essential condition, the manifestation in
space of the things about which we reason ; that they would
remain applicable, with equal strictness of demonstration, to
forms of existence, if such there be, which lie beyond the
realm of sensible extension. It is a fact, perhaps, in some
degree analogous to this, that we are able in many known
examples in geometry and dynamics, to exhibit the formal

186 THE LAWS OF THOUGHT

analysis of problems founded upon some intellectual con-
ception of space different from that which is presented to us
by the senses, or which can be realized by the imagination. *
I conceive, therefore, that the idea of space is not [176]
essential to the development of a theory of primary proposi-
tions, but am disposed, though desiring to speak with dif-
fidence upon a question of such extreme difficulty, to think
that the idea of time is essential to the establishment of a
theory of secondary propositions. There seem to be grounds
for thinking, that without any change in those faculties which
are concerned in reasoning, the manifestation of space to the
human mind might have been different from what it is, but
not (at least the same) grounds for supposing that the mani-

festation of time could have been otherwise than we perceive —

it to be. Dismissing, however, these speculations as possibly
not altogether free from presumption, let it be affirmed that
the real ground upon which the symbol 1 represents in primary
propositions the universe of things, and not the space they
occupy, is, that the sign of identity = connecting the mem-
bers of the corresponding equations, implies that the things
which they represent are identical, not simply that they are
found in the same portion of space. Let it in like manner
be affirmed, that the reason why the symbol 1 in secondary
propositions represents, not the universe of events, but the
eternity in whose successive moments and periods they are
evolved, is, that the same sign of identity connecting the
logical members of the corresponding equations implies, not

* Space is presented to us in perception, as possessing the three dimensions
of length, breadth, and depth. But in a large class of problems relating to the
properties of curved surfaces, the rotations of solid bodies around axes, the vibra-
tions of elastic media, &c., this limitation appears in the analytical investigation
to be of an arbitrary character, and if attention were paid to the processes of
solution alone, no reason could be discovered why space should not exist in four
or in any greater number of dimensions. ‘The intellectual procedure in the im-
aginary world thus suggested can be apprehended by the clearest light of analogy.

The existence of space in three dimensions, and the views thereupon of the
religious and philosophical mind of antiquity, are thus set forth by Aristotle: —
Μέγεθος δὲ τὸ μὲν ἐφ ἕν, γραμμή, τὸ δ᾽ ἐπὶ δύο ἐπίπεδον, τὸ δ᾽ ἐπὶ τρία σῶμα:
Καὶ παρὰ ταῦτα οὐκ ἔστιν ἄλλο μέγεθος, διὰ τὸ τρία πάντα εἵναι καὶ τὸ τρὶς πάντῃ.
Κάθαπερ γάρ φασι καὶ οἱ Πυθαγόρειοι, τὸ πᾶν καὶ τὰ πάντα τοῖς τρισὶν ὥρισται.
Τελευτὴ γὰρ καὶ μέσον καὶ ἀρχὴ τὸν ἀριθμὸν ἔχει τὸν τοῦ παντός" ταῦτα δὲ τὸν τῆς
τριάδος. Διὸ παρὰ τῆς φύσεως εἰληφότες ὥσπερ νόμους ἐκείνης, καὶ πρὸς τὰς
ἀγιστείας χρώμεθα τῶν θεῶν τῷ ἀριθμῷ τούτῷ.--- 206 Cela, τ.

i Ee ee Pe TES Ee νας, ΟΝ

OF SECONDARY PROPOSITIONS 187

that the events which those members represent are identical,
but that the times of their occurrence are the same. These
reasons appear to me to be decisive of the immediate question
of interpretation. Ina former treatise on this subject (Mathe-
matical Analysis of Logic, p. 49), following the theory of
Wallis respecting the Reduction of Hypothetical Proposi-
tions, 1 was led to interpret the symbol 1 in secondary pro-
positions as the universe of ‘‘cases” or ‘‘conjunctures of
circumstances”; but this view involves the necessity of a
definition of what is meant by a ‘‘case,” or ‘‘ conjuncture
of circumstances” ; and it is certain, that whatever is involved
in the term beyond the notion of time is alien to the objects,
and restrictive of the processes, of formal Logic.

[177] CHAPTER XII

OF THE METHODS AND PROCESSES TO BE ADOPTED IN
THE TREATMENT OF SECONDARY PROPOSITIONS

1. IT has appeared from previous researches (XI. 7) that
the laws of combination of the literal symbols of Logic are
the same, whether those symbols are employed in the ex-
pression of primary or in that of secondary propositions, the
sole existing difference between the two cases being a differ-
ence of interpretation. It has also been established (V. 6),
that whenever distinct systems of thought and interpreta-
tion are connected with the same system of formal laws, z.e.,
of laws relating to the combination and use of symbols, the
attendant processes, intermediate between the expression of
the primary conditions of a problem and the interpretation
of its symbolical solution, are the same in both. Hence, as
between the systems of thought manifested in the two forms
of primary and of secondary propositions, this community
of formal law exists, the processes which have been estab-
lished and illustrated in our discussion of the former class of
propositions will, without any modification, be applicable to
the latter.

2. Thus the laws of the two fundamental processes of
elimination and development are the same in the system of
secondary as in the system of primary propositions. Again,
it has been seen (Chap. VI. Prop. II.) how, in primary pro-
positions, the interpretation of any proposed equation devoid
of fractional forms may be effected by developing it into a
series of constituents, and equating to o every constituent
whose coefficient does not vanish. To the equations of
secondary propositions the same method is applicable, and

the interpreted result to which it finally conducts us is, as
188

EMETHODS IN SECONDARY PROPOSITIONS 189

in the former case (VI. 6), a system of co-existent denials.
- But while in the former case the force of those denials is ex-
pended upon the existence of certain classes of things, in the
latter it relates to the truth of certain combinations of the:
ele-[178]mentary propositions involved in the terms of the
given premises. And as in primary propositions it was seen
that the system of denials admitted of conversion into various
other forms of propositions (VI. 7), &c., such conversion will
be found to be possible here also, the sole difference consist-
ing not in the forms of the equations, but in the nature of
their interpretation.

3. Moreover, as in primary propositions, we can find the
expression of any element entering into a system of equations,
in terms of the remaining elements (VI. 10), or of any selected
number of the remaining elements, and interpret that ex-
pression into a logical inference, the same object can be
accomplished by the same means, difference of interpretation
alone excepted, in the system of secondary propositions.
The elimination of those elements which we desire to banish
from the final solution, the reduction of the system toa single
equation, the algebraic solution and the mode of its develop-
ment into an interpretable form, differ in no respect from the
corresponding steps in the discussion of primary propositions,

To remove, however, any possible difficulty, it may be de-
sirable to collect under a general Rule the different cases which
present themselves inthe treatment of secondary propositions.

RULE. — Express symbolically the gtven propositions
BXI. 11).

Eliminate separately from each equation in which it ts found
the indefinite symbol v (VII. 5).

Eliminate the remaining symbols which tt ts desired to
banish from the final solution: always before elimination re-
ducing to a single equation those equations in whtch the symbol
or symbols to be eliminated are found (V\II. 7). Collect the
resulting equations into a single equation V =o,

Then proceed according to the particular form tn which tt ts
desired to express the final relation, as—

190 THE LAWS OF THOUGHT

ist. 77 in the form of a denial, or system of dentals, develop
)
coefficients do not vanish.
andly. [fin the form of a disjunctive proposition, equate
to 1 the sum of those constituents whose coeffictents vanish.

the function V, and equate to 0 all those constituents whose

3

4

grdly. If in the form of a conditional proposition having a
sim-[179] ple element, as x or 1—4, for its antecedent, deter-
mine the algebraic expression of that element, and develop —

that expression.

4athly. 17 in the form of a conditional proposition having a
compound expression, as XY, aty+(1—a*)\(1-y), &e., for tts
antecedent, equate that expression to a new symbol t, and
determine t as a developed function of the symbols which are
to appear in the consequent, either by ordinary methods or by
the spectal method (1X. 9).

sthly. Juterpret the results by (ΧΙ. 13, 14).

Tf it only be desired to ascertain whether a particular ele-
mentary proposition x ts true or false, we must eliminate all
the symbols but «x; then the equation x=1 will indtcate that
the proposition ts true, x=0 that it ἐς false, o=0 that the pre-
mises are insufficient to determine whether tt is true or false.

4. Ex. 1.—The following prediction is made the subject
of a curious discussion in Cicero’s fragmentary treatise, De
Fato :—‘‘Si quis (Fabius) natus est oriente Canicula, is in
mari non morietur.” I shall apply to it the method of this
chapter. Let represent the proposition, ‘‘ Fabius was born
at the rising of the dogstar”; x the proposition, ‘‘ Fabius
will die in the sea.” In saying that x represents the proposi-
tion, ‘‘ Fabius, &c.,” it is only meant that x is a symbol so
appropriated (XI. 7) to the above proposition, that the equa-
tion += 1 declares, and the equation r=0 denies, the truth of
that proposition. The equation we have to discuss will be

γτεσίι -- <x). (1)
And first, let it be required to reduce the given proposition
to a negation or system of negations (XII. 3). We have, on

transposition,
y-U1-—X)=0.,

_METHODS IN SECONDARY PROPOSITIONS τοι

Eliminating v,
7» {ὦ - x)}=0,

or, Pele) 9;
or, i =O; (2)
The interpretation of this result 15 :—‘‘It is not true that

Fabius was born at the rising of the dogstar, and will die in
the sea.” [180] Cicero terms this form of proposition, ‘‘Con- |
junctio ex repugnantibus ” ; and he remarks that Chrysippus
thought in this way to evade the difficulty which he imagined
to exist in contingent assertions respecting the future : ‘‘ Hoc
loco Chrysippus zstuans falli sperat Chaldaos ceterosque
divinos, neque eos usuros esse conjunctionibus ut ita sua
percepta pronuntient : Si quis natus est oriente Canicula is
in mari non morietur ; sed potius ita dicant: Non et natus
est quis oriente Canicula, et in mari morietur. O licentiam
jocularem!. . . . Multa genera sunt enuntiandi, nec ullum
distortius quam hoc quo Chrysippus sperat Chaldzos con-
tentos Stoicorum causa fore.”—Czc. De Fato, 7, 8.

5. To reduce the given proposition to a disjunctive form.

The constituents not entering into the first member of
(2) are

x(1 -7) (1 — x), (1 — x)(1 ἘΣ
Whence we have
y(t εχ αὐ) 1. (3)

The interpretation of which is :—Lzther Fabius was born at
the rising of the dogstar, and will not perishin the sea, or he
was not born at the rising of the dogstar, and will perish in
the sea; or he was not born at the rising of the dogstar, and
will not perish in the sea.

In cases like the above, however, in which there exist con-
stituents differing from each other by a single factor, it is,
as we have seen (VII. 15), most convenient to collect such
constituents into a single term. If.we thus connect the first
and third terms of (3), we have

(1-y)e+1-X=1;

192 THE LAWS OF THOUGHT

and if we similarly connect the second and third, we have

y(I-—X)+FI-YH=t.
These forms of the equation severally give the inter-
pretations—

Either Fabius was not born under the dogstar, and will die
in the sea, or he will not ate in the sea.

Either Fabius was born under the dogstar, and will not
die in the sea, or he was not born under the dogstar.

[181] It is evident that these interpretations are strictly
equivalent to the former one.

Let us ascertain, in the form of a conditional proposition,
the consequences which flow from the hypothesis, that
‘¢Fabius will perish in the sea.”

In the equation (2), which expresses the result of the elimi-
nation of v from the original equation, we must seek to
determine x as a function of γ.

We have

ο ο ;
x= — =oy+-(1I —y) on expansion,
oy 1?)

Oo
ἡ ὅ5:- = Es
the interpretation of which is, —// Fabius shall dee in the sea,
he was not born at the rising of the dogstar.

These examples serve in some measure to illustrate the
connexion which has been established in the previous sections
between primary and secondary propositions, a connexion of
which the two distinguishing features are identity of process
and analogy of interpretation.

6. Ex, 2.—There is a remarkable argument in the second
book of the Repudlic of Plato, the design of which is to prove
the immutability of the Divine Nature. It is a very fine
example both of the careful induction from familiar instances
by which Plato arrives at general principles, and of the clear
and connected logic by which he deduces from them the
particular inferences which it is his object to establish. The
argument is contained in the following dialogue :

Pa ae see Ροο ἐν »

METHODS IN SECONDARY PROPOSITIONS το:

‘*Must not that which departs from its proper form be
changed either by itself or by another thing? Necessarily
so, Are not things which are in the best state least changed
and disturbed, as the body by meats and drinks, and labours,
and every species of plant by heats and winds, and such like
affections? Is not the healthiest and strongest the least
changed? Assuredly. And does not any trouble from
without least disturb and change that soul which is strongest
and wisest? And as to all made vessels, and furnitures,
and garments, according to the same [182] principle, are
not those which are well wrought, and in a good condition,
least changed by time and other accidents? Even so. And
whatever is in a right state, either by nature or by art, or
by both these, admits of the smallest change from any other
thing. So it seems. But God and things divine are in every
sense in the best state. Assuredly. In this way, then, God
should least of all bear many forms? Least, indeed, of all.
Again, should He transform and change Himself? Mani-
festly He must do so, if He is changed at all. Changes He
then Himself to that which is more good and fair, or to that
which is worse and baser? Necessarily to the worse, if He
be changed. For never shall we say that God is indigent
of beauty or of virtue. You speak most rightly, said I,
and the matter being so, seems it to you, O Adimantus,
that God or man wzllingly makes himself in any sense
worse? Impossible, said he. Impossible, then, it is, said
I, that a god should wish to change himself: but ever being
fairest and best, each of them ever remains absolutely in the
same form.”

The premises of the above argument are the following :

ist. If the Deity suffers change, He is changed either by
Himself or by another.

2nd. If He is in the best state, He is not changed by
another.

3rd. The Deity is in the best state.

4th. If the Deity is changed by Himself, He is changed
to a worse state.

13

194 THE LAWS OF THOUGAT

sth. If He acts willingly, He is not changed to a worse ἢ
state.
6th. The Deity acts willingly.
Let us express the elements of these premises as follows :
Let x represent the proposition, ‘‘The Deity suffers
change.”
y, He is changed by Himself.
z, He is changed by another.
s. Pes in the: best. state.
t, He is changed to a worse.
w, He acts willingly.

Then the premises expressed in symbolical language yield,
after elimination of the indefinite class symbol v, the
following equations :

[183] xyz + x(1 —y)(1 — 2) =0, (1)
SZ =0, (2)
oo τὶ (3)
(1 —t)=0, (4)
wt = 0, (5)
w=t. (6)

Retaining x, I shall eliminate in succession 2, s, y, ¢, and w
(this being the order in which those symbols occur in the
above system), and interpret the successive results.

Eliminating z from (1) and (2), we get

xs(I —y)=0. (7)

Eliminating s from (3) and (7),
x(t - 9) =o. (8)

Eliminating y from (4) and (8),
x(1 -?t)=0. (9)

Eliminating Ζ from (5) and (9),
7. τ 0. (10)

Eliminating w from (6) and (10),

oe (11)

These equations, beginning with (8), give the following
results :

METHODS IN SECONDARY PROPOSITIONS τὸς

From (8) we have “τις, therefore, /f the Detty suffers
change, He τς changed by Himself.

From (9), r=At, [f the Deity suffers change, He is changed
to a worse state.

From (10), r=(1 —w). Ifthe Deity suffers change, He

does not act willingly.

From (11), The Detty does not suffer change. This is
Plato’s result.

Now I have before remarked, that the order of elimination
is indifferent. Let us in the present case seek to verify this
fact by eliminating the same symbols in a reverse order,
beginning with w. The resulting equations are,

. [184] f=0, y=0, x(1-z)=0, Ζξο, x=0;
yielding the following interpretations :

God ts not changed to a worse state.
He is not changed by Himself.
Lf He suffers change, He ts changed by another.
He ts not changed by another.
He ἐξ not changed.

We thus reach by a different route the same conclusion.

Though as an exhibition of the power of the method, the
above examples are of slight value, they serve as well as
more complicated instances would do, to illustrate its nature
and character.

7, It may be remarked, as a final instance of analogy
between the system of primary and that of secondary pro-
positions, that in the latter system also the fundamental
equation,

x(1 —x)=0,

admits of interpretation, It expresses the axiom, A pro-
postition cannot at the same time be true and false. Let this

be compared with the corresponding interpretation (111. 15).
Solved under the form

196 THE LAWS OF THOUGHT

ο
ΤΞ 6

x=

Ὁ
= —X,
οΟ

by development, it furnishes the respective axioms: ‘‘A
thing is what it is”: ‘‘Ifa proposition is true, it is true”:
forms of what has been termed ‘‘ The principle of identity.”
Upon the nature and the value of these axioms the most
opposite opinions have been entertained. Some have re-
garded them as the very pith and marrow of philosophy,
Locke devoted to them a Chapter, headed, ‘‘On Trifling
Propositions.” * In both these views there seems to have
been a mixture of truth and error. Regarded as supplant-
ing experience, or as furnishing materials for the vain and
wordy janglings of the schools, such propositions are worse
than trifling. Viewed, on the other hand, as intimately
allied with the very laws and conditions of thought, they
rise into at least a speculative importance.

* Essay on the Human Understanding, Book IV. Chap. viii.

a

[185] CHAPTER XIII

ANALYSIS OF A PORTION OF DR. SAMUEL CLARKE’S
‘DEMONSTRATION OF THE BEING AND ATTRIBUTES
OF ΘΟ AND: ΟἿ A. >PORTION OF : THE “GETHIGA
ORDINE GEOMETRICO DEMONSTRATA” OF SPINOZA

1. THE general order which, in the investigations of the
following Chapter, I design to pursue, is the following. I
shall examine what are the actual premises involved in the
demonstrations of some of the general propositions of the
above treatises, whether those premises be expressed or
implied. By the actual premises I mean whatever proposi-
tions are assumed in the course of the argument, without
being proved, and are employed as parts of the foundation
upon which the final conclusion is built. The premises thus
determined, I shall express in the language of symbols, and
I shall then deduce from them by the methods developed in
the previous chapters of this work, the most important in-
ferences which they involve, in addition to the particular
inferences actually drawn by the authors, I shall in some
instances modify the premises by the omission of some fact
or principle which is contained in them, or by the addition
or substitution of some new proposition, and shall determine
how by such change the ultimate conclusions are affected.
In the pursuit of these objects it will not devolve upon me
to inquire, except incidentally, how far the metaphysical
principles laid down in these celebrated productions are
worthy of confidence, but only to ascertain what conclusions
may justly be drawn from given premises ; and in doing
this, to exemplify the perfect liberty which we possess as
concerns both the choice and the order of the elements of

the final or concluding propositions, viz., as to determining
197

198 EAE LAWS OF 2A OC GH Tf.

what elementary propositions are true or false, and what
are true or false under given restrictions, or in given
combinations.

2. The chief practical difficulty of this inquiry will con-
sist, [186] not in the application of the method to the
premises once determined, but in ascertaining what the
premises are. In what are regarded as the most rigorous
examples of reasoning applied to metaphysical questions, it
will occasionally be found that different trains of thought
are blended together ; that particular but essential parts of
the demonstration are given parenthetically, or out of the
main course of the argument ; that the meaning of a premiss
may be in some degree ambiguous ; and, not unfrequently,
that arguments, viewed by the strict laws of formal reason-
ing, are incorrect or inconclusive. The difficulty of deter-
mining and distinctly exhibiting the true premises of a
demonstration may, in such cases, be very considerable.
But it is a difficulty which must be overcome by all who
would ascertain whether a particular conclusion is proved
or not, whatever form they may be prepared or disposed to
give to the ulterior process of reasoning. It is a difficulty,
therefore, which is not peculiar to the method of this work,
though it manifests itself more distinctly in connexion with
this method than with any other. So intimate, indeed, is
this connexion, that is impossible, employing the method
of this treatise, to form even a conjecture as to the validity
of a conclusion, without a distinct apprehension and exact
statement of all the premises upon which it rests. In the
more usual course of procedure, nothing is, however, more
common than to examine some of the steps of a train of
argument, and thence to form a vague general impression
of the scope of the whole, without any such preliminary and
thorough analysis of the premises which it involves.

The necessity of a rigorous determination of the real
premises of a demonstration ought not to be regarded as an
evil; especially as, when that task is accomplished, every
source of doubt or ambiguity is removed. In employing

CLARKE AND SPINOZA 199

the method of this treatise, the order in which premises are
arranged, the mode of connexion which they exhibit, with
every similar circumstance, may be esteemed a matter of
indifference, and the process of inference is conducted with
a precision which might almost be termed mechanical.

3. The Demonstration of the Being and Attributes of
God consists of a series of propositions or theorems, each
[187] of them proved by means of premises resolvable, for
the most part, into two distinct classes, viz., facts of
observation, such as the existence of a material world, the
phenomenon of motion, &c., and hypothetical principles,
the authority and universality of which are supposed to be
recognised a priort. It is, of course, upon the truth of the
latter, assuming the correctness of the reasoning, that the
validity of the demonstration really depends. But what-
ever may be thought of its claims in this respect, it is un-
questionable that, as an intellectual performance, its merits
are very high. Though the trains of argument of which it
consists are not in general very clearly arranged, they are
almost always specimens of correct Logic, and they exhibit
a subtlety of apprehension and a force of reasoning which
have seldom been equalled, never perhaps surpassed. We
see in them the consummation of those intellectual efforts
which were awakened in the realm of metaphysical inquiry,
at a period when the dominion of hypothetical principles
was less questioned than it now is, and when the rigorous
demonstrations of the newly risen school of mathematical
physics seemed to have furnished a model for their direction.
They appear to me for this reason (not to mention the
dignity of the subject of which they treat) to be deserving of
high consideration ; and I do not deem it a vain or super-
fluous task to expend upon some of them a careful analysis.

4. The Ethics of Benedict Spinoza is a treatise, the object
of which is to prove the identity of God and the universe,
and to establish, upon this doctrine, a system of morals and
of philosophy. The analysis of its main argument is ex-
tremely difficult, owing not to the complexity of the separate

200 THE LAWS. OF 1 BOUGHT

propositions which it involves, but to the use of vague defini-
tions, and of axioms which, through a like defect of clearness,
it is perplexing to determine whether we ought to accept or
to reject. While the reasoning of Dr. Samuel Clarke is in
part verbal, that of Spinoza is so in a much greater degree ;
and perhaps this is the reason why, to some minds, it has
appeared to possess a formal cogency, to which in reality it
possesses no just claim. These points will, however, be con-
sidered in the proper place.

[188] CLARKE’S DEMONSTRATION

PROPOSITION I

5. ‘* Something has existed from eternity.”

The proof is as follows :—

‘‘ For since something now is, ’tis manifest that something
always was. Otherwise the things that now are must have
risen out of nothing, absolutely and without cause. Which
is a plain contradiction in terms. For to say a thing is pro-
duced, and yet that there is no cause at all of that production,
is to say that something is effected when it is effected by
nothing, that is, at the same time when it is not effected at
all. Whatever exists has a cause of its existence, either in
the necessity of its own nature, and thus it must have been of
itself eternal: or in the will of some other being, and then
that other being must, at least in the order of nature and
causality, have existed before it.”

Let us now proceed to analyze the above demonstration.
Its first sentence is resolvable into the following propositions :

Ist. Something is.

2nd. If something is, either something always was, or the
things that now are must have risen out of nothing,

The next portion of the demonstration consists of a proof
that the second of the above alternatives, viz., ‘‘ The things
that now are have risen out of nothing,” is impossible, and
it may formally be resolved as follows:

3rd. If the things that now are have risen out of nothing,

CLARKE AND SPINOZA 201

something has been effected, and at the same time that some-
thing has been effected by nothing.

4th. If that something has been effected by nothing, it
has not been effected at all.

The second portion of this argument appears to be a mere
assumption of the point to be proved, or an attempt to make
that point clearer by a different verbal statement.

The third and last portion of the demonstration contains
a distinct proof of the truth of either the original proposition
to be proved, viz., ‘‘Something always was,” or the point
proved in the second part of the demonstration, viz., the
untenable nature [189] of the hypothesis, that ‘‘ the things
that now are have risen out of nothing.” It is resolvable as

follows :—

5th. If something is, either it exists by the necessity of its
own nature, or it exists by the will of another being.

6th. If it exists by the necessity of its own nature, some-
thing always was.

7th. If it exists by the will of another being, then the
proposition, that the things which exist have arisen out of
nothing, is false.

The last proposition is not expressed in the same form in
the text of Dr. Clarke ; but his expressed conclusion of the
prior existence of another Being is clearly meant as equivalent
to a denial of the proposition that the things which now are
have arisen out of nothing.

It appears, therefore, that the demonstration consists of
two distinct trains of argument: one of those trains compris-
ing what I have designated as the firs¢ and second parts of the
demonstration ; the other comprising the firs¢ and ¢hzrd parts,
Let us consider the latter train.

The premises are :—

Ist. Something is.

2nd. If something is, either something always was, or the
things that now are have risen out of nothing.

3rd. If something is, either it exists in the necessity of its
own nature, or it exists by the will of another being.

202 THE LAWS OF THOUGHT,

4th. lf it exists in the necessity of its own nature, some- —

thing always was.

5th. If it exists by the will of another being, then the |
hypothesis, that the things which now are have risen out of —

nothing, is false.
We must now express symbolically the above proposition.

τοι a= Something is.
y =Something always was.
g= The things which now are, have risen from
nothing.
p=lIt exists in the necessity of its own nature
(2.4. the something spoken of above).
g=It exists by the will of another Being.

[190] It must be understood, that by the expression, Let

x= ‘‘Something is,” is meant no more than that # is the repre-

sentative symbol of that proposition (XI. 7), the equations
x=1, x=0, respectively declaring its truth and its falsehood.
The equations of the premises are :—

Ist; “x= 1 ;

2nd. x=v{y(1 -—2)+2(1-y)} ;

3rd. x=o{p(1-¢g)+9(1-p)};

4th. p=y;

sth. g=v(1-2);

and on eliminating the several indefinite symbols v, we have

c= 407; (1)
a{yz+(1—y)(1 - 2)} =0; (2)
x{pg+(1 -p)(1-9)}=0; (3)

P(r -y)=0; (4)
gz=0. (5)

6. First, I shall examine whether any conclusions are de-
ducible from the above, concerning the truth or falsity of the
single propositions represented by the symbols y, 2, 2, g, viz.,
of the propositions, ‘‘ Something always was” ; ‘‘ The things
which now are have risen from nothing” ; ‘‘ The something
which is exists by the necessity of its own nature”; ‘‘ The
something which is exists by the will of another being.”

ee ee a

CLARKE AND SPINOZA 203

For this purpose we must separately eliminate all the
symbols but y, all these but z, &c. The resulting equation
will determine whether any such separate relations exist.

To eliminate x from (1), (2), and (3), it is only necessary
to substitute in (2) and (3) the value of x derived from (1).
We find as the results,

Pot (HO ἡ), Ὁ (6)
pat (1 -p)(t- 9) Ὁ: (7)
To eliminate 2 we have from (4) and (7), by addition,
p(t —y) + q+ (1 -P)t- 9) =03 (8)
whence we find,
(1 -y)(1-9) .0. (9)

[191] To eliminate g from (5) and (9), we have

gz+(1—-y)(1-g)=0;
whence we find

φ(ι —y) =o. (10)
There now remain but the two equations (6) and (10),
which, on addition, give
yeti-y=o,
Eliminating from this equation z, we have
I-y=o0, or, y=I. (11)
Eliminating from the same equation y, we have
Z=0., {12
The interpretation of (11) is
Something always was.
The interpretation of (12) is
The things which are have not risen from nothing.
Next resuming the system (6), (7), with the two equations
(4), (5), let us determine the two equations involving 2 and 0

respectively.
To eliminate y we have from (4) and (6),

P(r —y) tye + (ι -)α ~ 2) =(0)5
whence, (p+1—z)z=0, οἵ, 22-:-ο. (13)

204 THE LAWS OCF THOUGHT

To eliminate z from (5) and (13), we have

g2t+ pz=0;
whence we get,
Ο-ΞΞο.
There remains then but the equation (7), from which
eliminating g, we have O=0 for the final equation, in 2.

Hence there ts no concluston derivable from the premtses
affirming the simple truth or falsehood of the proposition, ‘The
something which ts exists tn the necessity of tts own nature.”
And as, on eliminating 2, there is the same result, o=0, for
the ultimate equation in g, it also follows, that there ts no
conclusion deductible from the premises as to the simple truth
or falsehood of the proposttion, ‘‘ The something which ts exists
by the will of another Being.”

[192] Of relations connecting more than one of the pro-
positions represented by the elementary symbols, it is need-
less to consider any but that which is denoted by the equation
(7) connecting 2 and g, inasmuch as the propositions repre-
sented by the remaining symbols are absolutely true or false
independently of any connexion of the kind here spoken of.
The interpretation of (7), placed under the form

Alt -g)+9(1-A)=1, is,
The something whtch ts, etther exists in the necesstty of its
own nature, or by the will of another being.

I have exhibited the details of the above analysis with a,
perhaps, needless fulness and prolixity, because in the ex-
amples which will follow, | propose rather to indicate the
steps by which results are obtained, than to incur the danger
of a wearisome frequency of repetition The conclusions
which have resulted from the above application of the method
are easily verified by ordinary reasoning.

The reader will have no difficulty in applying the method
to the other train of premises involved in Dr. Clarke’s first
Proposition, and deducing from them the two first of the
conclusions to which the above analysis has led.

CLARKE AND SPINOZA 205

PROPOSITION II

7. Some one unchangeable and independent Being has
existed from eternity.

The premises from which the above proposition is proved
are the following :

Ist. Something has always existed.

2nd. If something has always existed, either there has
existed some one unchangeable and independent being, or
the whole of existing things has been comprehended in a
succession of changeable and dependent beings.

3rd. If the universe has consisted of a succession of change-
able and dependent beings, either that series has had a cause
from without, or it has had a cause from within.

4th. It has not had a cause from without (because it in-
cludes, by hypothesis, all things that exist).

[193] 5th. It has not had a cause from within (because no
part is necessary, and if no part is necessary, the whole cannot
be necessary).

Omitting, merely for brevity, the subsidiary proofs con-
tained in the parentheses of the fourth and fifth premiss, we
may represent the premises as follows :

eet «= Something has always existed.
y = There has existed some one unchangeable and
independent being.
= There has existed a succession of changeable
and dependent beings.
p = That series has had a cause from without.
g = That series has had a cause from within.

Then we have the following system of equations, viz. :

1St,> X45
and, x«=v{y(1—2) + 2(1 -¥)} 5
3rd. Ζεῦ p(1-g)+(1-P)9};
4th. ~=0;
sth. g=o;

206 THER DAWS Of THOUGHT

which, on the separate elimination of the indefinite symbols '

Vv, gives

I—-x=0; (1)

x{ yz+(1 —y)(1 - 2} =0; (2)
2{pgt+(1-p)(t-g)} Ξο; (3)
2Ξο; (4)

g=0; (5)

The elimination from the above system of x, 2, g, and y,
conducts to the equation
go.
And the elimination of x, 23, g, and z, conducts in a similar
manner to the equation
γε τ.
Of which equations the respective interpretations are:

ist. The whole of existing things has not been compre-
hended in a succession of changeable and dependent beings.

2nd. There has existed some one unchangeable and inde-
pendent being.

[194] The latter of these is the proposition which
Dr. Clarke proves. As, by the above analysis, all the pro-
positions represented by the literal symbols 4, y, 2, 2, g, are
determined as absolutely true or false, it is needless to en-
quire into the existence of any further relations connecting
those propositions together.

Another proof is given of Prop. II., which for brevity I
pass over. It may be observed, that the ‘‘ impossibility of in-
finite succession,” the proof of which forms a part of Clarke’s
argument, has commonly been assumed as a fundamental
principle of metaphysics, and extended to other questions
than that of causation. Aristotle applies it to establish the
necessity of first principles of demonstration ; * the necessity
of an end (the good), in human actions, &c. | There is,
perhaps, no principle more frequently referred to in his
writings. By the schoolmen it was similarly applied to prove
the impossibility of an infinite subordination of genera and

* Metaphystes, 111. 4; Anal. Post. 1. 19, et seq.
+ Nite. Ethics, Book I. cap. ii.

CLARKE AND SPINOZA 207

species, and hence the necessary existence of universals.
Apparently the impossibility of our forming a definite and
complete conception of an infinite series, z.e. of comprehend-
ing it as a whole, has been confounded with a logical incon-
sistency, or contradiction in the idea itself.

8. The analysis of the following argument depends upon
the theory of Primary Propositions,

PROPOSITION III

That unchangeable and independent Being must be self-
existent.

The premises are :—

I. Every being must either have come into existence out
of nothing, or it must have been produced by some external
cause, or it must be self-existent.

2. No being has come into existence out of nothing.

3. The unchangeable and independent Being has not been
produced by an external cause.

For the symbolical expression of the above, let us assume,

[195] += Beings which have arisen out of nothing.
y= Beings which have be produced by an external
cause,
z= Beings which are self-existent.
w= The unchangeable and independent Being.

Then we have

x(t —y)(1 -- 2) + y(t — Δ) -- 2) +2(1 -- x(t -- ))Ξ τ, (1)
X=0, . (2)
w=v(I ~ 4), (3)

from the last of which eliminating v,

wy =0. (4)

Whenever, as above, the value of a symbol is given as Oor 1,

it is best eliminated by simple substitution. Thus the elimi-
nation of 4 gives

Ἅγ.τ 2 1 901 νὺ 1; (5)

or, γε ( -)α - 2) ο. (6)

208 THE LAWS OF THOUGHT

Now adding (4) and (6), and eliminating γ, we get

w(I — 5) =0,

W = UZ}

the interpretation of which is, —The unchangeable and inde-

pendent being is necessarily self-existing.

Of (5), in its actual form, the interpretation is,—Hvery ὦ
being has either been produced by an external cause, or it ts |

self-existent.
9g. In Dr. Samuel Clarke’s observations on the above pro-

position occurs a remarkable argument, designed to prove

that the material world is not the self-existent being above ~

spoken of. The passage to which I refer is the following :
‘Tf matter be supposed to exist necessarily, then in that

necessary existence there is either included the power of

gravitation, or not. If not, then in a world merely material,
and in which no intelligent being presides, there never could
have been any motion ; because motion, as has been already
shown, and is now granted in the question, is not necessary
of itself. But if the [196] power of gravitation be included
in the pretended necessary existence of matter: then, it
following necessarily that there must be a vacuum (as the
incomparable Sir Isaac Newton has abundantly demonstrated
that there must, if gravitation be an universal quality or
affection of matter), it follows likewise, that matter is not a
necessary being. For if a vacuum actually be, then it is
plainly more than possible for matter not to be.” —(pp. 25, 26).

It will, upon attentive examination, be found that the
actual premises involved in the above demonstration are the
following :

ist. If matter is a necessary being, either the property of
gravitation is necessarily present, or it is necessarily absent.

2nd. If gravitation is necessarily absent, and the world is
not subject to any presiding Intelligence, motion does not
exist.

3rd. If the property of gravitation is necessarily present,
the existence of a vacuum is necessary.

Ἐς ΡΥ ΡΝ «σὺν ΝΟΣ,

CLARKE AND: SPINOZA 209

4th. If the existence of a vacuum is necessary, matter is
not a necessary being.

5th. If matter is a necessary being, the world is not sub-
ject to a presiding Intelligence.

6th. Motion exists.

Of the above premises the first four are expressed in the
demonstration ; the fifth is implied in the connexion of its
first and second sentences; and the sixth expresses a fact,
which the author does not appear to have thought it necessary
to state, but which is obviously a part of the ground of his
reasoning. Let us represent the elementary propositions in
the following manner :

Let x= Matter is a necessary being.
y =Gravitation is necessarily present.
¢= Gravitation is necessarily absent.
z= The world is merely material, and not subject
to any presiding Intelligence.
τὺ = Motion exists.
v= A vacuum is necessary.

Then the system of premises will be represented by the
following equations, in which g is employed as the symbol
of time indefinite :

[197] a =g{y(1 -- ἢ + (1 -9)Z}-
tz =9(1 — w).
y= QD.
v=g(I—%X).
coe
w =I.

From which, if we eliminate the symbols g, we have the
following system, viz.°

x{ytt+ (1 —y)(1t -2)} =o. (1)
tzw =0. (2)
Wi —v)=0. (3)
Ux = 0. | (4)
χᾷι — 2) =0. (5)
I-w=o. (6)

14

210 EE LAWS OF THOCG RE

Now if from these equations we eliminate w, v, z, y, and ¢,
we obtain the equation
a,

which expresses the proposition, Matter 7s not a necessary
being. This is Dr. Clarke's conclusion. If we endeavour
to eliminate any other set of five symbols (except the set
v, z, y, t, and %, which would give w=1), we obtain a result
of the form o=o. It hence appears that there are no other
conclusions expressive of the absolute truth or falsehood of any
of the elementary propositions designated by single symbols.

Of conclusions expressed by equations involving two
symbols, there exists but the following, viz.:—J/ the world
is merely material, and not subject to a presiding Intelligence,
gravitation is not necessarily absent, ‘This conclusion is ex-
pressed by the equation

το, whence Ζ-φίι -- ἢ.

If in the above analysis we suppress the concluding premiss,
expressing the fact of the existence of motion, and leave the
hypothetical principles which are embodied in the remaining
premises untouched, some remarkable conclusions follow.
To these I shall direct attention in the following chapter.

10. Of the remainder of Dr. Clarke’s argument I shall
briefly state the substance and connexion, dwelling only on
certain por-[198]tions of it which are of a more complex
character than the others, and afford better illustrations of
the method of this work.

In Prop. IV. it is shown that the substance or essence of
the self-existent being is incomprehensible. The tenor of
the reasoning employed is, that we are ignorant of the essen-
tial nature of all other things,—much more, then, of the
essence of the self-existent being.

In Prop. V. it is contended that ‘‘though the substance or
essence of the self-existent being is itself absolutely incompre-
hensible to us, yet many of the essential attributes of his
nature are strictly demonstrable, as well as his existence.”

In Prop. VI. it is argued that ‘‘ the self-existent being

ΨΥ

πὸ δὲ τος ee ee δι. ον ee — 7

CLARKE AND SPINOZA 2ὉῚ1

must of necessity be infinite and omnipresent ” ; and it is con-
tended that his infinity must be ‘‘an infinity of fulness as
well as of immensity.” The ground upon which the demon-
stration proceeds is, that an absolute necessity of existence
must be independent of time, place, and circumstance, free
from limitation, and therefore excluding all imperfection,
And hence it is inferred that the self-existent being must be
‘¢a most simple, unchangeable, incorruptible being, without
parts, figure, motion, or any other such properties as we find
in matter.”

The premises actually employed may be exhibited as
follows :

1. If a finite being is self-existent, it is a contradiction to
suppose it not to exist.

2. A finite being may, without contradiction, be absent
from one place.

3. That which may without contradiction be absent from
one place may without contradiction be absent from all places.

4. That which may without contradiction be absent from
all places may without contradiction be supposed not to exist.

Let us assume

x= Finite beings.

y= Things self-existent.

g= Things which it is a contradiction to suppose not to
exist,

w= Things which may be absent without contradiction
from one place.

t=Things which without contradiction may be absent
from every place.

[199] We have on expressing the above, and eliminating the
indefinite symbols,

χγᾷ(ι —z)=0. (1)
x(I — 20) ΞΞ ο. (2)
w(t -- ἢ =o. (3)
fz =0. (4)

Eliminating in succession ¢, w, and z, we get

212 THE LAWS Of THOUGHT
i ay -- Ο,

ο
“νι x) ;

the interpretation of which is,— Whatever ts self-existent ts —

infinite.

In Prop. VII. it is argued that the self-existent being must —

of necessity be One. The order of the proof is, that the self-
existent being is ‘‘necessarily existent,” that ‘‘ necessity

absolute in itself is simple and uniform, and without any ©

possible difference or variety,” that all ‘‘ variety or difference
of existence” implies dependence ; and hence that ‘‘ what-
ever exists necessarily is the one simple essence of the self-
existent being.”

The conclusion is also made to flow from the following pre-
mises :—

1. If there are two or more necessary and independent
beings, either of them may be supposed to exist alone.

2. If either may be supposed to exist alone, it is not a con-
tradiction to suppose the other not to exist.

3. If it is nota contradiction to suppose this, there are not
two necessary and independent beings.

Let us represent the elementary propositions as follows :—

“= there exist two necessary independent beings.
y =either may be supposed to exist alone.
#= it is not a contradiction to suppose the other not toexist.

We have then, on proceeding as before,

x(1 -y)=0. (1)
(I —Z)=0. (2)
2X = 0. (3)

[200] Eliminating y and z, we have
xXx=0,

Whence, There do not exist two necessary and independent
beings.
11. ΤῸ the premises upon which the two previous proposi-

tions rest, it is well known that Bishop Butler, who at the time:

eg
:

CLARKE ΑἸΥΣΒΡΙΝΟΖΑ 213

of the publication of the Demonstration, was a student in
a nonconformist academy, made objection in some celebrated
letters, which, together with Dr. Clarke’s replies to them, are
usually appended to editions of the work. The real question
at issue is the validity of the principle, that ‘‘ whatsoever is
absolutely necessary at all is absolutely necessary in every

_ part of space, and in every point of duration,’—a principle

assumed in Dr. Clarke’s reasoning, and explicitly stated in
his reply to Butler’s first letter. In his second communica-
tion Butler says: ‘‘ I do not conceive that the idea of ubiquity
is contained in the idea of self-existence, or directly follows
from it, any otherwise than as whatever exists must exist
somewhere.” hat is to say, necessary existence implies ex-
istence in some part of space, but notin every part. It does
not appear that Dr. Clarke was ever able to dispose effectually
of this objection. The whole of the correspondence is ex-
tremely curious and interesting. The objections of Butler
are precisely those which would occur to an acute mind
impressed with the conviction, that upon the sifting of first
principles, rather than upon any mechanical dexterity of
reasoning, the successful investigation of truth mainly de-
pends. Andthe replies of Dr. Clarke, although they cannot
be admitted as satisfactory, evince, in a remarkable degree,
that peculiar intellectual power which is manifest in the work
from which the discussion arose.

12. In Prop. VIII. it is argued that the self-existent and
original cause of all things must be an Intelligent Being.

The main argument adduced in support of this proposition
is, that as the cause is more excellent than the effect, the self-
existent being, as the cause and original of all things, must
contain in itself the perfections of all things ; and that Intelli-
gence is one of the perfections manifested in a part of the
creation. It is further argued that this perfection is not a
modification of [201] figure, divisibility, or any of the known
properties of matter ; for these are not perfections, but Zmz¢a-
tions. To this is added the ὦ posterzort argument from the
manifestation of design in the frame of the universe.

214 THE LAWS OF THOUGHT

There is appended, however, a distinct argument for the
existence of an intelligent self-existent being, founded upon
the phenomenal existence of motion in the universe. I shall
briefly exhibit this proof, and shall apply to it the method of
the present treatise. |

The argument, omitting unimportant explanations, is as_
follows :—

‘“’Tis evident there is some such a thing as motion in the
world ; which either began at some time or other, or was
eternal. Ifit began in time, then the question is granted
that the first cause is an intelligent being. ... On the con-
trary, if motion was eternal, either it was eternally caused by —
some eternal intelligent being, or it must of itself be necessary —
and self-existent, or else, without any necessity in its own
nature, and without any external necessary cause, it must_
have existed from eternity by an endless successive communi- ©
cation. If motion was eternally caused by some eternal in-
telligent being, this also is granting the question as to the -
present dispute. If it was of itself necessary and self-existent,
then it follows that it must be a contradiction in terms to sup- |
pose any matter to be at rest. And yet, at the same time,
because the determination of this self-existent motion must
be every way at once, the effect of it would be nothing else
but a perpetual rest. . . . But if it be said that motion, with-
out any necessity in its own nature, and without any external
necessary cause, has existed from eternity merely by an end-
less successive communication, as Spinoza inconsistently
enough seems to assert, this I have before shown (in the |
proof of the second general proposition of this discourse) to_
be a plain contradiction. It remains, therefore, that motion -
must of necessity be originally caused by something that is
intelligent.” |

The premises of the above argument may be thus disposed :

I. If motion began in time, the first cause is an intelligent
being. |

[202] 2. If motion has existed from eternity, either it has’
been eternally caused by some eternal intelligent being, or it

i.

CLARKE AND SPINOZA 215

is self-existent, or it must have existed by endless successive
communication.

3. 1f motion has been eternally caused by an eternal in-
telligent being, the first cause is an intelligent being.

4. If it is self-existent, matter is at rest and not at rest.

5. That motion has existed by endless successive communi-
cation, and that at the same time it is not self-existent, and
has not been eternally caused by some eternal intelligent
being, is false.

To express these propositions, let us assume—

x= Motion began in time (and therefore)
1 —x= Motion has existed from eternity.

y =The first cause is an intelligent being.

p =Motion has been eternally caused by some eternal in-
telligent being.

g = Motion is self-existent.

y= Motion has existed by endless successive communica-
tion.

S=— Matter τῷ αἴ rest.

The equations of the premises then are—

x= Uy.
ι - ατεϑίζᾷ - 4)(1 -- 7) Ἐφίι - 2). -- 7) +7(1 ~ A) -4)}:
p=vyy.

g=US(I -- 5) ΞΞ ο.
(τ 11 2) ΞῸ
Since, by the fourth equation, g =0, we obtain, on substitut-
ing for g its value in the remaining equations, the system
x= UY, 1-x=o{p(1—7)+7(1-P)},
p=y, πα 3). "0.
from which eliminating the indefinite symbols v, we have the
final reduced system,

x(I — J) =O, (1)
(x —x){pr+(1-p)(t -7)} =o, (2)
P(r -y) =0- (3)
r(1—p)=0. (4)

[203] We shall first seek the value of y, the symbol involved

216 THE LAWS OF THOUGHT

in Dr. Clarke’s conclusion. First, eliminating x from (1) ᾿
and (2), we have x
(1 -y) {pr + (1 -p)(1 -7)} =0. (5).
Next, to eliminate 7 from (4) and (5), we have
(1 — p) + (1 —y) {or + (1 - p)(1 -7)} ἡ
oP i ChE Ὁ ὁ.
whence UP) ac: (6)
Lastly, eliminating p from (3) and (6), we have
I-y=0,
EID γ»,1
which expresses the required conclusion, The first cause ts an —
intelligent being.

Let us now examine what other conclusions are deducible
from the premises.

If we substitute the value just found for y in the equations
(1), (2), (3), (4), they are reduced to the following pair of
equations, viz.,

Cea LPT = A) γα ep) =O: (7)

Eliminating from these equations x, we have

7(1-—p)=0, whence r=zp,

which expresses the conclusion, 27 motion has existed by end-
less successive communication, tt has been eternally caused by
an eternal intelligent being.

Again eliminating, from the given pair, 7, we have

(1 —x)(1-p)=0,

or, I-—XxX=U),
which expresses the conclusion, /f motion has existed from
eternity, it has been eternally caused by some eternal intellc-
gent being.

Lastly, from the same original pair eliminating 2, we get

(1-x)r=0,

which, solved in the form

I-x=U1-7),

CLARKE AND SPINOZA 217

[204] gives the conclusion, /f motion has existed from
eternity, tt has not extsted by an endless successive com-
munication.
Solved under the form
Fan)

the above equation leads to the equivalent conclusion, /f
motion exists by an endless successive communication, it began
in time.

13. Now it will appear to the reader that the first and last
of the above four conclusions are inconsistent with each other.
The two consequences drawn from the hypothesis that motion
exists by an endless successive communication, viz., Ist, that
it has been eternally caused by an eternal intelligent being ;
2ndly, that it began in time, —are plainly at variance. Never-
theless, they are both rigorous deductions from the original
premises. The opposition between them is not of a dogzcad,
but of what is technically termed a materzal, character. This
opposition might, however, have been formally stated in the
premises. We might have added to them a formal proposi-
tion, asserting that ‘‘ whatever is eternally caused by an eter-
nal intelligent being, does not begin in time.” Had this been
done, no such opposition as now appears in our conclusions
could have presented itself. Formal logic can only take ac-
count of relations which are formally expressed (VI. 16) ;
and it may thus, in particular instances, become necessary
to express, in a formal manner, some connexion among the
premises which, without actual statement, is involved in the
very meaning of the language employed.

To illustrate what has been said, let us add to the equations
(2) and (4) the equation

px=o,
which expresses the condition above adverted to. We have
(τ x){pr+ (1 τ Ὁ} +71 2) + px=0. (8)

Eliminating 2 from this, we find simply

r=0,

218 THE LAWS OF THOUGHT

which expresses the proposition, A/otzon does not extst by an
endless successive communication. If now we substitute for
γ its value in (8), we have

(t-—x)(1-p)+px=0, Or, I1-x=f;

[205] whence we have the interpretation, /f motion has
existed from eternity, tt has been eternally caused by an eternal ©
intelligent being ; together with the converse of that pro-—
position. ;

In Prop. 1X. it is argued, that ‘‘the self-existent and original —
cause of all things is not a necessary agent, but a being en- —
dued with liberty and choice.” The proof is based mainly —
upon his possession of intelligence, and upon the existence —
of final causes, implying design and choice. To the objection —
that the supreme cause operates by necessity for the produc- Ἷ
tion of what is best, it is replied, that this is a necessity of
fitness and wisdom, and not of nature. :

14. In Prop. Χ. it is argued, that ‘‘ the self-existent being,
the supreme cause of all things, must of necessity have infinite —
power.” The ground of the demonstration is, that as ‘‘all —
the powers of all things are derived from him, nothing can ~
make any difficulty or resistance to the execution of his will.” Ὁ
It is defined that the infinite power of the self-existent being —
does not extend to the ‘‘ making of a thing which implies a _
contradiction,” or the doing of that ‘‘ which would imply —
imperfection (whether natural or moral) in the being to whom
such power is ascribed,” but that it does extend to the
creation of matter, and of an immaterial, cogitative substance,
endued with a power of beginning motion, and with a liberty
of will or choice. Upon this doctrine of liberty it is con-
tended that we are able to give a satisfactory answer to ‘‘ that
ancient and great question, πόθεν τὸ κακὸν, what is the cause
and original of evil?” The argument on this head I shall
briefly exhibit.

‘* All that we call evil is either an evil of imperfection, as the
want of certain faculties or excellencies which other creatures
have ; or natural evil, as pain, death, and the like ; or moral .

CLARKE AND SPINOZA 219

evil, as all kinds of vice. The first of these is not properly

an evil; for every power, faculty, or perfection, which any

creature enjoys, being the free gift of God, . . it is plain
the want of any certain faculty or perfection in any kind of
creatures, which never belonged to their natures is no more
an evil to them, than their never having been created or
brought into being at all could properly have been called an
evil. The second kind of evil, which we call natural evil, is
either a necessary consequence of the [206] former, as death
to acreature on whose nature immortality wasnever conferred;
and then it is no more properly an evil than the former. Or
else it is counterpoised on the whole with as great or greater
good, as the afflictions and sufferings of good men, and then
also it is not properly an evil; or else, lastly, it is a punish-
ment, and then it is a necessary consequence of the third
and last kind of evil, viz., moralevil. And this arises wholly
from the abuse of liberty which God gave to His creatures
for other purposes, and which it was reasonable and fit to
give them for the perfection and order of the whole creation.
Only they, contrary to God’s intention and command, have
abused what was necessary to the perfection of the whole,
to the corruption and depravation of themselves. And thus
all sorts of evils have entered into the world without any
diminution to the infinite goodness of the Creator and
Governor thereof.”—p. I12.

The main premises of the above argument may be thus
stated :

ist. All reputed evil is either evil of imperfection, or
natural evil, or moral evil.

2nd. Evil of imperfection is not absolute evil.

3rd. Natural evil is either a consequence of evil of imper-
fection, or it is compensated with greater good, or it isa
consequence of moral evil.

4th. That which is either a consequence of evil of imper-
fection, or is compensated with greater good, is not absolute
evil.

5th. All absolute evils are included in reputed evils.

220 THE LAWS OF THOUGHT

To express these premises let us assume—

w =reputed evil.

ax=evil of imperfection.

y =natural evil.

g=moral evil.

p =consequence of evil of imperfection.
g =compensated with greater good.
r=consequence of moral evil.
t=absolute evil.

Then, regarding the premises as Primary Propositions, οὗ

which [207] all the predicates are particular, and the οοη-

junctions eéther, or, as absolutely disjunctive, we have the

following equations :

w=o{a(t—y)(t— 9) τ) -- α)ὰ -- 2) +2(1— x(x -))) σία — ὃ.
y=v{ p(t —g)(1—r) + 94 —p)(t —7) +71 “01 ὩΣ

P(t -- φ) +91 -p)=0(1 - 2). ἢ

t= VW. |

From which, if we separately eliminate the symbol v, we have ~

τοί τ - χίι -y)(1 -- 5) -γίι -- x)(1 — 2) -2(1—x)(r-y)}=0, (1) |

xt=0, (2) —
Vite Gr) υἱὲ 2) 17) = A Pt 7), Gp
{2(1-- 9) + 9(1 -- fp) }t=0, (4)
t(1 — τ) =o. (5)

Let it be required, first, to find what conclusion the premises
warrant us in forming respecting absolute evils, as concerns
their dependence upon moral evils, and the consequences of
moral evils.

For this purpose we must determine ¢in terms of zand~y.
The symbols w, +, y, 2, g must therefore be eliminated. The
process is easy, as any set of the equations is reducible to a
single equation by addition.

Eliminating w from (1) and (5), we have

ἦτ —a(1-y)(t—2) -y(1 —a)(1-2) -2(2-a\(1-y)}=0. (6)
The elimination of 2 from (3) and (4) gives
Jar + ygt+ yi —r)(t — 9) οἱ 3 (7)

CLARKE AND SPINOZA 221

The elimination of g from this gives

y(t -- γ)Ξ ο. (8)
The elimination of + between (2) and (6) gives
tye + (1 —y)(1 -- 2)} =0. (9)

The elimination of y from (8) and (9) gives
t(1 -- 2). —r)=0.
This is the only relation existing between the elements Ζ, <,
and 7.
[208] We hence get

“απ - τ)

Ἐπ κι τς ir) ee τ Our)
a 0 ο
Ὁ

=-g+-(1-2)r;
eS τὶ 5) ᾿

the interpretation of which is, Adsolute evil is etther moral
evil, or 11 1s, of not moral evil, a consequence of moral evil.

Any of the results obtained in the process of the above
solution furnishes us with interpretations. Thus from (8) we
might deduce

ere et ie ey
whence, Adsolute evils are either natural evils, which are the
consequences of moral evils, or they are not natural evils at all.

A variety of other conclusions may be deduced from the
given equations in reply to questions which may be arbitrarily
proposed. Of such I shall give a few examples, without ex-
hibiting the intermediate processes of solution.

Quest. 1.—Can any relation be deduced from the premises
connecting the following elements, viz. : absolute evils, con-
sequences of evils of imperfection, evils compensated with
greater good ?

Ans.—WVo relation exists. 1{ we eliminate all the symbols
Hut 2, 2, g, the result is o=0.

Quest. 2.—Is any relation implied between absolute
_ evils, evils of imperfection, and consequences of evils of im-
perfection ?

722 THE LAWS OF ΤΠΜΟΘΘΗ͂Σ

Ans.—The final relation between 4, ¢, and 2 is

xt + pt=o;
whence

oat — p)(1 -- x).

Therefore, Absolute evils are neither evils of imperfection, nor :
consequences of evils of imperfection. 2
[209] Quest. 3.—Required the relation of natural evils to
evils of imperfection and evils compensated with greater good.

We find

PY = 0,
“τ 3 ἢ
Therefore, Natural evils are either consequences of evils of tm-
perfection which are not compensated with greater good, or they
are not consequences of evils of tmperfection at all.

Quest. 4.—In what relation do those natural evils which
are not moral evils stand to absolute evils and the conse-
quences of moral evils ?

If (1 —z)=s, we find, after elimination,

ts(1-—r)=0;3

A

Ο oO Ο

ἘΠ το τ Π - ἢ).

Therefore, Natural evils, which are not moral evils, are etther
absolute evils, which are the consequences af moral evils, or
they are not absolute evils at all.

The following conclusions have been deduced in a similar
manner. The subject of each conclusion will show of what
particular things a description was required, and the predicate
will show what elements it was designed to involve :—

Absolute evils, which are not consequences of moral evils, are
moral and not natural evils.

Absolute evils which are not moral evils are natural evils,
which are the consequences of moral evils.

Natural evils which are not consequences of moral evils are
not absolute evils,

CLARKE AND SPINOZA 223

Lastly, let us seek a description of evils which are not ab-
solute, expressed in terms of natural and moral evils.
We obtain as the final equation,

I —tayst+—y(t τ) 42(1 —y)s+(1-y)(t -- 2).

The direct interpretation of this equation is a necessary truth,
but the reverse interpretation is remarkable. Hvzls which are
both [210] natural and moral, and evils which are neither
natural nor moral, are not absolute evils.

This conclusion, though it may not express a truth, is cer-
tainly involved in the given premises, as formally stated.

15. Let us take from the same argument a somewhat fuller
system of premises, and let us in those premises suppose that
the particles, ezther, or, are not absolutely disjunctive, so that
in the meaning of the expression, ‘‘either evil of imperfection,
or natural evil, or moral evil,” we include whatever possesses
one or more of these qualities.

Let the premises be—

1. All evil (w) is either evil of imperfection (x), or natural
evil (y), or moral evil (2).

2. Evil of imperfection (1) is not absolute evil (2).

3. Natural evil (y) is either a consequence of evil of imper-
fection (2), or it is compensated with greater good (4), or it
is a consequence of moral evil (7).
~ 4. Whatever is a consequence of evil of imperfection (2) is
not absolute evil (2).

5. Whatever is compensated with greater good (Φ) is not
absolute evil (2).

6. Moral evil (z) is a consequence of the abuse of liberty (2).

7, That which is a consequence of moral evil (7) is a conse-
quence of the abuse of liberty (2).

8. Absolute evils are included in reputed evils.

The premises expressed in the usual way give, after the eli-
mination of the indefinite symbols v, the following equations:

w(x -- x)(1 —9)(t — 2) =0, (1)
Leo. (2)

224 THE LAWS OF THOUGHT

γα -2)α -g)(1-7) =0, (3)

pt=9, (4)

φί -- Ο, (5)
2(I -- 2) =o, (6)
r(1 —%)=0, (7)
{ι —w)=o. (8)

Each of these equations satisfies the condition V(1 — V)=o0.
[211] The following results are easily deduced—

Natural evil ts either absolute evil, which ts a consequence
of moral evil, or it ts not absolute evil at all.

All evils are either absolute evils, which are consequences
of the abuse of liberty, or they are not absolute evils.

Natural evils are either evils of imperfection, which are not
absolute evils, or they are not evils of tmperfection at all.
Absolute evils are either natural evils, which are conse-

guences of the abuse of liberty, or they are not natural evils,
and at the same time not evils of imperfection.

Consequences of the abuse of liberty include all natural evils
which are absolute evils, and are not evils of tmperfection, with
an indefinite remainder of natural evils which are not absolute,
and of evils which are not natural.

16. These examples will suffice for illustration. The
reader can easily supply others if they are needed. We
proceed now to examine the most essential portions of the
demonstration of Spinoza.

DEFINITIONS

1. By a cause of ttself (causa suz), 1 understand that of
which the essence involves existence, or that of which the
nature cannot be conceived except as existing.

2. That thing is said to be finite or bounded in its own
kind (22 suo genere finita) which may be bounded by another
thing of the same kind; e.g. Body is said to be finite, be-
cause we can always conceive of another body greater than
a givenone. Sothought is bounded by other thought. But
body is not bounded by thought, nor thought by body.

CLARKE AND SPINOZA 228

3. By substance, [ understand that which is in itself (zx se),
and is conceived by itself (per se conctpitur), t.e., that whose
conception does not require to be formed from the conception
of another thing.

4. By attribute, I understand that which the intellect per-
ceives in substance, as constituting its very essence.

5. By mode, I understand the affections of substance, or
that which is in another thing, by which thing also it is
conceived.

6. By God, I understand the Being absolutely infinite,
that [212] is the substance consisting of infinite attributes,
each of which expresses an eternal and infinite essence,

Explanation.—I say absolutely infinite, not infinite in its
own kind. For to whatever is only infinite in its own kind
we may deny the possession of (some) infinite attributes.
But when a thing is absolutely infinite, whatsoever expresses
essence and involves no negation belongs to its essence.

7. That thing is termed free, which exists by the sole
necessity of its own nature, and is determined to action by
itself alone ; mecessary, or rather constrained, which is deter-
mined by another thing to existence and action, ina certain
and determinate manner.

8. By eternity, | understand existence itself, in so far as
-it is conceived necessarily to follow from the sole definition
of the eternal thing.

Explanation.—For such existence, as an eternal truth, is
conceived as the essence of the thing, and therefore cannot
be explained by mere duration or time, though the latter
should be conceived as without beginning and without end.

AXIOMS

1. All things which exist are either in themselves (zx se)
or in another thing.

2. That which cannot be conceived by another thing ought
to be conceived by itself.

3. From a given determinate cause the effect necessarily

15

226 THE LAWS OF THOUGHT

follows, and, contrariwise, if no determinate cause be granted,
it is impossible that an effect should follow.

cl

:

4. The knowledge of the effect depends upon, and in-—

volves, the knowledge of the cause.

5. Things which have nothing in common cannot be under- |
stood by means of each other ; or the conception of the one—

does not involve the conception of the other.

6. A true idea ought to agree with its own object. (/dea

vera debet cum suo tdeato conventre. )

7, Whatever can be conceived as non-existing does not
involve existence in its essence.

[213] Other definitions are implied, and other axioms are

virtually assumed, in some of the demonstrations. Thus, in

Prop. I., ‘‘Substance is prior in nature to its affections,”

the proof of which consists in a mere reference to Defs. 3°
and 5, there seems to be an assumption of the following |
axiom, viz., ‘‘ That by which a thing is conceived is prior |
in nature to the thing conceived.” Again, in the demon-
stration of Prop. V. the converse of this axiom is assumed
to be true. Many other examples of the same kind occur. |

It is impossible, therefore, by the mere processes of Logic,

to deduce the whole of the conclusions of the first book of
the Ethics from the axioms and definitions which are prefixed
to it, and which are given above. In the brief analysis which
will follow, I shall endeavour to present in their proper order
what appear to me to be the real premises, whether formally |

stated or implied, and shall show in what manner they i in-
volve the conclusions to which Spinoza was led. ;

17. I conceive, then, that in the course of his demonstra-
tion, Spinoza effects several parallel divisions of the univers€
of possible existence, as,

Ist. Into things which are in themselves, x, and things!
which are in some other thing, x’; whence, as these classes
of things together make up the universe, we have |

noe (Ax. 1)
or, =I-—X’,

2nd. Into things ΠΕΣ are conceived by themselves, »

lai Ae 8

oes

CLARKE AND SPINOZA 227

and things which are conceived through some other thing,
y ; whence
y=1-y. (Ax. 2)
3rd. Into substance, z, and modes, 2’; whence
z=1-2. (Def. 3, 5)
4th. Into things free, 4 and things necessary, /’; whence
f=t-f'. (Def. 7)
5th. Into things which are causes and self-existent, 6, and
things caused by some other thing, e’ ; whence
ἐξ τ πὶ. (Def. 1, Ax. 7)

[214] And his reasoning proceeds upon the expressed or
assumed principle that these divisions are not only parallel,
but equivalent. Thus in Def. 3, Substance is made equivalent
with that which is conceived by itself ; whence

Z=Y,
Again, Ax. 4, as it is actually applied by Spinoza, es-
tablishes the identity of cause with that by which a thing is

conceived ; whence
yrHe.

Again, in Def. 7, things free are identified with things

self-existent ; whence ;
ΞΞ é,

Lastly, in Def. 5, mode is made identical with that which

is in another thing ; whence 2’=2’, and therefore,
iit

All these results may be collected together into the follow-

ing series of equations, viz. :
eay=goafmemt-x =1-y=1~-f'=1-7 =1-¢,

And any two members of this series connected together by
the sign of equality express a conclusion, whether drawn by

Spinoza or not, which is a legitimate consequence of his
system. Thus the equation

Z=1-€,

β expresses the sixth Proposition of his system, viz., One

228 THE LAWS OF THOUGHT

substance cannot be produced by another. Similarly the

equation
Z=e,

expresses his seventh Proposition, viz., ‘‘It pertains to the
nature of substance to exist.” This a of deduction it is

en

4
q
P

unnecessary to pursue. Spinoza applies it chiefly to the de- :

duction according to his views of the properties of the Divine
Nature, having first endeavoured to prove that the only sub-
stance is God. In the steps of this process, there appear to
me to exist some fallacies, dependent chiefly upon the am-
biguous use of words, to which it will be necessary here to
direct attention.

[215] 18. In Prop. Ν᾿. itis endeavoured to show, that
‘‘ There cannot exist two or more substances of the same

nature or attribute.” The proof is virtually as follows: If —

there are more substances than one, they are distinguished
either by attributes or modes ; if by attributes, then there is
only one substance of the same attribute ; if by modes, then,

laying aside these as non-essential, there remains no real ©

ground of distinction. Hence there exists but one substance
of the same attribute. The assumptions here involved are in-

consistent with those which are found in other parts of the —
treatise. Thus substance, Def. 4, is apprehended by the

intellect through the means of attribute. By Def. 6 it may
have many attributes. One substance may, therefore, con-
ceivably be distinguished from another by a difference in some
of its attributes, while others remain the same.

In Prop. VIII. it is attempted to show that, All substance

is necessarily infinite. The proof is as follows. There ex- Ὁ

ists but one substance, of one attribute, Prop. V. ; and it per-
tains to its nature to exist, Prop. VII. It will, therefore, be
of its nature to exist either as finite or infinite. But not as
finite, for, by Def. 2 it would require to be bounded by
another substance of the same nature, which also ought to
exist zecessarily, Prop. VII. Therefore, there would be two
substances of the same attribute, which is absurd, τῶν Ve
Substance, therefore, is infinite,

CLARKE AND SPINOZA 229

In this demonstration the word ‘‘ finite” is confounded with
the expression, ‘‘ Finite in its own kind,” Def. 2. It is thus
assumed that nothing can be finite, unless it is bounded by
another thing of the same kind. This is not consistent with
the ordinary meaning of the term. Spinoza’s use of the term
finite tends to make space the only form of substance, and all
existing things but affections of space, and this, I think, is
really one of the ultimate foundations of his system.

The first scholium applied to the above Proposition is re-
markable. I give it in the original words: ‘‘Quum finitum
esse revera sit ex parte negatio, et infinitum absoluta affir-

_ matio existentiz alicujus nature, sequitur ergo ex sola Prop.

VII. omnem substantiam debere esse infinitam.” Now this
is in reality an assertion of the principle affirmed by Clarke,
and controverted by [216] Butler (XIII. 11), that necessary
existence implies existence in every part of space. Probably
this principle will be found to lie at the basis of every attempt
to demonstrate, a priorz, the existence of an Infinite Being.

From the general properties of substance above stated, and
the definition of God as the substance consisting of infinite
attributes, the peculiar doctrines of Spinoza relating to the
Divine Nature necessarily follow. As substance is self-ex-
istent, free, causal in its very nature, the thing in which other
things are, and by which they are conceived ; the same pro-
perties are also asserted of the Deity. He is self-existent,
Prop. XI. ; indivisible, Prop. XIII. ; the only substance, Prop.
XIV. ; the Being in which all things are, and by which all
pmings are conceived, Prop. XV. ; free, Prop. XVII. ; the
immanent cause of all things, Prop. XVIII. The proof that
God is the only substance is drawn from Def. 6, which is
interpreted into a declaration that ‘‘God is the Being ab-
solutely infinite, of whom no attribute which expresses the
essence of substance can be denied.” Every conceivable
attribute being thus assigned by definition to Him, and it
being determined in Prop. V. that there cannot exist two
substances of the same attribute, it follows that God is the
only substance.

230 THE LAWS OF THOUGHT

Though the E¢hics of Spinoza, like a large portion οὔ.
his other writings, is presented in the geometrical form, it —
does not afford a good praxis for the symbolical method οὔ
this work. Of course every train of reasoning admits, when ~
its ultimate premises are truly determined, of being treated
by that method ; but in the present instance, such treatment
scarcely differs, except in the use of letters for words, from
the processes employed in the original demonstrations.
Reasoning which consists so largely of a play upon terms
defined as equivalent, is not often met with ; and it is rather
on account of the interest attaching to the subject, than of
the merits of the demonstrations, highly as by some they
are esteemed, that I have devoted a few pages here to their
exposition.

19. It is not possible, I think, to rise from the perusal of
the arguments of Clarke and Spinoza without a deep con-
viction of the futility of all endeavours to establish, entirely
a priort, the existence [217] of an Infinite Being, His attri-
butes, and His relation to the universe. The fundamental
principle of all such speculations, viz., that whatever we can
clearly conceive, must exist, fails to accomplish its end, even
when its truth is admitted. For how shall the finite com-
prehend the infinite? Yet must the possibility of such con-
ception be granted, and in something more than the sense
of a mere withdrawal of the limits of phenomenal existence,
before any solid ground can be established for the knowledge,
a priori, of things infinite and eternal. Spinoza’s affirma-
tion of the reality of such knowledge is plain and explicit:
‘*Mens humana adequatum habet cognitionem externe et
infinite essentie Dei” (Prop. XLVII., Part 2nd). Let this
be compared with Prop. XXXIV., Part 2nd: ‘‘ Omnis idea
que in nobis est absoluta sive adzquata et perfecta, vera
est”; and with Axiom 6, Part Ist, ‘‘ Idea vera debet cum
suo ideato convenire.” Moreover, this species of knowledge
is made the essential constituent of all other knowledge :
‘“De natura rationis est res sub quadam eternitatis specie
percipere”” (Prop. XLIV., Cor. II., Part 2nd). Were it said,

CLARKE AND SPINOZA 231

that there is a tendency inthe human mind to rise in con-
templation from the particular towards the universal, from
the finite towards the infinite, from the transient towards
the eternal; and that this tendency suggests to us, with
_ high probability, the existence of more than sense perceives
or understanding comprehends ; the statement might be ac-
cepted as true for at least a large number of minds. There
is, however, a class of speculations, the character of which
must be explained in part by reference to other causes, —
impatience of probable or limited knowledge, so often all
that we can really attain to; a desire for absolute certainty
where intimations sufficient to mark out before us the path
of duty, but not to satisfy the demands of the speculative
intellect, have alone been granted to us; perhaps, too, dis-
satisfaction with the present scene of things. With the
undue predominance of these motives, the more sober pro-
cedure of analogy and probable induction falls into neglect.
Yet the latter is, beyond all question, the course most
adapted to our present condition. To infer the existence
of an intelligent cause from the teeming evidences of sur-
_ rounding design, to rise to the conception of a moral Governor
~of the world, from the study of [218] the constitution and
the moral provisions of our own nature ;—these, though but
the feeble steps of an understanding limited in its faculties
and its materials of knowledge, are of more avail than the
ambitious attempt to arrive at a certainty unattainable on
the ground of natural religion. And as these were the most
ancient, so are they still the most solid foundations, Revela-
tion being set apart, of the belief that the course of this world
is not abandoned to chance and inexorable fate.

[219] CHAPTER XIV

EXAMPLE OF THE ANALYSIS OF A SYSTEM OF EQUATIONS
BY THE METHOD OF REDUCTION TO A SINGLE EQUI-
VALENT EQUATION V=0, WHEREIN V SATISFIES THE
CONDITION V(1—V)=0

1. LeET us take the remarkable system of premises employed
in the previous Chapter, to prove that ‘‘ Matter is not a
necessary being”; and suppressing the 6th premiss, viz.,
Motion exists,—examine some of the consequences which
flow from the remaining premises. This is in reality to
accept as true Dr. Clarke’s hypothetical principles ; but to
suppose ourselves ignorant of the fact of the existence of
motion. Instances may occur in which such a selection of
a portion of the premises of an argument may lead to interest-
ing consequences, though it is with other views that the
present example has been resumed. The premises actually
employed will be—

1 If matter is a necessary being, either the property of
gravitation is necessarily present, or it is necessarily absent.

2. If gravitation is necessarily absent, and the world is
not subject to any presiding intelligence, motion does not
exist,

3. If gravitation is necessarily present, a vacuum is
necessary.

4. If a vacuum is necessary, matter is not a necessary
being.

5. If matter is a necessary being, the world is not subject
to a presiding intelligence.

If, as before, we represent the elementary propositions by

the following notation, viz. :
232

Ἰ

EXAMPLE OF ANALYSIS 233

+= Matter is a necessary being,
y=Gravitation is necessarily present,
w = Motion exists,
t= Gravitation is necessarily absent,
z= The world is merely material, and not subject to
a presiding intelligence,
v= A vacuum is necessary,
[220] we shall on expression of the premises and elimina-

tion of the indefinite class symbols (Φ), obtain the following
system of equations:

xyt + xyf=o,
lz7 = 0,
YU=0,

UX =O,
xZ=0,

in which for brevity 7 stands for 1—y, ¢ for 1—Z, and so on;
whence, also 1—f=¢4, 1—y=y, &c.

As the first members of these equations involve only
positive terms, we can form a single equation by adding
Meer tocether (Vill. Prop. Ils); vize:

xyt + xy + γὸ τ Ux +x5 + ἔστ =o,

and it remains to reduce the first member so as to cause it

to satisfy the condition V(1 — V)=0.

For this purpose we will first obtain its development with
reference to the symbols # and y. The result is—

(¢+d+0+F+cew)xy+(F+u+ 2+ tew)xy
+ (d+ fsw)xy + tzwxy =o.

And our object will be accomplished by reducing the four
coefficients of the development to equivalent forms, them-
selves satisfying the condition required.

Now the first coefficient is, since v-+v=I,

1+74+2+ few,

which reduces to unity (IX. Prop. I.).
The second coefficient is

f+u+zst+lew ;

234 THE LAWS OF THOUGHT

and its reduced form (X. 3) is
Ζ- ἐσ τ 02 + tozw.

The third coefficient, 7-+¢zw, reduces by the same method |
to d+¢gwv ; and the last coefficient ¢zw needs no reduction.
Hence the development becomes

[221] xy t+ (F+ τὺ Ὁ 102 + lisw) xy + (Ὁ + ἐσευῦ)ὰν + L2wxy =o ; (1)

and this is the form of reduction sought.

2. Now according to the principle asserted in Prop. III.
Chap. X., the whole relation connecting any particular set —
of the symbols in the above equation may be deduced by
developing that equation with reference to the particular
symbols in question, and retaining in the result only those —
constituents whose coefficients are unity. Thus, if 2 and y
are the symbols chosen, we are immediately conducted to

the equation
XY =O,
whence we have

pA — 2);
Θ

with the interpretation, // gravitation is necessarily present,
matter 15 not a necessary being.

Let us next seek the relation between and w. Develop-
ing (1) with respect to those symbols, we get
( + ty + toy + tozy + tosy)xw + (ν + % + toy + C057) x

+ (ay + tsvy + tay) xw + DyXW =.

The coefficient of rw, and it alone, reduces to unity. For
2027 + tozy = toy, and tvy+ toy =ty, and ty+ty=y, and lastly,
y+y=1. This is always the mode in which such reductions
take place. Hence we get

XxW = 0,

ο
aa — x),

of which the interpretation is, 17 motion extsts, matter ts not
a necessary being. |

EAAM PEER OH ANALYSTS 510

If, in like manner, we develop (1) with respect to # and

2, we get the equation
XZ =

x=

ΘΟ ΘΟ 0

with the interpretation, /f matter ts a necessary being, the
world ts merely material, and without a presiding intelligence.

[222] This, indeed, is only the fifth premiss reproduced,
but it shows that there is no other relation connecting the
two elements which it involves.

If we seek the whole relation connecting the elements 7,
w, and y, we find, on developing (1) with reference to those
symbols, and proceeding as before,

XY + xWY =O.
Suppose it required to determine hence the consequences of
_ the hypothesis, ‘‘ Motion does not exist,” relatively to the

questions of the necessity of matter, and the necessary pre-
sence of gravitation. We find

a I — W = Ah +5, With, αν — 0:
The direct interpretation of the first equation is, 17} motzon
does not exist, either matter is a necessary being, and gravt-
tation ts not necessarily present, or matter ts not a necessary
being.
The reverse interpretation is, /f matter ts a necessary
being, and gravitation not necessary, motion does not exist.
In exactly the same mode, if we sought the full relation
between 2, 2, and w, we should find

xX2W+XZ=0.

From this we may deduce

Εἰ καθ
Z=xw+—x, with xw=o.
ο

236 THE CAW ΟΣ Δ ΙΕ fF

Therefore, 77 the world ts merely matertal, and not subject
to any presiding intelligence, either matter ts a necessary
being, and motion does not exist, or matter ts not a necessary
being.

Also, reversely, If matter ἐς a necessary being, and there
7s no such thing as motion, the world ts merely material.

3. We might, of course, extend the same method to the
de-[223]termination of the consequences of any complex
hypothesis #, such as, ‘‘ The world is merely material, and
without any presiding intelligence (z), but motion exists ”
(w), with reference to any other elements of doubt or specula-
tion involved in the original premises, such as, ‘‘ Matter is a
necessary being” (%), ‘‘ Gravitation is a necessary quality
of matter” (y).' We should, for this purpose, connect with
the general equation (1) a new equation,

2;

reduce the systems thus formed to a single equation V =o,
in which V satisfies the condition V(1 — V)=o0, and proceed
as above to determine the relation between w, x, and y, and
finally « as a developed function of x and y. But it is very
much better to adopt the methods of Chapters VIII. and
IX. I shall here simply indicate a few results, with the
leading steps of their deduction, and leave their verification
to the reader’s choice.

In the problem last mentioned we find, as the relation
connecting 4, ἡ, w, and 2,

XW + XWY + XWYZ =O.
And if we write «=2y, and then eliminate the symbols +
and y by the general problem, Chap. IX., we find
XU + XU =O;

whence I Roe
uU=—-XV +OxXy4+—-X ;

[9] O

wherefore Cas
“5. XW LY —= 0,
ο

Hence, [f the world ts merelv material and without a presida-

EXAMPLE OF ANALYSIS 237

ing intelligence, and at the same time motion exists, matter
ἦς not a necessary being.

Now it has before been shown that z/ motion exists, matter
ἧς not a necessary being, so that the above conclusion tells
us even less than we had before ascertained to be (infer-

_ entially) true. Nevertheless, that conclusion is the proper

and complete answer to the question which was proposed,
which was to determine simply the consequences of a certain
complex hypothesis.

[224] 4. It would thus be easy, even from the limited
system of premises before us, to deduce a great variety of
additional inferences, involving, in the conditions which are
given, any proposed combinations of the elementary pro-
positions. If the condition is one which is inconsistent
with the premises, the fact will be indicated by the form of
the solution. The value which the method will assign to
the combination of symbols expressive of the proposed con-
dition will be o. If, on the other hand, the fulfilment of
the condition in question imposes no restriction upon the

- propositions among which relation is sought, so that every

combination of those propositions is equally possible, —the

~fact will also be indicated by the form of the solution. Ex-
amples of each of these cases are subjoined.

If in the ordinary way we seek the consequences which
would flow from the condition that matter ts a necessary
being, and at the same time that sotzon exists, as affecting
the Propositions, The world ts merely matertal, and with-
out a presiding intelligence, and Gravitation is necessarily
present, we shall obtain the equation

XI = 0,

which indicates that the condition proposed is inconsistent
with the premises, and therefore cannot be fulfilled.

If we seek the consequences which would flow from the
condition that Matter ἐς not a necessary being, and at the
same time that Motion does exist, with reference to the
same elements as above, viz., the absence of a presiding

238 THE LAWS OF THOU GH rT.

intelligence, and the necessity of gravitation,—we obtain

the following result,
(1 — xt = 2 ys + —9( 1 — a) +2(1 —y) e+ 2(1 - 3) -- 2),

which might literally be interpreted as follows :

47 matter ts not a necessary being, and motion exists, then
either the world ts merely material and without a presiding
intelligence, and gravitation ts necessary, or one of these two
results follows without the other, or they both fail of being

true. Wherefore of the four possible combinations, of which.

some one is true of necessity, and of which of necessity one
only can be true, it is [225] affirmed that any one may be
true. Such a result is a truism—a mere necessary truth.
Still it contains the only answer which can be given to the
question proposed.

I do not deem it necessary to vindicate against the charge
of laborious trifling these applications. It may be requisite
to enter with some fulness into details useless in themselves,
in order to establish confidence in general principles and
methods. When this end shall have been accomplished in
the subject of the present inquiry, let all that has contri-

buted to its attainment, but has afterwards been found super- —

fluous, be forgotten.

[226] CHAPTER XV

THE ARISTOTELIAN LOGIC AND ITS MODERN EXTENSIONS,
EXAMINED BY THE METHOD OF THIS TREATISE

I. THE logical system of Aristotle, modified in its details,
but unchanged in its essential features, occupies so important
a place in academical education, that some account of its
nature, and some brief discussion of the leading problems
which it presents, seem to be called for in the present work.
It is, I trust, in no narrow or harshly critical spirit that I
approach this task. My object, indeed, is not to institute
any direct comparison between the time-honoured system
of the schools and that of the present treatise ; but, setting
truth above all other considerations, to endeavour to exhibit
the real nature of the ancient doctrine, and to remove one
or two prevailing misapprehensions respecting its extent
—and sufficiency.

That which may be regarded as essential in the spirit and
procedure of the Aristotelian, and of all cognate systems of
Logic, is the attempted classification of the allowable forms
of inference, and the distinct reference of those forms, col-
lectively or individually, to some general principle of an
axiomatic nature, such as the ‘‘ dictum of Aristotle”: What-
soever is affirmed or denied of the genus may in the same
sense be affirmed or denied of any species included under
that genus. Concerning such general principles it may, I
think, be observed, that they either state directly, but in an
abstract form, the argument which they are supposed to
elucidate, and, so stating that argument, affirm its validity ;
or involve in their expression technical terms which, after
definition, conduct us again to the same point, viz., the

abstract statement of the supposed allowable forms of infer-
239

240 TLE EA SO a OC Gre

ence. The idea of classification is thus a pervading element
in those systems. Furthermore, they exhibit Logic as
resolvable into two great branches, the one of which is
occupied with the treatment of categorical, the other with
that of hypothetical or [227] conditional propositions. The
distinction is nearly identical with that of primary and
secondary propositions in the present work. The discussion
of the theory of categorical propositions is, in all the ordinary
treatises of Logic, much more full and elaborate than that
of hypothetical propositions, and is occupied partly with
ancient scholastic distinctions, partly with the canons of de-
ductive inference. To the latter application only is it neces-
sary to direct attention here. :

2. Categorical propositions are classed under the four
following heads, viz :—

TYPE
rst. Universal affirmative Propositions: All Y’s are X’s.
2nd. Universal negative ᾿ No Y’s are X’s.
3rd. Particular affirmative = SOMeC 2s atecy s.
4th. Particular negative Fr Some Y’s are not X’s.

To these forms, four others have recently been added, so
as to constitute in the whole eight forms (see the next article)
susceptible, however, of reduction to six, and subject to
relations which have been discussed with great fulness and
ability by Professor De Morgan, in his Formal Logic. A
scheme somewhat different from the above has been given to
the world by Sir W. Hamilton, and is made the basis of a
method of syllogistic inference, which is spoken of with very
high respect by authorities on the subject of Logic. *

The processes of Formal Logic, in relation to the above
system of propositions, are described as of two kinds, viz.,
‘“Conversion” and ‘‘Syllogism.” By Conversion is meant
the expression of any proposition of the above kind in an
equivalent form, but with a reserved order of terms. By
Syllogism is meant the deduction from two such propositions
having ἃ common term, whether subject or predicate, of some

* Thomson’s Outlines of the Laws of Thought, p. 177.

OF THE ARISTOTELIAN ΣΟ. 241

third proposition inferentially involved in the two, and form-
ing the ‘‘ conclusion.” It is maintained by most writers on
_ Logic, that those processes, and according to some, the single
process of Syllogism, furnish the universal types of reason-
ing, and that it is the business of the mind, in any train of
demonstration, to conform itself, whether [228] consciously
or unconsciously, to the particular models of the processes
which have been classified in the writings of logicians.

3. The course which I design to pursue is to show how
these processes of Syllogism and Conversion may be con-
ducted in the most general manner upon the principles of the
present treatise, and, viewing them thus in relation to a
system of Logic, the foundations of which, it is conceived,
have been laid in the ultimate laws of thought, to seek to
determine their true place and essential character.

The expressions of the eight fundamental types of proposi-
tion in the language of symbols are as follows :

Pom YS are ἃς, Y =Ux.
2c NO δὲ ατὸ Xs, y=U(1—X).
3. Some Y’s are X’s, 2), ON.
4. Some Y’s are not-X’s, vy =v(1 -- x).
A 5. All not- Y’s are X’s, I—y=Ux. (1)
6. No not-Y’s are X’s, I-y=v(1-x).
7. Some not- Y’s are X’s, v(I —y) = vx.
8. Some not-Y’s are not-X’s, v(1 —y)=v(1—2).

In referring to these forms, it will be convenient to apply,
in a sense shortly to be explained, the epithets of logical
quantity, ‘‘universal” and ‘‘ particular,” and of quality,
‘‘ affirmative ” and ‘‘ negative,” to the terms of propositions,
and not to the propositions themselves. We shall thus con-
sider the term ‘‘ All Y’s,” as universal-affirmative ; the term
M.Y’s,” or ‘‘Some Y’s,” as particular-affirmative ; the term
“ΑἹ! not- Y’s,” as universal-negative ; the term ‘‘ Some not-
Y's,” as particular-negative. The expression ‘‘ No Y's,” is
not properly a term of a proposition, for the true meaning of
the proposition, ‘‘ No ΚΡ are _X’s,” is “" All Y’s are not-X’s.”

The subject of that proposition is, therefore, universal-
16

242 Lie LAWS Of 7 OCUCT 7.

affirmative, the predicate particular-negative. That there is
a real distinction between the conceptions of ‘‘men” and
τ ΠΟΥ men” is manifest. — [his distinction ts all: that | con-
template when applying as above the designations of affirma-
tive and negative, without, however, insisting upon the
etymological propriety of the application to the terms of pro-
positions. The designations positive and privative would
have been more ap-[229]propriate, but the former term is
already employed in a fixed sense in other parts of this work.

4. From the symbolical forms above given the laws of
conversion immediately follow. Thus from the equation

πε
representing the proposition, ‘‘ All Y’s are X’s,” we deduce,
on eliminating v,
γα - x) =0,

which gives by solution with reference to 1 -- 2,

1-x=<(1-y);
oO
the interpretation of which is,
All not-X’s are not- Y’s.

ΤΠ is anexample of what is called ‘‘ negative conver-
sion.” In like manner, the equation

sme v(1 τ
representing the proposition, ‘‘No Y’s are X’s” gives

()
saa =)

the interpretation of which is, ‘‘No X’s are Y's.” This is
an example of what is termed simple conversion ; though it
is in reality of the same kind as the conversion exhibited in
the previousexample. All the examples of conversion which
have been noticed by logicians are either of the above kind,
or of that which consists in the mere transposition of the
terms of a proposition, without altering their quality, as
when we change

vy =vx, representing, Some Y’s are X’s,
into vx = vy, representing, Some X’s are Y’s ;

OF THE ARISTOTELIAN LOGIC 243

or they involve a combination of those processes with some
auxiliary process of limitation, as when from the equation

y=vx, representing, All Y’s are X’s,
we deduce on multiplication by Ὁ,

vy = vx, representing, Some Y’s are X’s,
and hence vx=vy, representing, Some X’s are Y’s.

[230] In this example, the process of limitation precedes
that of transposition.

From these instances it is seen that conversion is a parti-
cular application of a much more general process in Logic,
of which many examples have been given in this work.
That process has for its object the determination of any
element in any proposition, however complex, as a logical
function of the remaining elements. Instead of confining
our attention to the subject and predicate, regarded as
simple terms, we can take any element or any combination
of elements entering into either of them ; make that element,
or that combination, the ‘‘ subject” of a new proposition ;
and determine what its predicate shall be, in accordance
with the data afforded to us. It may be remarked, that even

the simple forms of propositions enumerated above afford

some ground for the application of such a method, beyond
what the received laws of conversion appear to recognise.
Thus the equation

y =vx, representing, All Y’s are X’s,
gives us, in addition to the proposition before deducted, the
three following :—

πὶ (1 — x) =0. There are no Y’s that are not-X’s.

2nd. τις. Things that are not-Y’s include all
ἥ things that are not-X’s, and an
indefinite remainder of things that

are X’s.

3rd. x=y+ mG - )). Things that are X’s include all things
Ἢ that are Y’s, and an indefinite re-
mainder of things that are not-Y’s.

244 THE LAWS. OF THOUGHT

These conclusions, it is true, merely place the given proposi- |
tion in other and equivalent forms,—but. such and no more
is the office of the received mode of ‘‘ negative conversion.” —

Furthermore, these processes of conversion are not elemen- —
tary, but they are combinations of processes more simple —
than they, more immediately dependent upon the ultimate
laws and axioms which govern the use of the symbolical ἢ
instrument of [231] reasoning. This remark is equally ap-
plicable to the case of Syllogism, which we proceed next to ~
consider.

5. The nature of syllogism is best seen in the particular
instance. Suppose that we have the propositions,

Alt Χ σέατο /-s,
All Xs are.Z's.

From these we may deduce the conclusion,
All X’s are 2’s.

This is a syllogistic inference. The terms Y and Z are@
called the extremes, and Y is called the middle term. The
function of the syllogism generally may now be defined.
Given two propositions of the kind whose species are tabu-
lated in (1), and involving one middle or common term Y, —
which is connected in one of the propositions with an extreme
&X, in the other with an extreme Z ; required the relation
connecting the extremes Y and Z. The term Y may appear
in its affirmative form, as, All Y’s, Some Y’s; or in its
negative form, as, All not-Y’s, Some not-Y’s; in either |
proposition, without regard to the particular form which it
assumes in the other.

Nothing is easier than in particular instances to resolve
the Syllogism by the method of this treatise. [ts resolution
is, indeed, a particular application of the process for the re-—

duction of systems of propositions. Taking the examples
above given, we have,

x= YY,

VHS;
whence by substitution,

Δ 27,5.

OF THE ARISTOTELIAN LOGIC 245

which is interpreted into
All Χχὺς are: 735.

Or, proceeding rigorously in accordance with the method
developed in (VIII. 7), we deduce

x(t-y)=0, γ(ι -- 2): ο.
Adding these equations, and eliminating y, we have
χει —2)=o;

eee 0 Ὁ All WR are σι
Ὁ

And in the same way may any other case be treated.

6. Quitting, however, the consideration of special ex-
amples, let us examine the general forms to which all syl-
logism may be reduced,

PROPOSITION I[

To deduce the general rules of Syllogism

_By the general rules of Syllogism, I here mean the rules
applicable to premises admitting of every variety both of
quantity and of quality in their subjects and predicates, ex-
cept the combination of two universal terms in the same pro-
position. The admissible forms of propositions are therefore
those of which a tabular view is given in (1).

Let X and Y be the elements or things entering into the
first premiss, Z and Y those involved in the second. Two
cases, fundamentally different in character, will then present
themselves. The terms involving Y will either be of lke
or of unlike quality, those terms being regarded as of like
quality when they both speak of ‘‘ Y’s,” or both of ‘‘ Not- Y’s,”
as of unlike quality when one of them speaks of ‘‘ Y’s,” and
the other of ‘‘Not-Y’s.” Any pair of premises, in which
the former condition is satisfied, may be represented by the
equations

Ox — 0, (1)
Ws=W'y ; (2)

246 THE LAWS OF THOUGHT

for we can employ the symbol y to represent either ‘‘ All
Y’s,” or “ΑἹ! not- Y’s,” since the interpretation of the symbol
is purely conventional. If we employ y in the sense of
«ΑἹ not- Y’s,” then 1—y will represent ‘‘ All Y's,” and no
other change will be introduced. An equal freedom is per-

᾿

2

:
a
Ἂν
ME bea
ΝΣ,
af:
3
ἣν

mitted with respect to the symbols x and 2, so that the :
equations (1) and (2) may, by properly assigning the inter- —

pretations of 4, 7, and z, be made to represent all varieties

in the combination of premises dependent upon the guality ᾿
of the respective terms. Again, by assuming proper inter- :

pretations to the symbols Ὁ, σ΄, w, w’, in those equations,

all varieties with reference to guanizty may also be [233] —

represented. Thus, if we take v=1, and represent by σ΄ a

class indefinite, the equation (1) will represent a universal —

proposition according to the ordinary sense of that term,
ζ.6., a proposition with universal subject and particular pre-

elite Rate arn iy

dicate. We may, in fact, give to subject and predicate in ἢ

either premiss whatever guantztzes (using this term in the

scholastic sense) we please, except that by hypothesis, they i

must not both be universal. The system (1), (2), repre-
sents, therefore, with perfect generality, the possible com-
binations of premises which have like middle terms.

7. That our analysis may be as general as the equations
to which it is applied, let us, by the method of this work,

eliminate y from (1) and (2), and seek the expressions for x,

Ι -- 2, and vx, in terms of g and of the symbols τ, ν΄, τὺ, w’.

The above will include all the possible forms of the subject —
of the conclusion. The form v(1 -- 1) is excluded, inasmuch —

as we cannot from the interpretation vr=Some X’s, given
in the premises, interpret v(1—%) as Some not-X’s. . The
symbol v, when used in the sense of ‘‘ some,” applies to that
term only with which it is connected in the premises.

The results of the analysis are as follows :

x =[vv'we"' + τίου —w)(1 -- w’) + ww'(1 -- v)(1 -- ΟἽ)

Ἐ( -ο)ᾷ -- }}]6 Ὁ Ξ(ονζα - ΟἽ + 1-0} (1 - 2), αν

OF THE ARISTOFELIAN: LOGIC 247
1-x«=[v(1-v'){ww'+(1-w)(1-w’)} +0(1 -w)w’

+ ={o0(1 —w)(1 —w’)+ww'(1 —v)(1 -v') + (1 -v)(1 —w)} ]2
+[o(1 — woo" + 2{00'(1 -w')+1-v}](1-3), (II.)
vx = {vv ww + =v (1 - w)(1- w!)}a+ (1 —w')(1 -- 2). (III.)

Each of these expressions involves in its second member
two terms, of one of which z is a factor, of the other 1 —z.
But syllogistic inference does not, asa matter of form, admit
of contrary classes in its conclusion, as of Z’s and not-Z’s
together. [234] We must, therefore, in order to determine
the rules of that species of inference, ascertain under what
conditions the second members of any of our equations are
reducible to a single term.

The simplest form is (III.), and it is reducible to a single
term if w’=1. The equation then becomes

Vx = UU'WE, (3)

the first member is identical with the extreme in the first
premiss ; the second is of the same quantity and quality as
“the extreme in the second premiss. For since w’=1, the
second member of (2), involving the middle term y, is uni-
versal; therefore, by the hypothesis, the first member is
particular, and therefore, the second member of (3), involv-
ing the same symbol w in its coefficient, is particular also.
Hence we deduce the following law.

CONDITION OF INFERENCE.—One middle term, at least,
universal.
RULE OF INFERENCE.—FE quate the extremes.

From an analysis of the equations (I.) and (II.), it will
further appear, that the above is the only condition of syl-
logistic inference when the middle terms are of like quality.
Thus the second member of (I.) reduces to a single term, if
w’=1 and v=1; and the second member of (II.) reduces to
a single term, if w’=1, v=1, w=1. Ineach of these cases,

248 THE LAWS OF THOUGAT

it is necessary that w’=1, the solely sufficient condition —

before assigned.

Consider, secondly, the case in which the middle terms .

are of unlike quality. The premises may then be represented ©

under the forms

Ὁ Ξ vy, (4) :

we=w'(1—Yy); (5)

and if, as before, we eliminate y, and determine the ex- |

pressions of 4, 1—%, and vz, we get

x =[vu'(1 —w)w' + = {z020'(1 —v)+(1-v)(1-v)(1 -w)

+u (1 —-w)(1 —w’)}|z

+ [vow +={(1 - γα -- σ) τ σα -- «ὐ}] -- 2. (IV)

[235] ,- x [ww'v +0(1 —v)(1 -- w) + = {zoe (1 - )

+(1-v)(1 -- σγ -- τ) +0 (1 -w)(1 —w’)} 2

+[o(1 -ογ)ετίσα -w')+(1-v)(1-v)} (1-2). (V.) !

vx = {vv' (τ — w)w’ + ~ov' (1 —w)(1 —w’)}z
+ {vv'w' + sou! (1 —w')}(1-2). (VI.)

Now the second member of (VI.) reduces to a single term
relatively to z, if w=1, giving

x= (υστυ' + 200'(t -- τυΎ}(ι -- 2);

the second member of which is opposite, both in quantity
and quality, to the corresponding extreme, wz, in the second
premiss. For since w=1, wz is universal. But the factor
vv’ indicates that the term to which it is attached is particular,
since by hypothesis v and v’ are not both equaltos. Hence

we deduce the following law of inference in the case of like
middle terms:

FIRST CONDITION OF INFERENCE. —Az least one universal
extreme.

RULE OF INFERENCE.—Change the quantity and quality
of that extreme, and equate the result to the other extreme.

OF THE ARISTOTELIAN LOGIC 240

Moreover, the second member of (V.) reduces to a single
ferm if v'=1, w =1; it then gives

heer {v2 + τα - v)w}s.

Now since v’=1, w’=1, the middle terms of the premises
are both universal, therefore the extremes uv, wz, are par-
ticular. But in the conclusion the extreme involving + is
opposite, both in quantity and quality, to the extreme vz in
the first premiss, while the extreme involving z agrees both
in quantity and quality with the corresponding extreme wz
in the second premiss. Hence the following general law:

[236] SECOND CONDITION OF INFERENCE.— Two univer-
sal middle terms.

RULE OF INFERENCE.—Change the quantity and quality
of etther extreme, and equate the result to the other extreme

unchanged.

There are in the case of unlike middle terms no other
conditions or rules of syllogistic inference than the above.
Thus the equation (1V.), though reducible to the form of
a syllogistic conclusion, when w=1 and v=1, does not

“thereby establish a new condition of inference; since, by
what has preceded, the single condition v=1, or w=1,
would suffice.

8. The following examples will sufficiently illustrate the
general rules of syllogism above given :

I. All Y’s are X’s.
AllZ’s are: V's:

This belongs to Case 1. All Y’s is the universal middle
term. The extremes equated give as the conclusion

All Z’s are X’s;

the universal term, All Z’s, becoming the subject ; the par-
ticular term (some) X’s, the predicate.

2. All Χ s are. V/s.
No Z’s are Y’s.

250 THE LAWS OF THOUGHT

The proper expression of these premises is

AliX’s are) ‘s;
All Z’s are ποι: Y’s.

They belong to Case 2, and satisfy the first condition of ἡ
inference. The middle term, Y’s, in the first premiss, is ἢ
particular-affrmative ; that in the second premiss, not- Y’s,
particular-negative. If we take all Z’s as the universal ex-
treme, and change its quantity and quality according to the
rule, we obtain the term Some not-Z’s, and this equated
with the other extreme, All X’s, gives,

All X’s are not-2’s, z.e., No X’s are Z’s.

If we commence with the other universal extreme, and pro-
ceed similarly, we obtain the equivalent result,

No Z’s are X’s.

[237] Be All Y's are Xs:
All not- Y’s are Z’s.

Here also the middle terms are unlike in quality. The pre-
mises therefore belong to Case 2, and there being two uni-
versal middle terms, the second condition of inference is
satisfied. If by the rule we change the quantity and quality
of the first extreme, (some) X’s, we obtain All not-X’s, which,
equated with the other extreme, gives

All not-X’s are Z’s.

The reverse order of procedure would give the equivalent
result,
All not-Z’s are X’s.

The conclusions of the two last examples would not be re-
cognised as valid in the scholastic system of Logic, which
virtually requires that the subject of a proposition should be
affirmative. They are, however, perfectly legitimate in them-
selves, and the rules by which they are determined form un-
doubtedly the most general canons of syllogistic inference.
The process of investigation by which they are deduced will
probably appear to be of needless complexity ; and it is cer-
tain that they might have been obtained with greater facility,

OF THE ARISTOTELIAN LOGIC ΖΕῚ

«

and without the aid of any symbolical instrument whatever.
It was, however, my object to conduct the investigation in
the most general manner, and by an analysis thoroughly
exhaustive. With this end in view, the brevity or prolixity
of the method employed is a matter of indifference. Indeed
the analysis is not properly that of the syllogism, but of a
much more general combination of propositions ; for we are
permitted to assign to the symbols Ὁ, σ΄, w, w’, any class-
interpretations that we please. To illustrate this remark, I
will apply the solution (I.) to the following imaginary case :

Suppose that a number of pieces of cloth striped with
different colours were submitted to inspection, and that the
two following observations were made upon them :

Ist. That every piece striped with white and green was
also striped with black and yellow, and vce versa.

2nd. That every piece striped with red and orange was
also striped with blue and yellow, and vice versa.

[238] Suppose it then required to determine how the
pieces marked with green stood affected with reference to the
colours white, black, red, orange, and blue.

Here (1) we assume 7— white, τ orecn. Φι black, 7 —
yellow, w=red, =orange, w’=blue, the expression of our
premises will be

IX =U,
Ws=W'y,

agreeing with the system (1) (2). The equation (I.) then
leads to the following conclusion :

Pieces striped with green are either striped with orange,
white, black, red, and blue, together, all pieces possessing
which character are included in those striped with green ; or
they are striped with orange, white, and black, but not with
red or blue; or they are striped with orange, red, and blue,
_ but not with white or black ; or they are striped with orange,
but not with white or red; or they are striped with white
and black, but not with blue or orange; or they are striped
neither with white nor orange.

Considering the nature of this conclusion, neither the

252 THE LAWS OF THOUGHT

symbolical expression (I.) by which it is conveyed, nor the
analysis by which that expression is deduced, can be con-
sidered as needlessly complex. |

9. The form in which the doctrine of syllogism has been
presented in this chapter affords ground for an important
observation. We have seen that in each of its two great
divisions the entire discussion is reducible, so far, at least,
as concerns the determination of rules and methods, to the
analysis of a pair of equations, viz., of the system (1), (2),
when the premises have like middle terms, and of the system
(4), (5), when the middle terms are unlike. Moreover, that
analysis has been actually conducted by a method founded
upon certain general laws deduced immediately from the con-
stitution of language, Chap. II., confirmed by the study of
the operations of the human mind, Chap. IILI., and proved
to be applicable to the analysis of all systems of equations
whatever, by which propositions, or combination of proposi-
tions, can be represented, Chap. VIII. Here, then, we have
the means of definitely resolving the question, whether
syllogism is indeed the fundamental type of reasoning, —
whether [239] the study of its laws is co-extensive with the
study of deductive logic. For if it be so, some indication
of the fact must be given in the systems of equations upon
the analysis of which we have been engaged. It cannot be
conceived that syllogism should be the one essential process
of reasoning, and yet the manifestation of that process present
nothing indicative of this high quality of pre-eminence. No
sign, however, appears that the discussion of all systems of
equations expressing propositions is involved in that of the
particular system examined in this chapter. And yet writers
on Logic have been all but unanimous in their assertion,
not merely of the supremacy, but of the universal sufficiency
of syllogistic inference in deductive reasoning. The language
of Archbishop Whately, always clear and definite, and on
the subject of Logic entitled to peculiar attention, is very ex-
press on this point. ‘‘ For Logic,” he says, ‘‘ which is, as
it were, the Grammar of Reasoning, does not bring forward

OR PHP ARTS OTELIAN LOGIC 253

the regular Syllogism as a dzstinct mode of argumentation, de-
signed to be substituted for any other mode ; but as the form
to which al/ correct reasoning may be ultimately reduced.” *
And Mr. Mill, in a chapter of his System of Logic, entitled,
‘“‘Of Ratiocination or Syllogism,” having enumerated the
ordinary forms of syllogism, observes, ‘‘ All valid ratiocina-
tion, all reasoning by which from general propositions pre-
viously admitted, other propositions, equally or less general,
are inferred, may be exhibited in some of the above forms.”
And again : ‘‘ Weare therefore at liberty, in conformity with
the general opinion of logicians, to consider the two ele-
mentary forms of the first figure as the universal types of all
correct ratiocination.” In accordance with these views it has
been contended that the science of Logic enjoys an immunity
from these conditions of imperfection and of progress to
which all other sciences are subject ;+ and its origin from the
- travail of one mighty mind of old has, by a somewhat daring
metaphor, been compared to the mythological birth of Pallas.

As Syllogism isa species of elimination, the question before
us manifestly resolves itself into the two following ones :—
ist. Whether all elimination is reducible to Syllogism ;
_ 2ndly. Whe-[240]ther deductive reasoning can with propriety
be regarded as consisting only of elimination. I believe,
upon careful examination, the true answer to the former
question to be, that it is always theoretically possible so to
resolve and combine propositions that elimination may sub-
sequently be effected by the syllogistic canons, but that the
process of reduction would in many instances be constrained
and unnatural, and would involve operations which are not
syllogistic. To the second question I reply, that reasoning
cannot, except by an arbitrary restriction of its meaning, be
confined to the process of elimination. No definition can
suffice which makes it less than the aggregate of the methods
which are founded upon the laws of thought, as exercised
upon propositions ; and among those methods, the process

* Elements of Logic, p. 13, ninth edition.
+ Introduction to Kant’s δορά.

254 THE LAWS OF THOCGH

of elimination, eminently important as it is, occupies only a —
place.

Much of the error, as I cannot but regard it, which pre-
vails respecting the nature of the Syllogism and the extent
of its office, seems to be founded in a disposition to regard all
those truths in Logic as przmary which possess the character
of simplicity and intuitive certainty, without inquiring into
the relation which they sustain to other truths in the Science,
or to general methods in the Art, of Reasoning. Arristotle’s
dictum de omni et nullo is a self-evident principle, but it is
not found among those #étzmate laws of the reasoning faculty
to which all other laws, however plain and self-evident, admit
of being traced, and from which they may in strictest order
of scientific evolution be deduced. For though of every
science the fundamental truths are usually the most simple
of apprehension, yet is not that simplicity the criterion by
which their title to be regarded as fundamental must be
judged. This must be sought for in the nature and extent
of the structure which they are capable of supporting. Tak-
ing this view, Leibnitz appears to me to have judged cor-
rectly when he assigned to the ‘‘ principle of contradiction ”
a fundamental place in Logic :* for we have seen the conse-
quences of that law of thought of which it is the axiomatic
expression (III. 15). But enough has been said upon the
nature of deductive inference and upon its constitutive ele-
ments. The subject of [241] induction may probably re-
ceive some attention in another part of this work.

10. It has been remarked in this chapter that the ordinary
treatment of hypothetical, is much more defective than that
of categorical, propositions. What is commonly termed the
hypothetical syllogism appears, indeed, to be no syllogism
at all. Let the argument—

A 5 τ ἀπ ἋΣ
But 4 is B,
Therefore C is D,

* Nouveaux Essais sur lentendement humain, Liv. IV. cap. 2; Théodicée,
Pt. I. sec. 44.

OF THE ARIS TOLPELIAN LOCTC 255

be put in the form—

If the proposition X is true, Υ is true,
But X is true,
Therefore Y is true ;

wherein by XY is meant the proposition A is B, and by Y,

the proposition Cis D. It is then seen that the premises
contain only two terms or elements, while a syllogism essen-
tially involves three. The following would be a genuine
hypothetical syllogism :

If X is true, Y is true;
Τὺ Υ is true, 21s tues
If X is true, Z is true.

After the discussion of secondary propositions in a former
part of this work, it is evident that the forms of hypothetical
syllogism must present, in every respect, an exact counter-
part to those of categorical syllogism. Particular Proposi-
tions, such as, ‘‘ Sometimes if XY is true, Y is true,” may be
introduced, and the conditions and rules of inference de-
duced in this chapter for categorical syllogisms may, with-
out abatement, be interpreted to meet the corresponding
cases in hypotheticals.

11. To what final conclusions are we then led respecting
the nature and extent of the scholastic logic? I think to the
following : that it is not a science, but a collection of scientific
truths, too incomplete to form a system of themselves, and
not sufficiently fundamental to serve as the foundation upon
which a perfect system may rest. It does not, however,
follow, that because the logic of the schools has been in-
vested with attributes to which it [242] has no just claim,
it is therefore undeserving of regard. A system which has
been associated with the very growth of language, which
has left its stamp upon the greatest questions and the most
famous demonstrations of philosophy, cannot be altogether
unworthy of attention. Memory, too, and usage, it must
be admitted, have much to do with the intellectual processes ;
and there are certain of the canons of the ancient logic which

256 THE LAWS OF THOUGHT

have become almost inwoven in the very texture of thought
in cultured minds. But whether the mnemonic forrns, in
which the particular rules of conversion and syllogism have

been exhibited, possess any real utility,—whether the very ~

skill which they are supposed to impart might not, with
greater advantage to the mental powers, be acquired by the
unassisted efforts of a mind left to its own resources, —are
questions which it might still be not unprofitable to examine.
As concerns the particular results deduced in this Chapter,
it is to be observed, that they are solely designed to aid the
inquiry concerning the nature of the ordinary or scholastic
logic, and its relation to a more perfect theory of deductive
reasoning.

ΠΡ ΓΉΡΕΦΗ ὍΝ:

[243] CREAT TEREX
OF THE THEORY OF PROBABILITIES

I. BEFORE the expiration of another year just two centuries
will have rolled away since Pascal solved the first known
question in the theory of Probabilities, and laid, in its solu-
tion, the foundations of a science possessing no common
share of the attraction which belongs to the more abstract
of mathematical speculations. The problem which the
Chevalier de Méré, a reputed gamester, proposed to the
recluse of Port Royal (not yet withdrawn from the interests
of science * by the more distracting contemplation of the
‘‘sreatness and the misery of man”), was the first of a
long series of problems, destined to call into existence new
methods in mathematical analysis, and to render valuable
service in the practical concerns of life. - Nor does the
interest of the subject centre merely in its mathematical
connexion, or its associations of utility. The attention is
repaid which is devoted to the theory of Probabilities as an
independent object of speculation,—to the fundamental
modes in which it has been conceived, —to the great second-
ary principles which, as in the contemporaneous science of
Mechanics, have gradually been annexed to it,—and, lastly,
to the estimate of the measure of perfection which has been
actually attained. I speak here of that perfection which
consists in unity of conception and harmony of processes.
Some of these points it is designed very briefly to consider
in the present Chapter.

* See in particular a letter addressed by Pascal to Fermat, who had solicited
his attention to a mathematical problem (/o7¢ Royal, par M. de Sainte Beuve) ;
also various passages in the collection of Fragments published by M. Prosper
Faugere.

257 | 17,

258 THE PAWS OF ΤΉ ΟΕ

2, A distinguished writer * has thus stated the funda-_
mental definitions of the science :

[244] ‘‘The probability of an event is the reason we —
have to believe that it has taken place, or that it will take ᾿

lace; ‘
: ‘©The measure of the probability of an event is the ratio
of the number of cases favourable to that event, to the total —
number of cases favourable or contrary, and all equally —
possible ” (equally like to happen).

From these definitions it follows that the word prodadzlity, —
in its mathematical acceptation, has reference to the state —
of our knowledge of the circumstances under which an event —
may happen or fail. With the degree of information which ~
we possess concerning the circumstances of an event, the :
reason we have to think that it will occur, or, to use a single —
term, our expectation of it, will vary. Probability is ex--
pectation founded upon partial knowledge. A perfect ac-
quaintance with αὐ πα circumstances affecting the occurrence
of an event would change expectation into certainty, and _
leave neither room nor demand for a theory of probabilities.

3. Though our expectation of an event grows stronger —
with the increase of the ratio of the number of the known
cases favourable to its occurrence to the whole number of
equally possible cases, favourable or unfavourable, it would —
be unphilosophical to affirm that the strength of that ex-
pectation, viewed as an emotion of the mind, is capable οἵ.
being referred to any numerical standard. The man off
sanguine temperament builds high hopes where the timid
despair, and the irresolute are lost in doubt. As subjects
of scientific inquiry, there is some analogy between opinion
and sensation. ‘The thermometer and the carefully prepared
photographic plate indicate, not the intensity of the sensa-
tions of heat and light, but certain physical circumstances
which accompany the production of those sensations. So
also the theory of probabilities contemplates the numerical
measure of the circumstances upon which expectation is

* Poisson, Recherches sur la Probabilité des Jugemens.

Care
oe

OF THE THEORY OF PROBABILITIES Στ

founded ; and this object embraces the whole range of its
legitimate applications. The rules which we employ in life-

_ assurance, and in the other statistical applications of the

theory of probabilities, are altogether independent of the
mental phenomena of expectation. They are founded
upon the assumption that the future will bear a resem-
[245]blance to the past; that under the same circum-
stances the same event will tend to recur with a definite
numerical frequency ; not upon any attempt to submit to
calculation the strength of human hopes and fears.

Now experience actually testifies that events of a given
species do, under given circumstances, tend to recur with
definite frequency, whether their true causes be known to us
or unknown. Of course this tendency is, in general, only
manifested when the area of observation is sufficiently large.
The judicial records of a great nation, its registries of births
and deaths, in relation to age and sex, &c., present a re-
markable uniformity from year to year. Ina given language,
or family of languages, the same sounds, and successions of
sounds, and, if it be a written language, the same characters
and successions of characters recur with determinate fre-

quency. The key to the rude Ogham inscriptions, found in

various parts of Ireland, and in which no distinction of words
could at first be traced, was, by a strict application of this
principle, recovered.* The same method, it is understood,
has been applied Ὁ to the deciphering of the cuneiform records
recently disentombed from the ruins of Nineveh by the
enterprise of Mr. Layard.

4. Let us endeavour from the above statements and
definitions to form a conception of the legitimate object of
the theory of Probabilities.

Probability, it has been said, consists in the expectation
founded upon a particular kind of knowledge, viz., the

* The discovery is due to the Rev. Charles Graves, Professor of Mathematics
in the University of Dublin. —Vide Proceedings of the Royal [rish Academy, Feb.
14, 1848. Professor Graves informs me that he has verified the principle by con-
structing sequence tables for all the European languages,

+ By the learned Orientalist, Dr. Edward Hincks.

260 THE LAWS CF ἀντ

knowledge of the relative frequency of occurrence of events.
Hence the probabilities of events, or of combinations of
events, whether deduced from a knowledge of the particular
constitution of things under which they happen, or derived
from the long-continued observation of past series of their
occurrences and failures, constitute, in all cases, our data.
The probability of some [246] connected event, or com- |
bination of events, constitutes the corresponding gu@eszttum,
or object sought. Now in the most general, yet strict
meaning of the term ‘‘ event,” every combination of events
constitutes also an event. The simultaneous occurrence of
two or more events, or the cccurrence of an event under
given conditions, or in any conceivable connexion with
other events, is still an event. Using the term in this liberty
of application, the object of the theory of probabilities
might be thus defined. Given the probabilities of any
events, of whatever kind, to find the probability of some
other event connected with them.

5. Events may be distinguished as simple or compound,
the latter term being applied to such events as consist in a
combination of simple events (I. 13). In this manner we
might define it as the practical end of the theory under con-
sideration to determine the probability of some event, simple
or compound, from the given probabilities of other events,
simple or compound, with which, by the terms of its defini-
tion, it stands connected.

Thus if it is known from the constitution of a die that
there is a probability, measured by the fraction Η͂ that the
result of any particular throw will be an ace, and if it is
required to determine the probability that there shall occur
one ace, and only one, in two successive throws, we may

state the problem in the order of its data and its guesitum,
as follows :

First DATUM.—Probability of the event that the first

throw will give an ace ==.

OF 2HE- THEORY OF PROBABILITIES: 261
SECOND DATUM.—Probability of the event that the second
throw will give an ace =5.

QuASITUM.—Probability of the event that either the first

throw will give an ace, and the second not an ace; or the

7

first will not give an ace, and the second will give one.

Here the two data are the probabilities of simple events
defined as the first throw giving an ace, and the second
throw giving an ace. The quesitum is the probability of a
compound event,—a certain disjunctive combination of the
simple events [247] involved or implied in the data. Pro-
bably it will generally happen, when the numerical con-
ditions of a problem are capable of being deduced, as above,
from the constitution of things under which they exist, that
the data will be the probabilities of szmple events and the
quzsitum the probability of a compound event dependent
upon the said simple events. Such is the case with a class
of problems which has occupied perhaps an undue share of
the attention of those who have studied the theory of
probabilities, viz., games of chance and skill, in the former
of which some physical circumstance, as the constitution of
a die, determines the probability of each possible step of the
game, its issue being some definite combination of those
meps; while in the latter, the relative. dexterity of the
players, supposed to be known ὦ priort, equally determines
the same element. But where, as in statistical problems,
the elements of our knowledge are drawn, not from the study
of the constitution of things, but from the registered ob-
servations of Nature or of human society, there is no reason
why the data which such observations afford should be the
probabilities of simple events. On the contrary, the occur-
rence of events or conditions in marked combinations (indi-
cative of some secret connexion of a causal character)

‘suggests to us the propriety of making such concurrences

profitable for future instruction by a numerical record of
their frequency. Now the data which observations of this
kind afford are the probabilities of compound events. The

262 LHE LAWS OF THOUGHT

solution, by some general method, of problems in which such
data are involved, is thus not only essential to the perfect
development of the theory of probabilities, but also a perhaps
necessary condition of its application to a large and practi-
cally important class of inquiries.

6. Before we proceed to estimate to what extent known
methods may be applied to the solution of problerns such
as the above, it will be advantageous to notice, that there is
another form under which all questions in the theory of
probabilities may be viewed ; and this form consists in sub-
stituting for evexts the propositions which assert that those
events have occurred, or will occur; and viewing the ele-
ment of numerical probability as having reference to the
truth of those propositions, not to the oc-[248 ]currence of the
events concerning which they make assertion. Thus, instead
of considering the numerical fraction 2. as expressing the pro-
bability of the occurrence of an event £, let it be viewed as
representing the probability of the truth of the proposition
“ΧΟ, which asserts that the event A will occur. Similarly,
instead of any probability, g, being considered as referring
to some compound event, such as the concurrence of the
events # and F, let it represent the probability of the truth
of the proposition which asserts that # and F will jointly
occur ; and in like manner, let the transformation be made
from disjunctive and hypothetical combinations of events to
disjunctive and conditional propositions. Though the new
application thus assigned to probability is a necessary con-
comitant of the old one, its adoption will be attended with
a practical advantage drawn from the circumstance that we
have already discussed the theory of propositions, have
defined their principal varieties, and established methods for
determining, in every case, the amount and character of
their mutual dependence. Upon this, or upon some equiva-
lent basis, any general theory of probabilities must rest. 1
do not say that other considerations may not in certain
cases of applied theory be requisite. The data may prove
insufficient for definite solution, and this defect it may be

Ee ne Re: are

a

ΨΥ ΤΠ

΄'-
μι

a se

ἘΝ

ee ee es eee ket See

OF THE Thon y OF PROBABILITIES. «262

thought necessary to supply by hypothesis. Or, where the
statement of large numbers is involved, difficulties may arise
after the solution, from this source, for which special methods
of treatment are required. But in every instance, some form
of the general problem as above stated (Art. 4) is involved,
and in the discussion of that problem the proper and peculiar
work of the theory consists. I desire it to be observed, that
to this object the investigations of the following chapters
are mainly devoted. It is not intended to enter, except
incidentally, upon questions involving supplementary hypo-
theses, because it is of primary importance, even with re-
ference to such questions (I. 17), that a general method,
founded upon a solid and sufficient basis of theory, be first
established.

7. The following is a summary, chiefly taken from Laplace,
of the principles which have been applied to the solution
of questions of probability. They are consequences of its
fundamental defini-[249]tions already stated, and may be
regarded as indicating the degree in which it has been found
possible to render those definitions available.

PRINCIPLE Ist. If p be the probability of the occurrence of
any event, 1 —/ will be the probability of its non-occurrence.

2nd. The probability of the concurrence of two inde-
pendent events is the product of the probabilities of those
events.

3rd. The probability of the concurrence of two depen-
dent events is equal to the product of the probability of one
of them by the probability that if that event occur, the other
will happen also,

4th. The probability that if an event, 55, take place, an
event, /, will also take place, is equal to the probability of
the concurrence of the events / and F, divided by the pro-
-bability of the occurrence of 2.

5th. The probability of the occurrence of one or the other
of two events which cannot concur is equal to the sum of
their separate probabilities.

6th. If an observed event can only result from some one

264. THE LAWS OF THOUGHT

of different causes which are a przort equally probable, the
probability of any one of the causes is a fraction whose |
numerator is the probability of the event, on the hypothesis
of the existence of that cause, and whose denominator is the —

sum of the similar probabilities relative to all the causes,

7th. The probability of a future event is the sum of the —
products formed by multiplying the probability of each cause —
by the probability that if that cause exists, the said future —

event will take place.

8. Respecting the extent and the relative sufficiency οὐ

these principles, the following observations may be made.

ist. It is always possible, by the due combination of these
principles, to express the probability of a compound event,
dependent in any manner upon independent simple events
whose distinct probabilities are given. A very large pro-
portion of the problems which have been actually solved are
of this kind, and the difficulty attending their solution has
not arisen from the insufficiency of the indications furnished
by the theory of probabilities, but from the need of an
analysis which should render [250] those indications avail-
able when functions of large numbers, or series consisting of
many and complicated terms, are thereby introduced. It
may, therefore, be fully conceded, that all problems having
for their data the probabilities of independent simple events
fall within the scope of received methods.

2ndly. Certain of the principles above enumerated, and
especially the sixth and seventh, do not presuppose that all
the data are the probabilities of simple events. In their
peculiar application to questions of causation, they do, how-
ever, assume, that the causes of which they take account are
mutually exclusive, so that no combination of them in the
production of an effect is possible. If, as before explained,
we transfer the numerical probabilities from the events with
which they are connected to the propositions by which those
events are expressed, the most general problem to which
the aforesaid principles are applicable may be stated in the
following order of data and guesita.

*
ht Se

OF THE THEORY OF PROBABILITIES 265

DATA
ist. The probabilities of the z conditional propositions :

If the cause A, exist, the event & will follow;
9 As ” i ”

3) ihe 2) » 2)
2nd. The condition that the antecedents of those pro-
positions are mutually conflicting,

REQUIREMENTS

The probability of the truth of the proposition which de-
clares the occurrence of the event / ; also, when that pro-
position is known to be true, the probabilities of truth of the
several propositions which affirm the respective occurrences
of the causes A,, A,. . A,.

Here it is seen, that the data are the probabilities of a
series of compound events, expressed by condttional pro-
positions. But the system is obviously a very limited and
particular one. For the antecedents of the propositions are
subject to the condition of being mutually exclusive, and
Ehere is but one consequent, the event 2, in the whole
system. It does not follow, from our [251] ability to dis-
cuss such a system as the above, that we are able to resolve
problems whose data are the probabilities of azy system of
conditional propositions ; far less that we can resolve pro-
blems whose data are the probabilities of any system of pro-
positions whatever. And, viewing the subject in its material
rather than its formal aspect, it is evident, that the hypothesis
of exclusive causation is one which is not often realized in
the actual world, the phenomena of which seem to be, usually,
the products of complex causes, the amount and character
of whose co-operation is unknown. Such is, without doubt,
the case in nearly all departments of natural or social inquiry
in which the doctrine of probabilities holds out any new
promise of useful applications.

g. To the above principles we may add another, which

266 THE TAWS Of FHOUGTT.

has been stated in the following terms by the Savilian Pro-
fessor of Astronomy in the University of Oxford.*

‘Principle 8. If there be any number of mutually ex-
clusive hypotheses, /,, ἅς, 43, . . of which the probabilities
relative to a particular state of information are 2., 24» 23» - -
and if new information be given which changes the proba-
bilities of some of them, suppose of %,,,, and all that follow,
without having otherwise any reference to the rest ; then the
probabilities of these latter have the same ratios to one
another, after the new information, that they had before,
that is,

Pit hot P's + +t BP m=Pi i Poi Pg + - +t Pm
where the accented letters denote the values after the new
information has been acquired.”

This principle is apparently of a more fundamental char-
acter than the most of those before enumerated, and perhaps
it might, as has been suggested by Professor Donkin, be
regarded as axiomatic. It seems indeed to be founded in
the very definition of the measure of probability, as ‘‘the
ratio of the number of cases favourable to an event to the
total number of cases favourable or contrary, and all equally
possible.” For, adopting this definition, it is evident that
in whatever proportion the number of equally [252] possible
cases is diminished, while the number of favourable cases
remains unaltered, in exactly the same proportion will the
probabilities of any events to which these cases have reference
be increased. And as the new hypothesis, viz., the diminu-
tion of the number of possible cases without affecting the
number of them which are favourable to the events in question,
increases the probabilities of those events in a constant ratio,
the relative measures of those probabilities remain unaltered.
If the principle we are considering be then, as it appears to
be, inseparably involved in the very definition of probability,
it can scarcely, of ztself, conduct us further than the attentive
study of the definition would alone do, in the solution of

* **On certain Questions relating to the Theory of Probabilities” ; by W. F.
Donkin, M.A., F.R.S., &c. Philosophical Magazine, May, 1851.

OF (he THEORY OF PROBABILITIES "δ᾽

problems. From these considerations it appears to be doubt-
ful whether, without some aid of a different kind from any
that has yet offered itself to our notice, any considerable
advance, either in the theory of probabilities as a branch of
speculative knowledge, or in the practical solution of its pro-
blems can be hoped for. And the establishment, soledy upon
the basis of any such collection of principles as the above, of
a method universally applicable to the solution of problems,
without regard either to the number or to the nature of the
propositions involved in the expression of their data, seems
to be impossible. For the attainment of such an object other
elements are needed, the consideration of which will occupy
the next Chapter.

ΠΤ aaa

[253] CHAPTER ov il

DEMONSTRATION OF A GENERAL METHOD FOR THE SOLUTION _
OF PROBLEMS IN THE THEORY OF PROBABILITIES |

1, IT has been defined (XVI. 2), that ‘‘ the measure of the
probability of an event is the ratio of the number of cases —
favourable to that event, to the total number of cases favour- |
able or unfavourable, and all equally possible.” In the ~
following investigations the term probability will be used in —
the above sense of ‘‘ measure of probability.” :
From the above definition we may deduce the following —
conclusions. .
I. When it is certain that an event will occur, the proba-_
bility of that event, in the above mathematical sense, is 1. _
For the cases which are favourable to the event, and the cases"
which are possible, are in this instance the same. |
Hence, also, if be the probability that an event 4 will
happen, 1—f will be the probability that the said event will
not happen. To deduce this result directly from the defini-
tion, let # be the number of cases favourable to the event |
+, nthe number of cases possible, then 2 — is the number of
cases unfavourable to the event 2. Hence, by definition,

κι = probability that x will happen.

see probability that x will not happen.
But 71 --- mM m
on a ee: 2.

II. The probability of the concurrence of any two events
is the product of the probability of either of those events

by the probability that if that event occur, the other will
occur also.
268

GENERAL METHOD IN PROBABILITIES 269

Let m be the number of cases favourable to the happen-
ing of the first event, and 7 the number of equally possible
cases unfavourable to it; then the probability of the first

Of the m cases favour-

: 772
event is, by defini-[254 ]tion,

y [254] oer
able to the first event, let 7 cases be favourable to the con-
junction of the first and second events, then, by definition,

3 is the probability that if the first event happen, the second

also will happen. Multiplying these fractions together, we

have
Cee Tce
m+n m m+n-

; rs
But the resulting fraction τεῖος has for its numerator the

number of cases favourable to the conjunction of events, and
for its denominator, the number 772-22 of possible cases.
Therefore, it represents the probability of the joint occurrence
of the two events.

Hence, if f be the probability of any event 4, and g the
probability that if # occur y will occur, the probability of the
conjunction xy will be pg.

Pile Phe probability thateit ancevent occur; the-event
will occur, is a fraction whose numerator is the probability
of their joint occurrence, and denominator the probability of
the occurrence of the event %.

This is an immediate consequence of Principle II.

IV. The probability of the occurrence of some one of a
series of exclusive events is equal to the sum of their separate
probabilities.

For let z be the number of possible cases ; #, the number
of those cases favourable to the first event ; mm, the number

of cases favourable to the second, &c. Then the separate

My,

ἜΡΟΝ 771 '
probabilities of the events are ΣᾺ: ae &c. Again, as the

events are exclusive, none of the cases favourable to one of
them is favourable to another ; and, therefore, the number of
cases favourable to some one of the series will be #z,+ mm, . . ,

270 THE LAWS OF THOUGHT 4
‘

and the probability of some one of the series happenings
will be M,+Ms-:* But this is the sum of the previous :
7 :

fractions, 72, “2, &c. Whence the principle is manifest.

[255] 2. DEFINITION.—Two events are said to be indedil ὃ
pendent when the probability of the happening of either of ©
them is unaffected by our expectation of the occurrence οὐ
failure of the other. |

From this definition, combined with Principle II., we have —
the following conclusion :

V. The probability of the concurrence of two independent
events is equal to the product of the separate probabilities of —
those events. A

For if p be the probability of an event 7, g that of an event ©
y regarded as quite independent of %, then is g also the pro- :
bability that if x occur y will occur. Hence, by Principle II., —
p@ is the probability of the concurrence of 4 and y.

Under the same circumstances, the probability that a will
occur and y not occur will be p(1—g). For 2 is the proba- :
bility that x will occur, and 1—g the probability that y will
not occur. In like manner (1—p)(1—g¢) will be the proba-
bility that both the events fail of occurring.

3. There exists yet another principle, different in kind from
the above, but necessary to the subsequent investigations of
this Chapter, before proceeding to the explicit statement of
which I desire to make one or two preliminary observations,

I would, in the first place, remark that the distinction be-
tween simple and compound events is not one founded in the ~
nature of events themselves, but upon the mode or connexion
in which they are presented to the mind. How many ©
separate particulars, for instance, are implied in the terms
‘© To be in health,” “Το prosper,” &c., each of which might
still be regarded as expressing a ‘‘simple event”? The
prescriptive usages of language, which have assigned to par-
ticular combination of events single and definite appellations,
and have left unnumbered other combinations to be expressed —

GENERAL METHOD IN PROBABILITIES 271

by corresponding combinations of distinct terms or phrases,
is essentially arbitrary. When, then, we designate as simple
events those which are expressed by a single verb, or by what
grammarians term a single sentence, we do not thereby imply
any real simplicity in the events themselves, but use the term
solely with reference to grammatical expression.

[256] 4. Now if this distinction of events, as simple or
compound, is not founded in their real nature, but rests upon
the accidents of language, it cannot affect the question of their
mutual dependence or independence. If my knowledge of
two simple events is confined to this particular fact, viz., that
the probability of the occurrence of one of them is f, and that
of the other g ; then I regard the events as independent, and
thereupon affirm that the probability of their joint occurrence
is fg. But the ground of this affirmation is not that the
events are simple ones, but that the data afford no infor-
mation whatever concerning any connexion or dependence
between them. When the probabilities of events are given,
but all information respecting their dependence withheld,
the mind regards them as independent. And this mode of
thought is equally correct whether the events, judged accord-
_ing to actual expression, are simple or compound, z.eé., whether
each of them is expressed by a single verb or by a combination
of verbs.

5. Let it, however, be supposed that, together with the
probabilities of certain events, we possess some definite in-
formation respecting their possible combinations, For ex-
ample, let it be known that certain combinations are excluded
from happening, and therefore that the remaining combina-
tions alone are possible. Then still is. the same general
principle applicable. The mode in which we avail ourselves
of this information in the calculation of the probability of any
‘conceivable issue of events depends not upon the nature of
the events whose probabilities and whose limits of possible
connexion are given. It matters not whether they are simple
or compound. It is indifferent from what source, or by what
methods, the knowledge of their probabilities and of their

272 THE LAWS OF THOUGHT

—

connecting relations has been derived. We must regard the
events as independent of any connexion beside that of which
we have information, deeming itt of no consequence whether

such information has been explicitly conveyed tous in the data, —

or thence deduced by logical inference. And this leads us to
the statement of the general principle in question, viz.:

VI. The events whose probabilities are given are to be
regarded as independent of any connexion but such as is
either expressed, or necessarily implied, in the data ; and the
mode in [257] which our knowledge of that connexion is to
be employed is independent of the nature of the source from
which such knowledge has been derived.

The practical importance of the above principle consists
in the circumstance, that whatever may be the nature of the
events whose probabilities are given,—whatever the nature
of the event whose probability is sought, we are always able,
by an application of the Calculus of Logic, to determine the
expression of the latter event as a definite combination of
the former events, and definitely to assign the whole of the
implied relations connecting the former events with each
other. In other words, we can determine what that com-
bination of the given events is whose probability is required,
and what combinations of them are alone possible. It follows
then from the above principle, that we can reason upon those
events as if they were simple events, whose conditions of
possible combination had been directly given by experience,
and of which the probability of some definite combination is
sought. The possibility of a general method in probabilities
depends upon this reduction.

6. As the investigations upon which we are about to enter
are based upon the employment of the Calculus of Logic, it
is necessary to explain certain terms and modes of expression
which are derived from this application.

By the event +, 1 mean that event of which the proposi-
tion which affirms the occurrence is symbolically expressed
by the equation

Ft
‘
ἕ
4
Z
re
&
4
es

GENERAL METHOD IN PROBABILITIES 273

By the event (4, y, 2, ..), I mean that event of which
_ the occurrence is expressed by the equation

Dy Vee) τ
Such an event may be termed a compound event, in relation
to the simple events x, 7, 2, which its conception involves.
Thus, if + represent the event ‘‘It rains,” y the event ‘‘ It

thunders,” the separate occurrences of those events being
expressed by the logical equations

x=T, yY=!I,

then will #(1—y)+y(1 —%) represent the event or state of
[258] things denoted by the Proposition, ‘‘It either rains
or thunders, but not both”; the expression of that state of
things being

x(1—y)+y(1 —x)=1.
If for brevity we represent the function (4%, », 2, . .), used
in the above acceptation by V, it is evident (VI. 13) that
the law of duality

Vir- V)=o

will be identically satisfied.

The simple events 4, y, 5. will be said to be ‘‘ conditioned ”
when they are not free to occur in every possible combina-
tion ; in other words, when some compound event depending
upon them is precluded from occurring. Thus the events
denoted by the propositions, ‘‘ It rains,” ‘‘ It thunders,” are
‘“conditioned” if the event denoted by the proposition,
“ΤῸ thunders, but does not rain,” is excluded from happen-
ing, so that the range of possible is less than the range of con-
ceivable combination. Simple unconditioned events are by
definition independent.

Any compound event is similarly said to be conditioned
if it is assumed that it can only occur under a certain con-
dition, that is, in combination with some other event consti-
tuting, by its presence, that condition.

7. We shall proceed in the natural order of thought, from
simple and unconditioned, to compound and conditioned

events.
18

274 THE LAWS OF THOUGHT

PROPOSITION I

ist. If p, φ, v are the respective probabilities of any uncon-
ditioned simple events x,y, 2, the probability of any compound
event V will be [V], this function [V] being formed ὄν
changing, in the function V, the symbols x, y, 2 tntop, Φ, γ, ὅς. :
andly. Under the same circumstances, the probability that —
if the event V occur, any other event V' will also occur, will :

be ae wherein [7 17] denotes the result obtained by multt-

plying together the logical functions V and V’, and changing ©
in the vesult x, y, 8, &c. tnto p, gq, r, Ge.

Let us confine our attention in the first place to the pos- ἢ
[259 ]sible combinations of the two simple events, x and y, of |
which the respective probabilities are and g. The primary :
combinations of those events (V. 11), and their corresponding
probabilities, are as follows :

EVENTS PROBABILITIES
XY, Concurrence of x andy, =f.
x(1 -- γ), Occurrence of x without y, Ζ(1 -- 9).
(ι -- x)y, Occurrence of y without x, (1 -- 2).

(1 —x)(1 ~y), Conjoint failure of « and y, (1 —p)(1 - 4).

We see that in these cases the probability of the compound
event represented by a constituent is the same function of
2 and g as the logical expression of that event is of rand y;
and it is obvious that this remark applies, whatever may be —
the number of the simple events whose probabilities are
given, and whose joint existence or fatlure is involved in the —
compound event of which we seek the probability.

Consider, in the second place, any dzsjunctzve combination
of the above constituents. The compound event, expressed
in ordinary language as the occurrence of ‘‘ either the event
« without the event y, or the event y without the event +,”
is symbolically expressed in the form x(1 —y)+9(1 —*), and |
its probability, determined by Principles IV. and V., is
PU-—g)+¢(1—-p). The latter of these expressions is the
same function of and qg as the former is of x and y. And
it is obvious that this is also a particular illustration of a

GENERAL METHOD IN PROBABILITIES 275

“

_ general rule. The events which are expressed by any two
or more constituents are mutually exclusive. The only
possible combination of them is a dzsyunctive one, expressed
in ordinary language by the conjunction ov, in the language
of symbolical logic by the sign +. Now the probability of
the occurrence of some one out of a set of mutually exclusive
events is the sum of their separate probabilities, and is ex-
pressed by connecting the expression for those separate pro-
babilities by the sign +. Thus the law above exemplified
is seen to be general. The probability of any unconditioned
event V will be found by changing in V the symbols x, y,
me. . into 2, 9, 7...
8. Again, by Principle III., the probability that if the
event V occur, the event V’ will occur with it, is expressed
by a frac-[260]tion whose numerator is the probability of
the joint occurrence of V and V’, and denominator the pro-
bability of the occurrence of V.
Now the expression of that event, or state of things, which
is constituted by the joint occurrence of the events V and V’,
will be formed by multiplying together the expressions V
and Γ᾽ according to the rules of the Calculus of Logic ; since
whatever constituents are found in both V and ”’ will appear
in the product, and no others. Again, by what has just
been shown, the probability of the event represented by that
product will be determined by changing therein x, y, z into
Pp, 7, *,.. Hence the numerator sought will be what [VV]
by definition represents. And the denominator will be [V],
wherefore
PAZ,
[7]

9g. For example, if the probabilities of the simple events
ΠῚ», 2 are 2, g, r respectively, and it is required to find the
probability that if either x or y occur, then either y or 2 will
occur, we have for the logical expressions of the antecedent
and consequent—

Probability that if V occur, V’ will occur with it =

Ist. Either x or y occurs, x(1 —y) +(1 — x).
2nd. Either y or z occurs, γ(1 — 2) + Ζᾷ -- y)-

276 THE LAWS OF THOUGHT

If now we multiply these two expressions together according
to the rules of the Calculus of Logic, we shall have for the
expression of the concurrence of antecedent and consequent,

x2(1 —y) ἘΚ — x)(1 -- 2).
Changing in this result x, y, z into 2, g, 7, and similarly
transforming the expression of the antecedent, we find for
the probability sought the value

prir—9) +91 - pit - 7) |

διτ 2) tote)

The special function of the calculus, in a case like the above,
is to supply the office of the reason in determining what are
the conjunctures involved at once in the consequent and the
antecedent. But the advantage of this application is almost
entirely [261] prospective, and will be made manifest in a
subsequent proposition.

PROPOSITION II

10. /¢t 7s known that the probabilities of certain simple
events, X y, 2,..arep, gq, r,.. respectively when a certain
condition V ts satisfied; V being in expression a function of
4,9, %,.. Regutred the absolute probabilities of the events
%, y, &,.., that ts, the probabilities of thetr respective occur-
rence independently of the condition V.

Let 7’, 7’, 7, &c., be the probabilities. required, z.e. the
probabilities of the events, x, y, z, . ., regarded not only as
simple, but as independent events. Then by Prop. I. the
probabilities that these events will occur when the condition
V, represented by the logical equation V = 1, is satisfied, are

[ΠΕ]. [ΠῚ we,

VAR eee
in which [¢V] denotes the result obtained by multiplying V
by 2, according to the rules of the Calculus of Logic, and
changing in the result x, y, z, into p’, g’, 7, &c. But the
above conditioned probabilities are by hypothesis equal to
p,q, 7%, .. respectively. Hence we have,

GENERAL METHOD IN PROBABILITIES ‘277

[x7] [»7] [2]
ἐπ 2 -- Ζ, =r, &c
i eee I
from which system of equations equal in number to the
- quantities 2, g, 7,.., the values of those quantities may

be determined.

Now +V consists simply of those constituents in V of
which xis a factor. Let this sum be represented by V,,, and
in like manner let yV be represented by V,, &c. Our equa-
tions then assume the form

el ΕΠ
where [17,] denotes the results obtained by changing in V,
the symbols %, y, 2, &c., into ~’, g’, 7, &c.

To render the meaning of the general problem and the
prin-[262]ciple of its solution more evident, let us take the
following example. Suppose that in the drawing of balls
from an urn attention had only been paid to those cases in
which the balls drawn were either of a particular colour,
‘‘white,” or of a particular composition, ‘‘ marble,” or were
marked by both these characters, no record having been kept
of those cases in which a ball that was neither white nor of
marble had been drawn. Let it then have been found, that
whenever the supposed condition was satisfied, there was a
probability 2 that a white ball would be drawn, and a pro-
bability g that a marble ball would be drawn: and from
these data alone let it be required to find the probability
that in the next drawing, without reference at all to the
condition above mentioned, a white ball will be drawn ; also
the probability that a marble ball will be drawn.

Here if x represent the drawing of a white ball, y that of
a marble ball, the condition V will be represented by the
logical function

xy χᾷ -—y)+ (1 —x)y.
Hence we have

Vi=xyta(r—y)=a, Vi=ayt(1-x)y=y;
whence [V.j=2, [Y,)=93

278 THE LAWS OF THOUGHT

and the final equations of the problem are

,

i 9 ἘΠῚ
pg t+p-g7)t7(t-P) eae +P(t-9)+9-f) *
from which we find
Pe) ων Eman) i Ca
2 αν 7 7

It is seen that 2’ and g’ are respectively proportional to 2

and g, as by Professor Donkin’s principle they ought to be.

The solution of this class of problems might indeed, by a .

direct application of that principle, be obtained.
To meet a possible objection, I here remark, that the
above reasoning does not require that the drawings of a

white and a marble ball should be independent, in virtue of —
the physical constitution of the balls. The assumption of ἢ

their independence is indeed involved in the solution, but it
does not rest upon any [263] prior assumption as to the

nature of the balls, and their relations, or freedom from ἢ

relations, of form, colour, structure, &c. It is founded
upon our total ignorance of all these things. Probability
always has reference to the state of our actual knowledge,
and its numerical value varies with varying information.

PROPOSITION III

11. 70 determine in any question of probabilities the logical
connexton of the quesitum with the data; that 75, to assign
the event whose probability ἐς sought, as a logical function of
the event whose probabilities are given.

Let S, 7, &c., represent any compound events whose —

probabilities are given, S and 7 being in expression known

functions of the symbols z, J, 2, &c., representing simple ὦ
events. Similarly let W represent any event whose proba- ©

bility is sought, W being also a known function of ὌΠ Pe: F
We As. δ W must satisfy the fundamental law of
duality, we are permitted to replace them by single logical
symbols, s, ¢,..w. Assume then

s= 5S, t=7, w=W.

fot

fein

χρυ gs Be

aD lia A ΜΟῚ ΟΡ es ci, Sy Sas

GENERAL METHOD IN PROBABILITIES 275

These, by the definition of S, 7,.. W, will bea series
οἵ logical equations connecting the symbols s, ¢, .. w, with
the symbols 4, y, 2

By the methods of the Calculus of Logic we can eliminate
-from the above system any of the symbols 4, y, 2, . ., re-
presenting events whose probabilities are not given, and de-
termine w as a developed function of 5, ¢, &c., and of such
of the symbols Ἐς», 2, &c., if any such there be, as corre-
spond to events whose probabilities are given. The result
will be of the form

ww dt oh ΟΣ
ο οΟ

where A, δ, C, and D comprise among them all the possible
constituents which can be formed from the symbols s, ¢, &c.,
z.e. from all the symbols representing events whose proba-
bilities are given.

The above will evidently be the complete expression of
the relation sought. For it fully determines the event W,
repre-[264]sented by the single symbol w, as a function or
combination of the events similarly denoted by the symbols
s, ¢, &c., and it assigns by the laws of the Calculus of Logic
the condition

3) Ξ
as connecting the events s, ¢, &c., among themselves. We
may, therefore, by Principle VI., regard s, ¢, &c., as semple
events, of which the combination w, and the condition with
which it is associated D, are definitely determined.

Uniformity in the logical processes of reduction being de-
sirable, I shall here state the order which will generally be
pursued,

12. By (VIII. 8), the primitive equations are reducible to
the forms

s(1 - S)+ S(1-5)=0;
eels 1. aed {τ|
ee W)+ ee ee

under which they can be added together without impairing

280 THE SAWS OF THOUGHT

their significance. We can then eliminate the symbols 4, y, 2,
either separately or together. If the latter course is chosen, ~
it is necessary, after adding together the equations of the —
system, to develop the result with reference to all the -
symbols to be eliminated, and equate to o the product of all
the coefficients of the constituents (VI1. 9).

As w is the symbol whose expression is sought, we may ~
also, by Prop. III. Chap. IX., express the result of elimina-
tion in the form

Ew+ E'(1 -w)=o.

E and E’ being successively determined by making in the
general system (1), w=1 and w=o, and eliminating the
symbols 4, y, z,.. Thus the single equations from which —
E and £’ are to be respectively determined become

si -—S)+ S(1-s)+41-7)+7U-4..+1-We=o0;

“( -- 5) Ὁ δά -- 5). Δα -- Ζ2: ΖΕ -δ- We=o. Ἷ
From these it only remains to eliminate x, y, z, &c., and to |
determine w by subsequent development.

[265] In the process of elimination we may, if needful,
avail ourselves of the simplifications of Props. I. and II.
Cha eX,

13. Should the data, beside informing us of the proba-
bilities of events, further assign among them any explicit
connexion, such connexion must be logically expressed, and
the equation or equations thus formed be introduced into the
general system.

PROPOSITION IV

14. Given the probabilities of any system of events; to
determine by a general method the consequent or derived pro-
bability of any other event.

As in the last Proposition, let S, 7, &c., be the events
whose probabilities are given, W the event whose proba-
bility is sought, these being known functions of x, Ve Se.
Let us represent the data as follows:

Probability of S=2; (1)
Probability of 7=g; ᾿

GENERAL METHOD IN PROBABILITIES 281

and so on, 2, g, &c., being known numerical values. If then
we represent the compound event S by s, 7 by ¢, and W by
w, we find by the last proposition,

w=A+oB4+°C4-4D; (2)
Ο Oo

A, B, C, and D being functions of s, ¢, &c. Moreover the
data (1) are transformed into

TOD. 5— 7, Proba7z—a, Se: (3)
Now the equation (2) is resolvable into the system
w=A+gC ;
D=0, (4)

g being an indefinite class symbol (VI. 12). But since by
the properties of constituents (V. Prop. III.), we have

A+B+C+D=1,

the second equation of the above system may be expressed
in the form
A+B+C=1.
[266] If we represent the function d+B+C by V, the

system (4) becomes
| w=A+9qC; (5)

V=1. (6)

Let us for a moment consider this result. Since V is the
sum of a series of constituents of s, ¢, &c., it represents the
compound event in which the simple events involved are those
denoted by s, ¢, &c. Hence (6) shows that the events de-
noted by s, z, &c., and whose probabilities are p, g, &c.,
have such probabilities not as zudependent events, but as
events subject to a certain condition V. Equation (5) ex-
presses w as a similarly conditioned combination of the
same events.

Now by Principle VI. the mode in which this knowledge
of the connexion of events has been obtained does not influ-
ence the mode in which, when obtained, it is to be employed.
We must reason upon it as if experience had presented to us
the events s, ¢, &c., as simple events, free to enter into every

282 THE PAWS OF [HOUGHT

combination, but possessing, when actually subject to the :
condition V, the probabilities ~, g, &c., respectively. {

Let then 7’, 7’, . ., be the corresponding probabilities of —
such events, when the restriction V is removed. Then by ἢ
Prop. II. of the present chapter, these quantities will be ἢ

determined by the system of equations,

yn ane cer. (7) :

and by Prop. I. the probability of the event w under the 4
same condition V will be

Prob. zw = ee ; (8) 7

wherein V, denotes the sum of those constituents in V of
which s is a factor, and [V,] what that sum becomes when ἢ
Ge. ake chanced into 7" 7... respectively. | ne am
constant ¢ represents the probability of the indefinite event
7; it is, therefore, arbitrary, and admits of any value from ~
oto, 7
Now it will be observed, that the values of #’, g’, &c., are
determined from (7) only in order that they may be sub-
stituted in (8), so as to render Prob. w a function of known
quantities, 2, g, [267] ὅς. It is obvious, therefore, that
instead of the letters 2’, g’, &c., we might employ any others
as Ss, ἢ &c., in the same quantitative acceptations. This
particular step would simply involve a change of meaning
of the symbols s, ¢, &c.—their ceasing to be logical, and ἴ

becoming quantitative. The systems (7) and (8) would then
become

Ve V,
Ὁ; ν᾽ “5 ἄτα. : (9)

Prob. a (10)

In employing these, it is only necessary to determine from
(9) 5, ¢, &c., regarded as quantitative symbols, in terms of
2, 9, &c., and substitute the resulting values in (10). It is

GENERAL METHOD IN PROBABILITIES 283

evident, that s, ¢, &c., inasmuch as they represent proda-
bilities, will be positive proper fractions.
The system (9) may be more symmetrically expressed in

the form
Ane Ζ
pip aa en ie
rae (11)

Or we may express both (9) and (10) together in the symme-
trical system

γι 2c A+cC

ai pene. _7.- = — OA RES Fae

- 9 7 : oa
wherein w represents Prob. w.

15. It remains to interpret the constant c assumed to re-
present the probability of the indefinite event g. Now the
logical equation

w=A+gC,

interpreted in the reverse order, implies that if either the
event A take place, or the event C in connexion with the
event g, the event w will take place, and not otherwise.
Hence g represents that condition under which, if the event
C take place, the event w will take place. But the proba-
bility of gis c. Hence, therefore, c= probability that if the
event C take place the event w will take place.
Wherefore by Principle II.,

ra Probability of concurrence of C and w
Probability of € :
[268] We may hence determine the nature of that new ex-
perience from which the actual value of c may be obtained.
For if we substitute in C for s, ¢, &c., their original expres-
sions as functions of the simple events x, y, 2, &c., we shall
form the expression of that event whose probability consti-
tutes the denominator of the above value of c; and if we
multiply that expression by the original expression of w, we
shall form the expression of that event whose probability con-
stitutes the numerator of c, and the ratio of the frequency of
this event to that of the former one, determined by new observa-
tions, will give the value of c. Let it be remarked here, that

284 THE LAWS OF THOCGHRT

the constant cdoes not xecessarily make its appearance in the —
solution of a problem. It is only when the data are insuffi- ᾿
cient to render determinate the probability sought, that this :
arbitrary element presents itself, and in this case it is seen
that the final logical equation (2) or (5) informs us how it is

to be determined. :
If that new experience by which c may be determined can- —

not be obtained, we can still, by assigning to c its limiting ᾿
values o and 1, determine the limits of the probability of τὺ.
These are
Minor limit of Prob. zw -%.
Superior limit oe i
Between these limits, it is certain that the probability sought ᾿
must lie independently of all new experience which does not ‘
absolutely contradict the past. ;
If the expression of the event C consists of many con-
stituents, the logical value of w being of the form ἢ

w= A+°C\+-C,+ &e.,

we can, instead of employing their aggregate as above, present :
the final solution in the form

Prob. w= A+qC; ow ce

[269] Here c,=probability that if the event C, occur, the
event w will occur, and so on for the others. Convenience
must decide which form is to be preferred.

16. The above is the complete theoretical solution of the
problem proposed. It may be added, that it is applicable -
equally to the case in which any of the events mentioned in
its original statement are conditioned. Thus, if one of the
data is the probability 2, that if the event + occur the event
y will occur ; the probability of the occurrence of x not being
given, we must assume Prob, x=c (an arbitrary constant),
then Prob. zy=cp, and these two conditions must be intro-

SCENEKAL METHOD TN PROBABILITIES § 285

duced into the data, and employed according to the previous
method. Again, if it is sought to determine the probability
that if an event x occur an event y will occur, the solution
will assume the form

ELov. ἂν

Prob. sought = >» 37
ee πος

the numerator and denominator of which must be separately
determined by the previous general method.

17. We are enabled by the results of these investigations
to establish a general rule for the solution of questions in
probabilities.

GENERAL RULE

CASE ].— When all the events are unconditioned.

Form the symbolical expression of the events whose pro-
babilities are given or sought.

Equate such of those expressions as relate to compound
events to a new series of symbols, s, ¢, &c., which symbols
regard as representing the events, no longer as compound
but simple, to whose expressions they have been equated.

Eliminate from the equations thus formed all the logical
symbols, except those which express events, 5, Z, &c., whose
respective probabilities ~, 7, &c., are given, or the event w
whose probability is sought, and determine w asa developed
function of s, ¢, &c. in the form

FEN elo see Oui BE
Oo = 70

[270] Let 4+ B+C= TV, and let V, represent the aggre-
gate of those constituents in V which contains s as a factor,
V, of those which contain ¢ as a factor, and thus for all the
symbols whose probabilities are given.

Then, passing from Logic to Algebra, form the equations

dD (1)

A+cC
2 (2)

286 THE LAWS, OF TACCGCH

from (1) determine s, ¢, &c. as functions of 2, g, &c., and ~
substitute their values in (2). The result will express the j

solution required.
Or form the symmetrical system of equations

where wz represents the probability sought.
If c appear in the solution, its interpretation will be

Prob. Cw
C= τς
Prop 2

and this interpretation indicates the nature of the experience %

which is necessary for its discovery.

CASE I].— When some of the events are conditioned.

If there be given the probability p that if the event Α΄ 3

occur, the event Y will occur, and if the probability of the

antecedent XY be not given, resolve the proposition into the —

two following, viz. :
Probability of X =,
Probability of Y Y= cp.
If the gu@situm be the probability that if the event W
occur, the event Ζ will occur, determine separately, by the
previous case, the terms of the fraction

Prob. WZ
Prob. W ’

and the fraction itself will express the probability sought.
[271] It is understood in this case that XY, Y, W, Z may

be any compound events whatsoever. The expressions X Y —

and WZ represent the products of the symbolical expressions
of X and Y and of W and Z, formed according to the rules
of the Calculus of Logic.

The determination of the single constant ¢ may in certain ©

cases be resolved into, or replaced by, the determination of
a series of arbitrary constants, Cy, Cg . . according to conveni-
ence, as previously explained.

ac te δι

—

; (3) ;

pean eek Me, | SN Rete

RP Dames

|
|
|
1

GENERAL METHOD IN PROBABILITIES 287

18. It has been stated (I. 12) that there exist two distinct

‘definitions, or modes of conception, upon which the theory

of probabilities may be made to depend, one of them being
connected more immediately with Number, the other more
directly with Logic. We have now considered the conse-
quences which flow from the numerical definition, and have
shown how it conducts us to a point in which the necessity
of a connexion with Logic obviously suggests itself. We
have seen to some extent what is the nature of that con-
nexion ; and further, in what manner the peculiar processes
of Logic, and the more familiar ones of quantitative Algebra,
are involved in the same general method of solution, each of
these so accomplishing its own object that the two processes
may be regarded as supplementary to each other. It remains
to institute the reverse order of investigation, and, setting
out from a definition of probability in which the logical re-
lation is more immediately involved, to show how the
numerical definition would thence arise, and how the same
general method, equally dependent upon both elements,
would finally, but by a different order of procedure, be
established.

That between the symbolical expressions of the logical
calculus and those of Algebra there exists a close analogy, is
a fact to which attention has frequently been directed in the
course of the present treatise. It might even be said that
they possess a community of forms, and, to a very consider-
able degree, a community of laws. With a single exception
in the latter respect, their difference is only one of inter-
pretation. Thus the same expression admits of a logical or
of a quantitative interpretation, according to the particular
meaning which we attach to the sym-[272]bols it involves.

The expression xy represents, under the former condition,

a concurrence of the events denoted by x and y; under the
latter, the product of the numbers or quantities denoted by
x and y. And thus every expression denoting an event,
simple or compound, admits, under another system of inter-
pretation, of a meaning purely quantitative. Here then

288 THE LAWS OF THOUGHT

arises the question, whether there exists any principle of
transition, in accordance with which the logical and the
numerical interpretations of the same symbolical expression
shall have an intelligible connexion. And to this question
the following considerations afford an answer.

19. Let it be granted that there exists such a feeling as
expectation, a feeling of which the object is the occurrence
of events, and which admits of differing degrees of intensity.
Let it also be granted that this feeling of expectation ac-
companies our knowledge of the circumstances under which
events are produced, and that it varies with the degree and
kind of that knowledge. Then, without assuming, or tacitly
implying, that the intensity of the feeling of expectation,
viewed as a mental emotion, admits of precise numerical
measurement, it is perfectly legitimate to inquire into the pos-
sibility of a mode of numerical estimation which shall, at least,
satisfy these following conditions, viz., that the numerical
value which it assigns shall increase when the known circum-
stances of an event are felt to justify a stronger expectation,
shall diminish when they demand a weaker expectation, and
shall remain constant when they obviously require an equal
degree of expectation.

Now these conditions at least will be satisfied, if we assume
the fundamental principle of expectation to be this, viz., that
the laws for the expression of expectation, viewed as a
numerical element, shall be the same as the laws for the ex-
pression of the expected event viewed as a logical element.
Thus if φᾷ, 7, 5) represent any unconditional event com-
pounded in any manner of the events 4, y, 5, let the same
expression ¢(%, 7, ), according to the above principle, de-
note the expectation oi that event: 7%, 7, 2 represciiting no
longer the simple events involved, but the expectations of
those events.

[273] For, in the first place, it is evident that, under this
hypothesis, the probability of the occurrence of some one of
a set of mutually exclusive events will be equal to the sum
of the separate probabilities of those events. Thus if the

GENERAL METHOD IN PROBABILITIES 289

alternation in question consist of 7 mutually exclusive events
whose expressions are

P(X, ¥, 2), PolX, », 2)» . . + » φ,(α, γ, 2),

the expression of that alternation will be
φ,(5,.», 2) + hol, Is 2)... + Φ,(α, γ, 2) = 75

the literal symbols x, y, z being logical, and relating to the
simple events of which the three alternatives are com-
pounded : and, by hypothesis, the expression of the proba-
bility that some one of those alternatives will occur is

P(x, ν, 2) + $,(x, J; 2) et + (αὶ ys 2),

x,y, 8 here denoting the probabilities of the above simple
events. Now this expression increases, ceteris paribus, with
the increase of the number of the alternatives which are in-
volved, and diminishes with the diminution of their number ;
which is agreeable to the condition stated.

Furthermore, if we set out from the above hypothetical
definition of the measure of probability, we shall be con-
ducted, either by necessary inference or by successive steps
of suggestion, which might perhaps be termed mecessary, to
the received numerical definition. We are at once led to
recognise unity (1) as the proper numerical measure of cer-
tainty. For it is certain that any event # or its contrary
1—2z will occur. The expression of this proposition is

x+(1—-%x)=1,

whence, by hypothesis, x+(1—-*), the measure of the pro-
bability of the above proposition, becomes the measure of
certainty. But the value of that expression is 1, whatever
the particular value of x may be. Unity, or 1, is therefore,
jon the hypothesis in question, the measure of certainty.

Let there, in the next place, be 22 mutually exclusive,
{but equally possible events, which we will represent by
t,, t,,..¢,. [274] The proposition which affirms that some

n

one of these must occur will be expressed by the equation

A+ .-+4= 1;
19

290 THe GAWS ΕΟ

and, as when we pass in accordance with the reasoning of
the last section to numerical probabilities, the same equation
remains true in form, and as the probabilities ¢,, ¢,,.. 7, are

equal, we have
nt, = 1,

oe! I I
whence A=, and similarly Ca le Suppose it then

required to determine the probability that some one event
Of the partial series τ, = ¢,, will occur; we have tor the
expression required

cee oe ch eae fe . to m terms

Hence, therefore, if there are 72 cases favourable to the oc-
currence of a particular alternation of events out of 2 possible
and equally probable cases, the probability of the occurrence

Tie
of that alternation will be expressed by the fraction oa

Now the occurrence of any event which may happen in
different equally possible ways is really equivalent to the
occurrence of an alternation, z.e., of some one out of a set of
alternatives. Hence the probability of the occurrence of any
event may be expressed by a fraction whose numerator
represents the number of cases favourable to its occurrence,
and denominator the total number of equally possible cases.
But this is the rigorous numerical definition of the measure
of probability. That definition is therefore involved in the
more peculiarly /ogzcal definition, the consequences of which;
we have endeavoured to trace.

20. From the above investigations it clearly appears, Ist,
that whether we set out from the ordinary numerical defini-
tion of the measure of probability, or from the definition which
assigns to the numerical measure of probability such a law of)
value as shall establish a formal identity between the logical
expressions [275] of events and the algebraic expressions of
their values, we shall be led to the same system of practical

GENERAL METHOD IN PROBABILITIES 291

results. 2ndly, That either of these definitions pursued to
its consequences, and considered in connexion with the rela-
tions which it inseparably involves, conducts us, by inference
or suggestion, to the other definition. To a scientific view
of the theory of probabilities it is essential that both prin-
ciples should be viewed together, in their mutual bearing and
dependence.

[276] CHAPTER XVIII

ELEMENTARY ILLUSTRATIONS OF THE GENERAL METHOD
IN PROBABILITIES

1. IT is designed here to illustrate, by elementary examples, —
the general method demonstrated in the last Chapter. The |
examples chosen will be chiefly such as, from their simplicity, ὦ
permit a ready verification of the solutions obtained. But |
some intimations will appear of a higher class of problems, ἢ
hereafter to be more fully considered, the analysis of which |
would be incomplete without the aid of a distinct method .
determining the necessary conditions among their data, in ἢ
order that they may represent a possible experience, and_
assigning the corresponding limits of the final solutions. The -
fuller consideration of that method, and of its applications,

Se

is-reserved ἥν the next Chapter:

2. Ex. 1.—The probability that it thunders upon a given
day is ~, the probability that it both thunders and hails is g,
but of the connexion of the two phenomena of thunder and
hail, nothing further is supposed to be known. Required —
the probability that it hails on the proposed day.

Let x represent the event—It thunders.

Let y represent the event—It hails.

Then xy will represent the event—It thunders and hails; —
and the data of the problem are |

Re oe τον ee a a ee eee oe Se eee ee ree

Prob. τυ Ξ 9, Probesy — a:

There being here but one compound event xy involved, ἢ
assume, according to the rule, .

sey =U, (1)
Our data then become δ

Prob. ἀξ Prob. x=: (2)

292

ELEMENTARY ILLUSTRATIONS 293
and it is required to find Prob. y. Now (1) gives
(2771 , =O = uae + “u(t -- a) +0(1 ~ ux + (1 — γᾷ - αν.

Hence (XVII. 17) we find

Γ =uxt+(1-u“)x+(1-u)(1 -- x),
VIS Uri wits, τ.

and the equations of the General Rule, viz.,

tia :
p >
Prob. y=4 Ἐς,
V

become, on substitution, and observing that A=az, C=
(1 -- 2}(1 --- 2), and that V reduces to r+ (1—u)(1—»*),

Fo (3)

Prop τ τς - 4)(1 -- x)

x+(1—u)(1—x)’ (4)

from which we readily deduce, by elimination of x and a,
Prob. y=¢ +¢(1 -- 2). (5)

In this result c represents the unknown probability that if
the event (1—%)(1—%) happen, the event y will happen.
Now (1 —4)(1—%)=(1—4y)(1 —4) =1—4, on actual multipli-
cation. Hence c is the unknown probability that if it do not
thunder, it will hail.

The general solution (5) may therefore be interpreted as
follows :—The probability that it hails is equal to the proba-
bility that it thunders and hails, g, together with the proba-
bility that it does not thunder, 1—f, multiplied by the
probability c, that if it does not thunder it will hail. And
- common reasoning verifies this result.

If ¢ cannot be numerically determined, we find, on assign-
ing to it the limiting values o and 1, the following limits of
wrod. 7, Viz. :

Inferior limit =¢.
Superior limit=g+1-—-f.

294 THE LAWS OF THOUGHT

[278] 3. Ex. 2.—The probability that one or both of two
events happen is 23, that one or both of them fail is g. What
is the probability that only one of these happens ?

Let x and y represent the respective events, then the data
are—

Prob. «y+ «(1 -—y)+(1- x)y=,
Prob. x(1 —y) + (1 -x)y+(1-x)(1 -- )) εξ φ;
and we are to find
Prob. «(1 -y) +y(1 - x).

Here all the events concerned being compound, assume
sy+x(i—-y)+(1-xyes,
a(1-y)+(1-x)y+(1-x)(1-y)=4,
χᾷ -—y)+(1 -—x)y=w.

Then eliminating x andy, and determining w as a developed
function of s and ¢, we find

w = st + ὁρᾷ τοῦ + 0(1 -- δ) “(1 ~5)(1 -- ἢ.
Hence A=st, C=0, V=st+s(1-—t)+(1—s)t=s+(1—s)é,

V,=s, V,=¢; and the equations of the General Rule (XVII.
17) become

Sore
oe a 2 τ S)Z, I
rer (1-5) (1)
Brel ye ene

S+(1—s)é

whence we find, on eliminating s and Ζ,
Prob. w=p+q-1.

Hence 2 -Ἐφ — 1 is the measure of the probability sought. This
result may be verified as follows :—Since p is the probability
that one or both of the given events occur, 1 -- 2 will be the
probability that they both fail ; and since g is the probability
that one or both fail, 1g is the probability that they both
happen. Hence 1—g+1~-g, or 2 —p—g, is the probability
that they either both happen or both fail. But the only
remaining alternative which is possible is that one alone of
the events happens. Hence the probability of this occur-
rence is --(2 --2 -- Φ), or +g —1, as above.

Se

PEEVE NTARY ILLUSTRATIONS 295

=

[279] 4. Ex. 3.—The probability that a witness A speaks
the truth is g, the probability that another witness B speaks
the truth is g, and the probability that they disagree in a
statement is y What is the probability that if they agree,
their statement is true?

Let x represent the hypothesis that A speaks truth ;
y that B speaks truth; then the hypothesis that A and B
disagree in their statement will be represented by +(1—y)
+y(1—2#); the hypothesis that they agree in statement by
ay+(1—#)(1—y), and the hypothesis that they agree in the
truth by xy. Hence we have the following data:

Prob. x=, Prob. y=g, Prob. x(1 -—y)+y(1 -x)=7,
from which we are to determine

Prob: ay
Prob. xy+(1 —x)(t—y)

But as Prob. χίι -- 7))-Ἐ 7 -- ξε σ, it is evident that Prob.
ay +(1—4+)(1—y) will be 1-7; we have therefore to seek

Prob. xy

7 - A

Now the compound events concerned being in expression,
x(1—y)+y(1 —*) and xy, let us assume
χᾷι -- ») Ἐγᾷ -- “)Ξ 9 ᾿ (1)
xy Ξ (ἢ
Our data then are Prob. =f, Prob. y=g, Prob. 5. -Ξ- 7, and
we are to find Prob. w.
The system (1) gives, on reduction,

{a(x —y) + (1 — x) }(1— δ) + 5{ay +(x -αλὰ -9)}
+ xy(1 -- Ww) +w(1 -- xy) =0;
whence
_ «(1 -y)(1 - 5) +y(1 —x)(1 -- 5) +sxy +5(1 — α)ᾷ —y) + x
2Χ} - ὦ

«υ
- =xys + ay(1— 5) τ ον ys + ταί τι Ὁ

Ἐσ(ι -- ᾽ν» Ὲ τι τα) —y)s + τα — 2 ~ 5)

4+ 0(1—x)(1—y)(1 -- 5). (2)

296 THE LAWS OF THOUGHT

[280] In the expression of this development, the coefficient
* has been made to replace every equivalent form (X. 6).
O

Here we have
Y= ay(1 ἡ χα —y)s+(1—axyys-+ (1 —a)(1 ~y\(r-5) 5
whence, passing from Logic to Algebra,
xy(I -—S)+x(1 —y)s_ xy —s)+(1 = x)ys _ x(1 —v)s+(1-—x)ys
2 7 εἴ
=ay(1 - 5) + a(t —y)s-+ τ x)yst (1 -a)(1 -y)(4-5).

xy(1 5)
F100. = 5) Fae ays EAMETS)

from which we readily deduce

Prob. w =P 59
whence we have
Prob. xy ptg-7
Teor, § 2(1 7) (3)

for the value sought.

If in the same way we seek the probability that if A and
& agree in their statement, that statement will be false, we
must replace the second equation of the system (1) by the
following, viz.:

ee) a,

the final logical equation will then be
τυ ΣΙ oxy(I -- 5) + 0x(1 —y)s + ται —y)(1- 5s)

+0(1 - ays + τα — x)y(1 — 8) 42(1 —x)(1 -- γ)ς
+(1-x)(1-y)(1-5); (4)

whence, proceeding as before, we finally deduce

Probes 22 ae ee (5)
2
Wherefore we have
[281] Prob. (1-x)(1-y) 2-p-g-r
eae = (6)
1-- 2(1 -- 2)

for the value here sought.

ELEMENTARY ILLUSTRATIONS 297

These results are mutually consistent. For since it is
certain that the joint statement of A and B must be either
true or false, the second members of (3) and (5) ought by
addition to make 1. Now we have identically,

Prete 7 5
2(1 -- 2) 2 Fe -

It is probable, from the simplicity of the results (5) and
(6), that they might easily be deduced by the application of
known principles ; but it is to be remarked that they do not
fall directly within the scope of known methods. The num-
ber of the data exceeds that of the simple events which they
involve. M. Cournot, in his very able work, Exposition de
la Théorte des Chances, has proposed, in such cases as the
above, to select from the original premises different sets of
data, each set equal in number to the simple events which
they involve, to assume that those simple events are inde-
pendent, determine separately from the respective sets of the
data their probabilities, and comparing the different values
thus found for the same elements, judge how far the assump-
tion of independence is justified. This method can only
approach to correctness when the said simple events prove,
according to the above criterion, to be nearly or quite inde-
pendent ; and in the questions of testimony and of judgment,
in which such an hypothesis is adopted, it seems doubtful
whether it is justified by actual experience of the ways
of men.

5. Ex. 4.—From observations made during a period of
general sickness, there was a probability 23. that any house
taken at random in a particular district was visited by fever,
a probability g that it was visited by cholera, and a pro-
bability ~ that it escaped both diseases, and was not in a
defective sanitary condition as regarded cleanliness and
ventilation. What is the probability that any house
taken at random was in a defective sanitary condition ἢ

With reference to any house, let us appropriate the symbols
4, y, 2, as follows, viz.:

298 THE LAWS OF THOUGHT

[282] The symbol to the visitation of fever.
y a cholera.
2 to defective sanitary condition.

The events whose probabilities are given are then denoted by ~
x, y, and (I1—x)(I1—y)(1—2), the event whose probability is —
sought is z. Assume then, ;
(1-2x)(1-y)(1 2) Ὁ,
then our data are, o-
Prob. x=/, Prob. y=g, Prob. w=rz,
and we are to find Prob. z Now
hd A Saat τ
ΕἸ ΠΥ Ὁ}

- ἔχγευ + Sxy( Ι -- 20) Ὁ χί I - ν)εῦ Ὁ 2e( I—y)(1 -- το)
ο ο ο ο

o—
- Ξ

+2(1-x)ywt τὰ - x)y(1 —w) + o(1 -- x)(1 —y)w
ο
+(1-2)(1 -- ν)α -- τὺ). (1) |
The value of V deduced from the above is

V=xv(i-w)tati-—y)(1 -—w) + (1 -- αὐνᾷι -- w)
+(1-x)(1 -y)w+(1-x)(1 -y)(1 -w) = 1 -- τὸ τ τῦ(ι -x)(1 -- ν);

and similarly reducing V,, V,, V., we get
V.=x(1-w), V,=y(1-w), Ve=w(1-2)(1 -9);

furnishing the algebraic equations

τι π᾿ κι πὴ σι τι) —-w+w(1—-x)(1-y). (2) |
2 ἐ Α

As respects those terms of the development characterized .

by the coefficients - , I shall, instead of collecting them into ~

a single term, present them, for the sake of variety (XVII.
18), in the form .

2x(1 ~w)+2(1—a)y(1 τα); (3)

the value of Prob. z will then be

ELEMENTARY ILLUSTRATIONS 299

283] Bob. maa - x) -- ")(ι -- 20) tex(1-w) τ ἐ(ι —x)y(1 πο)
I-w+w(1—x)(1-y)

(4)
From (2) and (4) we deduce

Prob. 8 ee i ae)
I

by --Σ

2

as the expression of the probability required. If in this re-
- sult we make c=0, and c’ =o, we find for an inferior limit of
ee and if we make c=1, ¢=1,
τ we obtain for its superior limit 1 -- 7.

6. It appears from inspection of this solution, that the
premises chosen were exceedingly defective. The constants
_¢ and ε΄ indicate this, and the corresponding terms (3) of
_ the final logical equation show how the deficiency is to be
supplied. Thus, since

x(1 -w)=x{1 -(1-x)(1 -y)(1 -- 2)} = 2,

(1—x)y(1 -w) =(1 -- xy {1 -(1 - x)(1 -- ))( -- 2)} ΞΞ (( --- σὴν,
_we learn that ς is the probability that if any house was
visited by fever its sanitary condition was defective, and that
_¢ is the probability that if any house was visited by cholera
without fever, its sanitary condition was defective.

If the terms of the logical development affected by the

its value

: O : ;
coefficient - had been collected together as in the direct

statement of the general rule, the final solution would have
assumed the following form :

ich. τ 2. Γλι -7-)..(2.9- 22

ἘΠ Σ

c here representing the probability that if a house was visited
by either or both of the diseases mentioned, its sanitary con-
dition was defective. This result is perfectly consistent with
_the former one, and indeed the necessary equivalence of the
different forms of solution presented in such cases may be
formally established.
ΕΓ The above solution may be verified in particular cases.
_ Thus, taking the second form, if c=1 we find Prob, z=1—- 7,

300 THEO LAWS OL T 70 fT.

acorrect result. For if the presence of either fever or cholera
certainly [284] indicated a defective sanitary condition,
the probability that any house would be in a defective
sanitary state would be simply equal to the probability that it
was not found in that category denoted by ΖΦ, the probability
of which would, by the data, be 1-7. Perhaps the general
verification of the above solution would be difficult.

The constants 2, g, and χ inthe above solution are subject
to the conditions x

χει σφύζει.

7, Ex. 5.—Given the probabilities of the premises of a
hypothetical syllogism, to find the probability of the
conclusion.

Let the syllogism in its naked form be as follows :

Major premiss: If the proposition Y is true X is true.
Minor premiss: If the proposition Z is true Vis true.
Conclusion : If the proposition Z is true X is true.

Suppose the probability of the major premiss to be 2, that
of the minor premiss g.

The data then are as follows, representing the proposition
X by #, &c., and assuming c and c’ as arbitrary constants :
Prob. y=c, Prob. xy=cp;

Probe ec," γ᾽; ζῶ:
from which we are to determine,

Prob. xz Prob. xz _
Prob. z τ
Let us assume,

LY=U, YZ=V, XZ=W;
then, proceeding according to the usual method to determine
w as a developed function of y, z, 24, and v, the symbols
corresponding to propositions whose probabilities are given,
we find

w= ξὺν + ομᾷι — 2)(1 -- v)y + 0(1 — 2)zvy
+ 5(1 = w)s(t (1-3) + (1 = N(x )α -- τὴν

+ 0(1—w)(1 -- 2)(t —v)(1 —y) + terms whose coefficients are ~
ο

ee ΔΑ

ELEMENTARY. ILLUSTRATIONS 301

[285] and passing from Logic to Algebra,

a ad) LS eA Sd A a) AS) LS 2,
3 0a)
_ usoy + (1 —u)zvy d
ε τ
_usoy + us — 2)(1 -- σὴν + (1 -- u)zvy + (1 —u)(1 -- 2)(1 =O s V.
; 4

isin ee _ usvy +a(t —u)2(1 —v)(1 -- γ)
. τὺ -- : ;

wherein
V=uzvy+u(i -- 2)(1 -v)y + (1 -- μ)δὸν + (1 — u)2(1 -- σ)( — γ)

+ (1 οὐ τε 2)(x — oy + (1 (τ - 2)(1 τ v)(1 }}
the solution of this system of equations gives

Prob. w=¢pg +ac'(1 -- 4),

aeaunes Prob. 99 pg +a(1~9)

the value required. In this expression the arbitrary constant
a is the probability that if the proposition Z is true and Y
false, X is true. In other words, it is the probability, that
if the minor premiss is false, the conclusion is true.

This investigation might have been greatly simplified by

’ assuming the proposition Z to be true, and then seeking the

probability of X. The data would have been simply
Prob, 7— 7, ΕΠ αὺ Ξ 70.

whence we should have found Prob. r=pg+a(1—q). Itis
evident that under the circumstances this mode of procedure
would have been allowable, but I have preferred to deduce
the solution by the direct and unconditional application of
the method. The result is one which ordinary reasoning
verifies, and which it does not indeed require a calculus to
obtain. General methods are apt to appear most cumbrous
when applied to cases in which their aid is the least required.

Let it be observed, that the above method is equally appli-
cable to the categorical syllogism, and not to the syllogism
only, [286] but to every form of deductive ratiocination.
Given the probabilities separately attaching to the premises of
any train of argument, it is always possible by the above

302 THE LAWS OF THOUGHT

method to determine the consequent probability of the truth |
of a conclusion legitimately drawn from such premises. It
is not needful to remind the reader, that the truth and the
correctness of a conclusion are different things.

8. One remarkable circumstance which presents itself in
such applications deserves to be specially noticed. It is,
that propositions which, when true, are equivalent, are ποῖ
necessary equivalent when regarded only as probable. This
principle will be illustrated in the following example.

Ex. 6.—Given the probability 2 of the disjunctive proposi-
tion ‘‘ Either the proposition Y is true, or both the proposi-
tions XY and Y are false,” required the probability of the
conditional proposition, ‘‘If the proposition Y is true, Y is
trues.

Let x and y be appropriated to the propositions XY and Y
respectively. Then we have

Prob. y+ (1 -x)(1 —y) =f,

from which it is required to find the value of Prob. zy
Prop. Ὁ
Assume yt(i-x\(1-y)=4 (1)
Eliminating y we get
(1-2)(1-A)=0,
whence ΘΑ, 12.
Ο

and proceeding in the usual way,

Prob, x=1-f/+¢, (2)

where c is the probability that if either Y is true, or Y and
V falses 7 45 truce.
Next to find Prob. xy. Assume

xy =W. (3)
Eliminating y from (1) and (3) we get
2(1—f)=0;
[287] whence, proceeding as above,

Prob. z=¢f,

ELEMENTARY ILLUSTRATIONS 303

c having the same interpretation as before. Hence

Prob: αν ὁ τπρῦ
Prob.x 1-pt+cp’

for the probability of the truth of the conditional proposi-
tion given.

Now in the science of pure Logic, which, as such, is con-
versant only with truth and with falsehood, the above dis-
junctive and conditional propositions are equivalent. They
are true and they are false together. It is seen, however,
from the above investigation, that when the disjunctive
proposition has a probability 23, the conditional proposition

has a different and partly indefinite probability eae Aare

I—pt+op
Nevertheless these expressions are such, that when ezther of
them becomes 1 or O, the other assumes the samevalue. The
results are, therefore, perfectly consistent, and the logical
transformation serves to verify the formula deduced from the
theory of probabilities.

The reader will easily prove by a similar analysis, that if
the probability of the conditional proposition were given as
p, that of the disjunctive proposition would be 1 --Φ- ᾧ,
where c is the arbitrary probability of the truth of the pro-
position X.

9. Ex. 7.—Required to determine the probability of an
event x, having given either the first, or the first and second,
or the first, second, and third of the following data, viz.:

ist. The probability that the event x occurs, or that it
alone of the three events, 2, y, 2, fails, 152.

2nd. The probability that the event y occurs, or that it
alone of the three events, 7, y, 2, fails, is Φ.

3rd. The probability that the event 2 occurs, or that it
alone of the three events 4, y, 2, fails, is 72.

SOLUTION OF THE FIRST CASE

Here we suppose that only the first of the above data is
given.

304 THE LAWS CF THOUGI YT

[288] We have then,

Prob. {x +(1 —x)yz} =f,
to find Ero, 4.

weet x+(I-x)ys=s,
then eliminating yz as a single symbol, we get
A(t -5) = 0:

oO
Erence pee. eet ah
τ δ Ae

whence, proceeding according to the rule, we have
Prob. x=¢, (1)

where c is the probability that if x occurs, or alone fails, the
former of the two alternatives is the one that will happen.
The limits of the solution are evidently o and 2.

This solution appears to give us no information beyond
what unassisted good sense would have conveyed. It is,
however, all that the single datum here assumed really
warrants us in inferring. We shall in the next solution see
how an addition to our data restricts within narrower limits
the final solution.

SOLUTION OF THE SECOND CASE
Here we assume as our data the equations

Prob. {x+(1 - x)yz} =f,
Prob. {y+(1 -y)xz} Ξε φ.
Let us write
x+(1-x)ys=s,
y+(i-y)xs=t;
from the first of which we have, by (VIII. 7),

{x+(1-x)yz}(1-s)+s{1-x—(1—x)vz}=0,
or (x + £y2)5 + sx(1 —ys)=0;
provided that for simplicity we write # for 1—%, 7 for 1—y,

and so on. Now, writing for 1—yzx its value in constituents,

we have
(x + xy2)5 + sx(yZ +92 +72) =0,

an equation consisting solely of positive terms.

|
|
|
|

BREPNMENTAKRY [LEUSTRATIONS 305

[289] In like manner we have from the second equation,
(y +yxz)t + ty(xF+*%2+X42)=0;
and from the sum of these two equations we are to eliminate
y and 2.

If in that sum we makey=1, z=1, we get the result +2,
If in the same sum we make y=1, z=0, we get the result

xF+5x+0,
If in the same sum we make γξξο, z=1, we get
x5 + SK + 01 + EX.
And if, lastly, in the same sum we make y =0, z=0, we find
xS+sx+intix, or X5+sx+0.

These four expressions are to be multiplied together. Now
the first and third may be multiplied in the following manner :
(5 Ἐ δίας + sX + xf + tx)
= x5 + 4t+(5+7)(sx+¢*%) by (IX. Prop. 11.)

= 05 + xb + St + sxt. (2)
Again, the second and fourth give by (IX. Prop. I.)
(x5 + sX + F)(x5 + 5x + ἢ)
= KF + SH. (3)
Lastly, (2) and (3) multiplied together give
(x5 + sX)(x5 + sXE + xt + 1X5)
= x5 + sx(sxt+ xf + x5)
= x5 + sxt.
Whence the final equation is
(1 -s)x+s(1 -- δα - x) Ξε ο,
which, solved with reference to %, gives

s(1 -Σὃ
wae —t)-(1-S)

most +51 —t)+4+0(1 -- 5) Ἐ o(1 —s)(1-?),

[2907 and, proceeding with this according to the rule, we

have, finally,
Prob. «= p(1 -- 97) + οὗ, (4)
20

306 THE LAWS Cf ΣΕ GH T.

where c is the probability that if the event s¢ happen, 2 will
happen. Now if we form the developed expression of s¢ by
multiplying the expressions for s and ¢ together, we find—

c= Prob. that if and y happen together, or x and z happen
together, and y fail, or y and z happen together, and ~ fail,
the event x will happen.

The limits of Prob. x are evidently 2(1 -- Φ) and 2.

This solution is more definite than the former one, inasmuch
as it contains a term unaffected by an arbitrary constant.

SOLUTION OF THE THIRD CASE
Here the data are—
Prob. {x+(1 -x)yz} =f,
Prob. {y+(1 -y)xz}=49,
Prob. {z+(1- z)xy} =r.

Let us, as before, write ¢ for 1—%, &c., and assume

X+XYZ=S,
yrjus=t,
Z+2xy=uU.

On reduction by (VIII. 8) we obtain the equation

(x + xyz)s + sx( V2 + 93 + 75)

+ (y+ yxz)t + ty( 2% + x3 + £7)

+ (2+ xy) + u3(xy + ἂν + XP) =0. (5)
Now instead of directly eliminating y and z from the above

equation, let us, in accordance with (IX. Prop. III.), assume
the result of that elimination to be

Ex+ FE (1—x)=0,
then £ will be found by making in the given equation x=1,
and eliminating y and z from the resulting equation, and Z’

will be found by making in the given equation +=0, and

eliminating y and z from the result. First, then, making
#%=1, we have

[291] SH (V+ ps)e + pF + (2 +.2)a + ΜΚ} =0,

and making in the first member of this equation successively

ELEMENTARY [LEUSTRATIONS 307

Meet, 2-1; y=, 3=0; ὡς and multiplying together the
results, we have the expression
(S+Z+a)(S+Z+a)F+7+H)(5+¢42),
which is equivalent to
(S+Z+2)(5+¢+4).
This is the expression for E. We shall retain it in its present
form. It has already been shown by example (VIII. 3), that
the actual reduction of such expressions by multiplication,
though convenient, is not necessary.
_ Again in (5), making +=0, we have
V2E+ SVE +S +IHZ) +i + bp +2 +uzZ=0;
from which, by the same process of elimination, we find for
Ε΄ the expression
(S+Z+a)(st+itu)st+t+a)(stt+u).
The final result of the elimination of y and z from (5) is
therefore
(S+7+a)\(5+¢+u)x
+(F+F%+2)(s4-7+u)(s+t+a)(st+t+u)(1-x)=0.
Whence we have

(§+7+a)/(st+f+u)(stt+a)(st+t+zu)
~ (F+F4+ay(s+itu)(s+i+a)(s+t+u) - σε ζει) τ)’

or, developing the second member,
x =o stu Ἂν 1 shu He 1 sta + sti
ce) 12) Oo
4 5tu + οδμ + O5ti + O50. (6)
O

Hence, passing from Logic to Algebra,

stu + sti _ Stut+sta _stut Slu

1
= stu + sti + Stu + Sti + Sti. (7)
[292] Pienee stit + cstu (8)
stu + stit + Stu + Sti + Sti
3 : : : ὅλας Ζ
To simplify this system of equations, change 2 into 5, :

308 THE LAWS OF THOUGHT

into 2, &c., and after the change let ἃ stand for sta+s+¢+1.
We then have

Prob. x = cae Ὶ (9)
with the relations
Ἔρις ἘΣ ἐπεπς ee ATH ee (10)
Ζ r
From these equations we get
stu+s=Ap, (11)

stu+s=X-—t-u-t,

\p=rX-u-f-1,

u+t=X(1-p)-1.

Similarly, “u+s=M1i-g)-1,
and s+t=X1i-7)-1.

From which equations we find

ee Ce 0) τοῦ pe Me eB)

ee ee ΣῪ -

2 2
ΡΣ δ ἐν as ὁ αὐ τὰ (12) |
2 i
Now, by (10),
stu=)p—s.

Substitute in this equation the values of s, ¢, and « above
determined, and we have

((1+p-g-r)A-1}{(1 +9g-p—r)r\-1}{(1+7-p-g)dr- 1}

=4{(p+g+r-1)d4 1}, (13)

an equation which determines Δ. The values of s, 2, and x,

are then given by (12), and their substitution in (g) completes .

the solution of the problem.

[293] 10. Now a difficulty, the bringing of which pro-
minently before the reader has been one object of this in-
vestigation, here arises. How shall it be determined, which

root of the above equation ought to be taken for the value of ΧΡ

To this difficulty some reference was made in the opening of
the present Chapter, and it was intimated that its fuller con-
sideration was reserved for the next one; from which the
following results are taken.

- εἰ χω,

ἐδ τόν ΚΣ

ELEMENTARY [LLUSTRATIONS 309

_ In order that the data of the problem may be derived from
a possible experience, the quantities 2, g, and ~ must be
subject to the following conditions :

ti Cae = Ο,

I+g-p-rso,

11 7--.2 -- 923 ο. (14)
Moreover, the value οἵ ἃ to be employed in the general
solution must satisfy the following conditions :

I
1+g-p-r’

N

I
ry ayer (15)

Now these two sets of conditions suffice for the limitation
of the general solution. It may be shown, that the central
equation (13) furnishes but one value of ), which does satisfy
these conditions, and that value cf ἃ is the one required.

Let 1-Ὁ2 --τἶῷ --- be the least of the three coefficients of ἃ

λ λ

Vil
Vil
Vil

given above, then will be the greatest of those

I
I+p—g-+7
values, above which we are to show that there exists but
one value of }. Let us write (13) in the form
π΄ Τὶ. τ|ἰτυς 2 ite 2. .-λ 1}

—4{(ptqgtr—-1)A+1}=0; (16)

and represent the first member by V/V.

Assume Ve : , then V becomes

which is negazzve.
[294] Let X= o, then V is poszézve and infinite.
Again,

Bie Stace

+ similar positive terms,

dn

I
I+p~-—q-r

which expression is positive between thelimits A=

nd \= ~,

310 THE LAWS OF THOUGHT

If then we construct a curve whose abscissa shall be
measured by Δ, and whose ordinates by V, that curve will,
between the limits specified, pass from below to above the
abscissa Δ, its convexity always being downwards. Hence
it will but once intersect the abscissa ἃ within those limits ;
and the equation (16) will, therefore, have but one root
thereto corresponding.

The solution is, therefore, expressed by (9), ἃ being that
root of (13) which satisfies the conditions (15), and s, Ζ, and z
being given by (12). The interpretation of c may be deduced
in the usual way.

It appears from the above, that the problem is, in all cases,
more or less indeterminate.

[295] CUAL Dine,
OF STATISTICAL CONDITIONS

1. By the term statistical conditions, I mean those con-
ditions which must connect the numerical data of a problem
in order that those data may be consistent with each other,
and therefore such as statistical observations might actually
have furnished. The determination of such conditions con-
stitutes an important problem, the solution of which, to an
extent sufficient at least for the requirements of this work,
I purpose to undertake in the present Chapter, regarding it
partly as an independent object of speculation, but partly also
as a necessary supplement to the theory of probabilities
already in some degree exemplified. The nature of the con-
nexion between the two subjects may be stated as follows :

2. There are innumerable instances, and one of the kind
presented itself in the last Chapter, Ex. 7, in which the solu-
tion of a question in the theory of probabilities is finally de-
pendent upon the solution of an algebraic equation of an
elevated degree. In such cases the selection of the proper
root must be determined by certain conditions, partly relat-
ing to the numerical values assigned in the data, partly to
the due limitation of the element required. The discovery
of such conditions may sometimes be effected by unaided
reasoning. For instance, if there is a probability 2 of the
occurrence of an event A, anda probability g of the con-
currence of the said event A and another event δ, it is
evident that we must have

2575.

But for the gexera/ determination of such relations, a distinct

method is required, and this we proceed to establish.
211

312 THE, LAWS OCF THOCGAT

As derived from actual experience, the probability of any
event is the result of a process of approximation. It is the
limit of the ratio of the number of cases in which the event
is observed to occur, to the whole number of equally possible —
cases which [296] observation records,—a limit to which we
approach the more nearly as the number of observations is
increased. Now let the symbol 2, prefixed to the expression
of any class, represent the number of individuals contained in
that class. Thus, x representing men, and y white beings,
let us assume

n(x) =number of men.
n( xy) = number of white men.
m{x(1 —y)} =number of men who are not white; and so on.

In accordance with this notation (1) will represent the
number of individuals contained in the universe of discourse,

and oe will represent the probability that any individual
being, selected out of that universe of being denoted by x(1),
is a man, If observation has not made us acquainted
with the Zo¢ad values of n(x) and μι), then the probability
: ΠΡ τς 62

in question is the limit to which aa approaches as the
number of individual observations is increased.

In like manner if, as will generally be supposed in this

chapter, + represent an event of a particular kind observed, —
n(x) will represent the number of occurrences of that event,

m1) the number of observed events (equally probable) of all

kinds, and met or its limit, the probability of the occurrence
of the event z.

Hence it is clear that any conclusions which may be
deduced respecting the ratios of the quantities (rv), n(y),
m1), &c. may be converted into conclusions respecting the
probabilities of the events represented by x, y, &c. Thus,

if we should find such a relation as the following, viz.,

n(x) + u( y)<n(1),

OF STATISTICAL CONDITIONS 313

expressing that the number of times in which the event x
occurs and the number of times in which the event y occurs,
are together less than the number of possible occurrences
m(1), we might thence deduce the relation,

ἘΠῚ a

(τ) κι)
ΟΥ̓ Prob. x + Prob. γε τ:

[297] dud generally any such statistical relations as the above
well be converted into relations connecting the probabilities of
the events concerned, by changing n(1) into 1, and any other
symbol n(x) into Prod, x.

3. First, then, we shall investigate a method of determin-
ing the numerical relations of classes or events, and more
particularly the major and minor limits of numerical value.
Secondly, we shall apply the method to the limitation of the
solutions of questions in the theory of probabilities.

It is evident that the symbol z is distributive in its opera-
tion. Thus we have

n{ay + (1 -- αλίι -- »)} =a(xy) Ἐπίᾳ -- x(t —y)f
n{ x(x — y)} = a(x) — a(xy),

and soon. The number of things contained in any class re-
solvable into distinct groups or portions is equal to the sum
of the numbers of things found in those separate portions.
It is evident, further, that any expression formed of the
logical symbols x, v, &c. may be developed or expanded in
any way consistent with the laws of the symbols, and the
symbol z applied to each term of the result, provided that
any constant multiplier which may appear, be placed outside
the symbol z ; without affecting the value of the result. The
expression 7(1), should it appear, will of course represent the
~ number of individuals contained in the universe. Thus,

n{(r—x)(1 —y)} ΠΧ y + xy)
=n(1) — n(x) -- n(y) + (xy).
Again, μίαν +(1 —x)(1 -—y)} ξ- πᾷ -- α -- Ὁ 257)
=n(1) -- n(x) -- π(ν) + 20(xy).

314 THE LAWS OF 1THOCGHT

ΤΥ του

In the last member the term 25(1:}) indicates ¢wzce the number

of individuals contained in the class ry.

4. We proceed now to investigate the numerical limits of |

classes whose logical expression is given. In this inquiry
the following principles are of fundamental importance :

Ist. If all the members of a given class possess a certain
property x, the total number of individuals contained in the
class x [298] will be a superior limit of the number of indi-
viduals contained in the given class.

2nd. A minor limit of the number of individuals in any
class y will be found by subtracting a major numerical limit
of the contrary class, 1—y, from the number of individuals
contained in the universe.

To exemplify these principles, let us apply them to the

following problem :

PROBLEM.—Given, (1), 2(#), and x(y), required the
superior and inferior limits of (vy).

Here our data are the number of individuals contained in
the universe of discourse, the number contained in the class 2,
and the number in the class y, and it is required to determine
the limits of the number contained in the class composed of

the individuals that are found at once in the class x and in >

the class y.

By Principle I. this number cannot exceed the number con-
tained in the class x, nor can it exceed the number contained
in the class y. Its major limit will then be the least of the
two values (4) and x(y).

By Principle II. a minor limit of the class xy will be given
by the expression

n(1) — major limit of #{x(1—y) +y(1 -- x) + (1 - x)(1 -}}} (1)

since #(I—y)+y(1—¥%)+(1—#)(1—y) is the complement of
the class xy, ὦ. 6. what it wants to make up the universe.
Now #(1-y)+(1—#)(1-y)=1-—y. We have therefore
for (1),
a(t) -- major limit of z{1 —y +y(1 — x)}
=n(1) — 2(1 —y) — major limit of z{ (1 -- x)}. (2)

OF STATISTICAL CONDITIONS 315

The major limit of z{y(1—~*)} is the least of the two values
my) and x(1—%). Let 2(y) be the least, then (2) becomes
n(x) - n(x 3) ~ 219)

=n(1) — (1) + 2(y) -- π΄) =0.
Secondly, let z(1—~*%) be less than w(y), then
major limit of z{y(1 —-x)}=x(1- <x);
therefore (2) becomes
[299] n(1) -- πα —y) -- καὶ -- x)
τε μ(1) -- 2(1) + κί) -- λ(1)-Ἐ x(x)
= n(x) +n(y) = n(1),
The minor limit of ~(vy) is therefore either 0 or x(4)+n(y)
-- π(1), according as (y) is less or greater than (1 --- 2), or,
which is an equivalent condition, according as x(*) is greater
or less than (1 — 7).

Now as Ο is necessarily a minor limit of the numerical
value of azy class, it is sufficient to take account of the second
of the above expressions for the minor limit of z(ry). We
have, therefore,

Major limit of (xy) =least of values 2(x) and x(y).
Minor limit of (xv) = (x) + 2(y) — 2(1).*

PROPOSITION I

5. 70 express the major and minor limits of a class repre-
sented by any constituent of the symbols x, y, 2, &t., having
given the values of n(x), u(y), n(2), &e., and n(1).

Consider first the constituent xyz.

It is evident that the major numerical limit will be the
least of the values (x), u(y), 2(2).

The minor numerical limit may be deduced as in the
previous problem, but it may also be deduced from the solu-
tion of that problem. Thus:

Minor limit of (ayz) = (xy) + 2(z) — 2(1). (1)

* The above expression for the minor limit of #(.y) is applied by Professor
De Morgan, by whom it appears to have been first given, to the syllogistic form :
Most men in a certain company have coats.
Most men in the same company have waistcoats.
Therefore some in the company have coats and waistcoats.

316 THE LAWS OF THOUGHT

Now this means that (xyz) is at least as great as the εχ--
pression a(xy)+(z)—{1). But z(@y) is at least as great as
n(x)+n(y)—n(1). Therefore x(xyz) is at least as great as
n(x) + ny) - κα) + (2) — (1),
or a(x) +2(y) + 2(z) — 2π(1).
[300] Hence we have
Minor limit of 2(xyz) = a(x) + 2(y) + (2) -- 2n(1).

By extending this mode of reasoning we shall arrive at the
following conclusions :

ist. The major numerical limit of the class represented by
any constituent will be found by prefixing z separately to
each factor of the constituent, and taking the least of the
resulting values.

2nd. The minor limit will be found by adding all the values
above mentioned together, and subtracting from the result
as many, less one, times the value of (1).

Thus we should have
Major limit of {xy(1 -- z)} =least of the values 2(x), 2(y), and (1 — 2).
Minor limit of 2{xy(1 -- 2)} =n(x) + 2(y) + a(t - 2) -- 20(1)

=x +n(y) — κ(2) -- 2(1).

In the use of general symbols it is perhaps better to regard
all the values (+), u(y), μι — 2), as major limitsofz{zy(1 -- 2}},
since, in fact, it cannot exceed any of them. I shall in the
following investigations adopt this mode of expression.

PROPOSITION II]

6. To determine the major numertcal limtt of a class ex-
pressed by a series of constituents of the symbols x, y, 2, &¢.,
the values of n(x), n(y), n(z), &c., and n(1), being given.

Evidently one mode of determining such a limit would be to
form the least possible sum of the major limits of the several
constituents. Thus a major limit of the expression

n{ xy + (1 -- α)ίι -y)}
would be found by adding the least of the two values πᾷ), κί),
furnished by the first constituent, to the least of the two values

OF STALIOTICAL CONDITIONS 317

ΜΑΙ --- 2), πᾷ —y), furnished by the second constituent. If we
do not know which is in each case the least value, we must form
the four possible sums, and reject any of these which are equal
_toorexceed (1). Thus in the above example we should have

[301] n(x) + πᾷ -- x) -Ξ π(ι).
mx) + πᾷι -- )) -- π(α) + n(x) - π( 3).
ny) -Ὁ π(ι -- γ) Ξε (1) +2(y) -- 2(x).
n(y) + πᾷ —y) =n(1).
Rejecting the first and last of the above values, we have
m(1)+n(x)—n(y), and x(1)+n(y) — n(x),

for the expressions required, one of which will, unless
m(x)=n(y), be less than 2(1), and the other greater. The
least must of course be taken.

When two or more of the constituents possess a common
factor, as x, that factor can only, as is obvious from Prin-
ciple I., furnish a single term 2(*) in the final expression of
the major limit. Thus if (7) appear as a major limit in
two or more constituents, we must, in adding those limits
together, replace 2(x)+ x(x) by (4), and so on. ‘Take, for
example, the expression z{ay+4(1—y)z}. The major limits
of this expression, immediately furnished by addition,

would be—
1 2X), 4. n(y)+n(x).
2. μ(χ)- πᾷ -- )). 5. n(y)+n(1 -- )).
3. n(x) + n(z). 6. 2(y)+2(z).

Of these the first and sixth only need be retained ; the
second, third, and fourth being greater than the first ; and
the fifth being equal to 2(1). The limits are therefore

a(x) and n(y)+n(z),

and of these two values the last, supposing it to be less than
m(1), must be taken.
These considerations lead us to the following Rule :

RULE. —TZake one factor from each constituent, and prefix
to 11 the symbol n, add the several terms or results thus formed

318 THE LAWS Of) FROCGH TT,

together, rejecting all repetitions of the same term, the sum |
thus obtained will be a major limit of the expression, and the
least of all such sums will be the major limit to be employed.

Thus the major limits of the expression

aye + x(t — y(t — 2) + (1 -- x)(r -- ))(1 -- 2)

would be
[302] m(x)+n(1-y), and x(x)+x(1 - 2), |
or n(x) +n(1)—x(y), and x(x) +x(1) - (2).

If we began with z(), selected from the first term, and
took (x) from the second, we should have to take (1 —/y)
from the third term, and this would give

my)+na(x)+n(1-yv), or μι) +(x).
But as this result exceeds 2(1), which is an obvious major
limit to every class, it need not be taken into account.

PROPOSITION III]

7. To find the minor numerical {27,111 of any class expressed
by constituents of the symbols x, y, 2, having given n(x), n(y),
Wes 1):

This object may be effected by the application of the pre-
ceding Proposition, combined with Principle II., but it is’
better effected by the following method :

Let any two constituents, which differ from one another
only by a single factor, be added, so as to form a single’
class term as #(1—y)+.y form x, and this species of aggre-
gation having been carried on as far as possible, z.e., there
having been selected out of the given series of constituents |
as many sums of this kind as can be formed, each such sum
comprising as many constituents as can be collected into a
single term, without regarding whether any of the said con-
stituents enter into the composition of other terms, let these
ultimate aggregates, together with those constituents which
do not admit of being thus added together, be written down ;
as distinct terms. Then the several minor limits of those
terms, deduced by Prop. I., will be the minor limits of the_

OR STATISTICAL CONDITIONS 319

expression given, and one only of those minor limits will at
the same time be positive.

Thus from the expression ry+(1—*+)y+(1—2)(I —y) we
can form the aggregates y and I —+, by respectively adding
the first and second terms together, and the second and
third. Hence x(y) and 2(1—~) will be the minor limits of
the expression given. Again, if the expression given were

[303] ays + x(t —y)et+ (1 -- x)yst (1 - x)(1 -y)s

+ay(1-2z)+(1-x)(1-y)(1 -- 2),
_ we should obtain by addition of the first four terms the single
term z, by addition of the first and fifth term the single term
ay, and by addition of the fourth and sixth terms the single
term (1 —%)(1—y); and there is no other way in which con-
stituents can be collected into single terms, nor are there
any constituents left which have not been thus taken account
of. The three resulting terms give, as the minor limits of
the given expression, the values

n(z), n(x) + a(y) - πα)
and πᾷι -- α) Ἐπᾷ -- γ) -- π(α), or πᾷ) —2(x) -- x(y).

8. The proof of the above rule consists in the proper
application of the following principles :—i1st. The minor
limit of any collection of constituents which admit of being
added into a single term, will obviously be the minor limit
of that single term. This explains the first part of the rule.
2nd. The minor limit of the sum of any two terms which
either are distinct constituents, or consist of distinct con-
stituents, but do not admit of being added together, will be
the sum of their respective minor limits, if those minor limits
are both positive ; but if one be positive, and the other nega-
tive, it will be equal to the positive minor limit alone. For
if the negative one were added, the value of the limit would
be diminished, z.e. it would be less for the sum of two terms
than for a single term. Now whenever two constituents
differ in more than one factor, so as not to admit of being
added together, the minor limits of the two cannot be both
positive. Thus let the terms be 272 and (1—*7)(1— 7)z,

320 THE LAWS OF TF ROUCH T

which differ in two factors, the minor limit of the first is
μ(α -Ἐ γ-Ὲ 4 -- 2), that of the second 211 -—r++1—y+z—2), or,
Ist. a{xt+y—1-(1-2)}. 2nd. z{1-x-y-(1-3)}.

If n(x+y—1) is positive, 2(1—x—y) is negative, and the
second must be negative. If w(v+y—1) is, negative, the
first is negative ; and similarly for cases in which a larger
number of factors are involved. It may in this manner be
shown that, according to the mode in which the aggregate
terms are formed in [304] the application of the rule, no
two minor limits of distinct terms can be added together,
for either those terms will involve some common constituent,
in which case it is clear that we cannot add their minor
limits together,—or the minor limits of the two will not be
both positive, in which case the addition would be useless.

PROPOSITION IV

9. Given the respective numbers of individuals comprised
wn any classes, 5, t, &c., logically defined, to deduce a system
of numerical limits of any other class w, also logically de-
jined.

As this is the most general problem which it is meant to
discuss in the present chapter, the previous inquiries being
merely introductory to it, and the succeeding ones occupied
with its application, it is desirable to state clearly its nature
and design.

When the classes s, ¢, . . w are said to be logically defined,
it is meant that they are classes so defined as to enable us
to write down their symbolical expressions, whether the
classes in question be simple or compound. ΒΥ the general
method of this treatise, the symbol w can then be determined
directly as a developed function of the symbols s, ¢, &c., in
the form

w= A+oB+-C+42D, (1)
ο

wherein A, B, C, and D are formed of the constituents of
s, ὦ, &c, How from such an expression the numerical limits

OF STATISTICAL CONDITIONS 301

of w may in the most general manner be determined, will
be considered hereafter. At present we merely purpose to
show how far this object can be accomplished on the principles
developed in the previous propositions ; such an inquiry being
sufficient for the purposes of this work. For simplicity, I
shall found my argument upon the particular development,

w= st+os(t --ἢ + (1 —sit+2(1 - s\(r— 2), (2)

in which all the varieties of coefficients present themselves.
Of the constituent (1—s)(1—#), which has for its co-

β ον. :
efficient a it is implied that some, none, or all of the class

evident that (w) will have its highest numerical value when
all the members of the class denoted by (1 — s)(1 —Z) are found
in w. Moreover, as none of the individuals contained in the
classes denoted by s(1—¢) and (1 —s)¢ are found in w, the
superior numerical limits of w will be identical with those of
the class s¢+(1—s)(1—¢). They are, therefore,

a(s)+n(1—t) and πᾷ) Ἐλᾷ -- 5).

denoted [305] by that constituent are found in w. It is
|

In like manner a system of superior numerical limits of the
development A+0B+°C +=D, may be found from those of
A+C by Prop. 71.
_ Again, any minor numerical limit of w will, by Principle
II., be given by the expression
| a(t) — major limit of (1 — 20),
but the development of w being given by (1), that of 1—w
will obviously be
1-w=04+B+°C+-D.
° ce)
This may be directly proved by the method of Prop. II.
Chap. X. Hence

Minor limit of 2(w) = 2(1) -- major limit of (B+ C)
= minor limit of #(4 + D),
21

322 THE LAWS OF THOUGHT

by Principle II., since the classes A+ D and B+C are sup-
plementary. Thus the minor limit of the second member of
(2) would be μά), and, generalizing this mode of reasoning,
we have the following result :

A system of minor limits of the development

Aopen Cy 67}
Ο oO

will be given by the minor limits of A+ 2).

This result may also be directly inferred. For of minor
numerical limits we are bound to seek the greatest. Now
we obtain in general a higher minor limit by connecting the
class D [306] with A in the expression of w, a combination
which, as shown in various examples of the Logic we are per-
mitted to make, than we otherwise should obtain.

Finally, as the concluding term of the development of w
indicates the equation D=o, it is evident that 2(D)=o.
Hence we have

Minor limit of #(D)<o,
and this equation, treated by Prop. III., gives the requisite
conditions among the numerical elements x(s), 2(¢), &c., in
order that the problem may be real, and may embody in its
data the results of a possible experience.

Thus from the term τὰ —s)¢ in the second member of (2)

we should deduce
μᾷι -- 5) Ὁ π(ῷ — 2(1)<0,
n(t)<n(s).
These conclusions may be embodied in the following rule :

10. RULE.—Determine the erpression of the class w as a
developed logical function of the symbols, 5, t, &c. in the form

AT OB Glee Dp.
[9] ©

Then will
Maj. lim. w= Maj. lim. 4+C.
Min. lim. w= Min. lim. 4 + DZD.

OF STATISTICAL CONDITIONS 323

The necessary numerical conditions among the data being given
by the inequality
Min. lim. Dex(1).

To apply the above method to the limitation of the solu-
tions of questions in probabilities, it is only necessary to
replace in each of the formule (+) by Prob. (x), 2(yv) by
Prob. y, &c., and, finally, 2(1) by 1. The application being,
however, of great importance, it may be desirable to exhibit
in the form of a rule the chief results of transformation.

11. Given the probabilities of any events, s, Ζ, &c., whereof
another event w is a developed logical function, in the form

w=A+o0B+°C+2D,
ie) re)

[307] required the systems of superior and inferior limits of
Prob. w, and the conditions among the data.

SOLUTION. —The superior limits of Prob. (4+C), and the
inferior limits of Prob. (4+) will form two such systems
as are sought. The conditions among the constants in the
data will be given by the inequality,

Inf. lim. Prob. D=o.

In the application of those principles we have always
anf. lim. Prob. x,x, . . x, = Prob. x, + Prob. x, .. + Prob. x, —(#- 1).

Moreover, the inferior limits can only be determined from
single terms, either given or formed by aggregation. Superior
limits are included in the form 2 Prob. 4, Prob. x apply-
'ing only to symbols which are different, and are taken
from different terms in the expression whose superior
limit is sought. Thus the superior limits of Prob. ryz+

4(1—y)(1—2) are
Prob. x, Prob. y+ Prob. (1-2), and Prob. z+ Prob. (1-7).

Let it be observed, that if in the last case we had taken Prob.
2 from the first term, and Prob. (1 -- 5) from the second,—a
connexion not forbidden,—we should have had as their sum
I, which as a result would be useless because ὦ przorz neces-

324 THE LAWS OF THOUGHT

sary. It is obvious that we may reject any limits which do :

not fall between o and 1.
Let us apply this method to Ex. 7, Case III., of the last

Chapter.
The final logical solution is

x a Ostu + lsiu +ista + sia
Oo οΟ Oo
« 5tu + osu + οὐδ + o5ti,
Ο
the data being

Prob. s=f, Prob. f=g, Prob. w=z.

We shall seek both the numerical limits of x, and the con-
ditions connecting 2, g, and 7.

[308] The superior limits of x are, according to the rule,
given by those of stu+sii. They are, therefore,

DO ty ft -: 7:
The inferior limits of # are given by those of

slu + οἰ + Sti + Stu,

We may collect the first and third of these constituents in —
the single term s/, and the second and third in the single term —

sz. The inferior limits of « must then be deduced separ-

ately from the terms s(1—Z), s(1—%), (1—s)tu, which give
Mri τῷ pl ον τ 2 7: 2,

or D-9, pf, and ¢+7-p—t.

Finally, the conditions among the constants 23, g, and r

are given by the terms
Nie, ὌΝ ea,
from which, by the rule, we deduce

pti-qgt+tr-2=0, p+qti-r-—-2<o, I-ptgtr-—2=0,
or I+g—p-r50, 1+7-p-gs0, 1+p-g-750.

These are the limiting conditions employed in the analysis.
of the final solution. The conditions by which in that
solution ἃ is limited, were determined, however, simply
from the conditions that the quantities s, ¢, and # should be

sare

OF SEALTISTICAL CONDITIONS 325

positive. Narrower limits of that quantity might, in all pro-
bability, have been deduced from the above investigation.

12. The following application is taken from an important
problem, the solution of which will be given in the next
Chapter. There are given,

Prob. x=¢,, Prob. y=c,, Prob. s=«¢,f,, Prob. t=cf,,
together with the logical equation
5 Ξε σίχγ + sixy + Stxy + oS
ae { stxy + stxy + stxy + stxy + stxy
οἱ +shey + Stay + δέχ» + δίχα;

[309] and it is required to determine the conditions among
the constants ¢,, ¢,, £1, f,, and the major and minor limits
of 2.

First let us seek the conditions among the constants.
Confining our attention to the terms whose coefficients are

=; we readily form, by the aggregation of constituents, the
following terms, viz.:
Si) ACR) Sele) 4);

nor can we form any other terms which are not included

under these. Hence the conditions among the constants are,

n(s) + πᾷ -- x) -- n(1)<0,
a(t) + κι -- γ) — n(1)=0,
ns) + ny) +2(1 -- ἢ -- 22(1)=0,
n(t) + n(x) + πᾷ - 5) - 2n(1)<o.
Now replace w(x) by cy, 2(y) by ¢,, m(s) by εἰς, u(¢Z) by
CoP, and κί} by 1, and we have, after slight reductions,
QPi<4, @po<lo,
yPy<I—O(1— py), ἀεςι - 4(1-D))-
Such are, then, the requisite conditions among the constants.
Again, the major limits of z are identical with those of

._ the expression

stxy +5(1 —f)x(t —y) + (1 - σα — x)y ;
which, if-we bear in mind the conditions

n(s)<n(x), a(t)<n(y),

326 THE LAWS OF THRCUGH T,

above determined, will be found to be
n(s) + πᾷ), Or, OP, +o po,
n(s) Ὁ μι — x), or, I —¢,(1 — py);
n(t)+n(1—y), OF, τ οὐ — fp).

Lastly, to ascertain the minor limits of z, we readily form —
: I
from the constituents, whose coefficients are I or ᾿ς {πε

single terms s and Ζ, nor can any other terms not included —

under these be [310] formed by selection or aggregation.
Hence, for the minor limits of 5 we have the values c.f,
and τος.

13. It is to be observed, that the method developed above
does not always assign the narrowest limits which it is pos-
sible to determine. But it in all cases, I believe, sufficiently
limits the solutions of questions in the theory of probabilities.

The problem of the determination of the zarrowest limits
of numerical extensions of a class is, however, always re-
ducible to a purely algebraical form.* Thus, resuming the
equations

w=A+o0B+°C+D,
τ

let the highest inferior numerical limit of w be represented
by the formula an(s)+ dn(2) . . +an(1), wherein a, ὁ, c,.. d
are numerical constants to be determined, and s, 2, ἄς. the
logical symbols of which A, Δ, C, D are constituents. Then

an(s)+bn(t) . . +dn(1)=minor limit of A subject to

: the condition D=o.
Hence if we develop the function

as+bt..+d,

reject from the result all constituents which are found in D,

*
ee ee Se ΡΨ eee

the coefficients of those constituents which remain, and are —

* The author regrets the loss of a manuscript, written about four years ago,

in which this method, he believes, was developed at considerable length. His —
recollection of the contents is almost entirely confined to the zmpresston that the ©

principle of the method was the same as above described, and that its sufficiency
was proved, The prior methods of this chapter are, it is almost needless to say,
easier, though certainly less general.

OF STATISTICAL CONDITIONS 327

found also in A, ought not individually to exceed unity in
value, and the coefficients of those constituents which remain,
and which are not found in A, should individually not exceed
Ο in value. Hence we shall have a series of inequalities of
the form f<1, and another series of the form g<o, fand g
being linear functions of a, ὦ, c, &c. Then those values of a,
ὦ, . . da, which, while satisfying the above conditions, give to
the function
an(s)+én(¢) . . +dn(1),

its highest value must be determined, and the highest value
in [311] question will be the highest minor limit of w. To
the above we may add the relations similarly formed for the
determination of the relations among the given constants
ms), u(t), .. 21).

14. The following somewhat complicated example will
show how the limitation of a solution is effected, when the
problem involves an arbitrary element, constituting it the
representative of a system of problems agreeing in their
data, but unlimited in their gue@szta.

PROBLEM. —Of z events %, 4, . . %,, the following particu-
lars are known:

Ist. The probability that either the event x, will occur,
or all the events fail, is δ

2nd. The probability that either the event x, will occur,
or all the events fail, is ~,. And so on for the others.

It is required to find the probability of any single event,
or combination of events, represented by the general func-
tional form ¢(%,, - . 4,), OF ¢.

Adopting a previous notation, the data of the problem are

Prob. (x, +x, ες oh mone Prob. (Xp ἘΠῚ 5. 6 5): 2...

And Prob. τ...) is required.

Assume generally
Np tH, 2. ΔῊ ΞΕ 5 (1)

p=w. (2)
We hence obtain the collective logical equation of the problem

U{(x, +H, απ .. H)}+ owt wh=o. (2)

328 THE LAWS OF THOUGHT

From this equation we must eliminate the symbols 4,,. . %,,
and determine was a developed logical function of 5,,.. s,.
Let us represent the result of the aforesaid elimination in
the form
£w+ E'(1-w)=0;
then will & be the result of the elimination of the same
symbols from the equation

BH (ys. χε τινα, — 4]. Ky) PEK G=O. (4)

Now £& will be the product of the coefficients of all the
constituents (considered with reference to the symbols %,,
4. . 2,) [312] which are found in the development of the
first member of the above equation. Moreover, ¢, and there-
fore 1— 4, will consist of a series of such constituents, having
unity for their respective coefficients. In determining the
forms of the coefficients in the development of the first
member of (4), it will be convenient to arrange them in the
following manner :

Ist. The coefficients of constituents found in 1 -- φ.

2nd. The cochicient Of 477, ἘΣ tf foundan d:

3rd. The coefficients of constituents found in @, excluding
ΓΟ ΞΕ ΕΗ wave) ts

The above is manifestly an exhaustive classification.

First then; the coefficient of any constituent found in
1— 4, will, in the development of the first member of (4),
be of the form

1 + positive terms derived from 2.

Hence, every such coefficient may be replaced by unity,
Propet: «Chap. 1X,

Secondly ; the coefficient of z,.. #,, if found in ¢, in the
development of the first member of (4) will be

Dee Or sae to

Thirdly ; the coefficient of any other constituent, 7, .. %;,

#,,,.. #,, found in g, in the development of the first member
OM(4) will be es es:
Now it is seen, that £ is the product of all the coefficients

OF STATISTICAL CONDITIONS 320

above determined ; but as the coefficients of those constituents
which are not found in ¢ reduce to unity, E may be regarded
as the product of the coefficients of those constituents which
are found in ¢@. From the mode in which those coefficients
are formed, we derive the following rule for the determina-
tion of A, viz., in each constituent found in g, except the
constituent 7,7, .. %,, for x, write 5,, for 4, write s,, and so
on, and add the results ; but for the constituent 7,7, .. %,,
if it occur in g, write 5,+5,..+5,; the product of all these
sums is &.

To find 2’ we must in (3) make wo, and eliminate
4, %, τὸ #, from the reduced equation. That equation

will be
S{(x, +4, .. +H,_)5,+5,(%,-%, . . &,)} +6 =0. (5)

[313] Hence £’ will be formed from the constituents in
I—¢, 2.6. from the constituents zo¢ found in @ in the same
way as # is formed from the constituents found in φ.

Consider next the equation

Ew + E'(1 ~-w)=o.
This gives
E

ees (6)

τὺ

νον απ: are functions Gk the symbals sos... ὁ
The expansion of the value of w will, therefore, consist
of all the constituents which can be formed out of those
symbols, with their proper coefficients annexed to them, as
determined by the rule of development.

Moreover, & and £’ are each formed by the multiplication
of factors, and neither of them can vanish unless some one
of the factors of which it is composed vanishes. Again, any
factor, as 5,.. +5, can only vanish when all the terms by
the addition of which it is formed vanish together, since in
development we attribute to these terms the values Oo and 1,
only. It is further evident, that no two factors differing
from each other can vanish together. Thus the factors
δι +5,..4+5,, ands,+5,..+5,, cannot simultaneously vanish,

330 THE LAWS OF FHOUGHT

for the former cannot vanish unless s,=0, or s,;=1; but the
latter cannot vanish unless s, =o.

First, let us determine the coefficient of the constituent
5,5, . . §, in the development of the value of w.

The simultaneous assumption 5,=1, 5,=1, .. 5,=1, would
cause the factor s,+5,..+5, to vanish if this should occur
in & or &’; and no other factor under the same assumption
would vanish; but s,+5,..-+5, does not occur as a factor
of either & or &’; neither of these quantities, therefore, can

/

vanish ; and, therefore, the expression is neither

E’ — E’
Ὁ,
¥, O, nor —.
(θ᾽
Wherefore the coeffictent of 5,5... . 5, in the expanded value
I
of w, may be represented by —.
O

Secondly, let us determine the coefficient of the constituent
σὺν
[111] ΤΠ asstinptions s;—1, s,—1,. .s.—14, would cause
the factor 5,+5,..+5, to vanish. Now this factor is found

in & and not in 4’ whenever ¢ contains both the constituents

a ios Ἐ
τ τὰ τὺ. τ. Ἔϊεγ then ΞΕ becomes iH
Oni 7.1 he factor s,-4-5,.. +s, is found in Ε΄ and not in Ὁ,
if @ contains neither of the constituents 42, .. x, and

,

tate ΤῈ εῖ then ᾿Ξ. becomes — τ Ὁ Ilastly,

the factor 5,+5,..+5, is contained in both & and £’, if
one or ime. constituents 7.7, 2°44 and wv... 2.2, 1s found in

7

: ia O
and one is not. Here then - becomes -.
ps | DD O

The coeffictent of the constituent s,s... 5, will therefore
O ͵ : ;
Tn τ: ἰοὺ Ἰὼ , according as ᾧ contains both the constituents
41%, .. X, ANA 4,%_.. %,, or netther of them, or one of them
and not the other.

Lastly, to determine the coefficient of any other con-
Slituent ἂς τ S07 ς

n°

OF STATISTICAL CONDITIONS 331

a he assumptions: αι τε S;— 1, 5,—0,. δ ΞΞῸ; would
cause the factor 5,..+5,+5,,,..+5, to vanish. Now this
οὐ is found’ in) if the constituent ἀρ 7,6. ον 18
found in ᾧ and in #’, if the said constituent is not found in

/ /

φ. In the former case we have ———. =1; in the latter

E’-E E’

Bee We: O
E’—E o-E

Hence the coeffictent of any other constituent s,.. 5,5,,,.. 5,
7s 1 or Oaccording as the stmtlar constituent x... #44... 4,
ts or ts not found in φ.

We may, therefore, practically determine the value of w
in the following manner. Rejecting from the given ex-
Bression, Ohio. the ΟΠ ΞΕ ΘΙ ον 401.) γε θην ae
should both or either of them be contained in it, let the
Byinbols4. ον inthe result be-changed ito 5,550. δι
pespectively. Let the coefficients of the constituents 5,5, κ᾿
and 5,5... 5, be determined [315] according to the special
rules for those cases given above, and let every other con-
Stituent have for its coefficient o. The result will be the
walue of wz as.adtunetion Of 5, f5.025,6

As a particular case, let ¢=2z,. It is required from the
_ given data to determine the probability of the event 7.

The symbol #,, expanded in terms of the entire series of
νυν ΠΟΙ ΤΑΣ ΤΠ sencrate allthe constituents of
those symbols which have #, as a factor. Among _ those
constituents will be found the constituent 2,4, . .%,, but not
Bre CONSULUCHE 7.25. sie.

Hence in the expanded value of +, as a function of the
ayinbols s,,5,, . « s,, the constituent s,s, . . s, will have the

= 0.

case we have

ee © ; ΓΑ : aie ot
coefficient —, and the constituent 5,5, .. 5, the coefficient 5
O

If from z, we reject the constituent 7,7,.. +,, the result
will be +,—+,%,. . #,, and changing therein +, into s,, &c.,
we have s,—5,5,.. 5, for the corresponding portion of the
sepression of 4, as a function of 5,, 5,, . - 5,,.

Hence the final expression for 7 is

332 THE LAWS OF THOUGAT

Hy = Sy — 548 Sy tb πο ᾿ς ἜΤΗ 5 :
ἐπ πα προ ποτ δα (7) %
+ constituents whose coefficients are o.
The sum of all the constituents in the above expansion —
: : O : ee
whose coefficients are either I, O, or x will be τ . S,.

We shall, therefore, have the following algebratc system ]
for the determination of Prob. 7, viz. :

ee ἢ aie id Mead Nana) LED (8) 4
{πὴ Ane i
with the relations a
oper ais
ral Ds ie “2. (9) a
=1-5,5,.. 5,=A. :

It will be seen, that the relations for the determination of :
$15... S, are quite independent of the form of the function]
φ, and the values of these quantities, determined once, will |
serve [316] for all possible problems in which the data αὔθ
the same, however the guc@szta of those problems may vary. —
The nature of that event, or combination of events, whose |
probability is sought, will affect only the form of the func- ;
tion in which the determined values of s,5,.. 5, are to be 7
substituted. ;

We have from (9) ‘

pies arty a ee

Sy =p,A, So Ξε 29λ, PE Gay Ξ-- 2,λ. :
Whence 1-(1-P,A)(1 -- pA) . . (1 - ppd) =A. :
Or, 1-A=(1-7,A)(1—-f,A) . . (1—-f,A); (10) ©

from which equation the value of ἃ is to be determined.

Supposing this value determined, the value of Prob. x, _
will be

ee PP PA :

r= (t= PAN(I= Ayr) - . ἃ —PyA)’ |

or, on reduction by (10),
Prob. x,=f,-(1-c)p,po 2. Ὁ (11) [

Let us next seek the conditions which must be fulfilled —

among the constants ~,, ~,,..,, and the limits of theq
Value Of PTOp. 4. |
;

OF STATISTICAL CONDITIONS 333

As there is but one term with the coefficient εἶ there is
O

but one condition among the constants, viz.,

Minor limit, (τ -- κ,)ἃ -- 2). .ὄ (@=—S, eo:

Or, μ(ι -- 51) -Ἐπᾷι -τ 2} . . Ἔπᾷ -- 5.) — (a -- 1)n(1)<o.
fOr, m(1) — 2(s,) — π(5.) . . — 2(S,)<0.

Whence Ppctiiate ct Pe et,

the condition required.
The major limit of Prob. 7, is the major limit of the sum

of those constituents whose coefficients are I or me But
O
δαὶ sum iss,. Hence
Major limit Prob. «,= major limit s,=/,.

[317] The minor limit of Prob. 2, will be identical with
the minor limit of the expression

BS Ss ony Sp) Es) Ee oe):
A little attention will show that the different aggregates,
terms which can be formed out of the above, each including

the greatest possible number of constituents, will be the
following, viz. :

Ἐς Sa) τῆν τὺ. τὸ):

From these we deduce the following expressions for the
minor limit, viz. :

2ι1.-- δὲ Pi- Ps ++ 2ι 2η: ae ere ps a oe

The value of Prob. x, will, therefore, not fall short of any
of these values, nor exceed the value of 2.

Instead, however, of employing these conditions, we may
directly avail ourselves of the principle stated in the demon-
stration of the general method in probabilities. The condi-
tion that s,, 55, . . 5, must each be less than unity, requires
that \ should be less than each of the quantities - ; εἰ eis os

1 23 n

And the condition that s,, νι Ss, must each be greater

334 LHe LAWS, Of; THOUGHT.

than Ο, requires that A should also be greater than 0. Now —
Pi» 22» - - 2, being proper fractions satisfying the condition

2. ἘΖ4 ἘΝ +?,>1,

it may be shown that but one positive value of ἃ can be
deduced from the central equation (10) which shall be less

Ts as | I
than each of the quantities —, —,..—. That value of X

Pi 2. 2;

is, therefore, the one required.
To prove this, let us consider the equation

(ι -- 2ιλ)( -- 22λ) .. (1 -- 2,λ) τ τ Ἔ λεο.

When \=O the first member vanishes, and the equation
is satisfied. Let us examine the variations of the first

member between the limits \=o and an supposing 2,
1

the greatest of the values 7,, ~.,. . ,.

[318] Representing the first member of the equation by
V, we have

a= Pile PA) ἃ P,A) - 2 {τ PAY a Pak) et,
which, when \=o, assumes the form —p,—p,..—p,+1,
and is negative in value.
Again, we have

ὩΣ

De =p, pI -- p,A)(1 — 2,λ) + &e.,
consisting of a series of terms which, under the given re-
strictions with reference to the value of A, are posttive.

Lastly, when a we have
1

Goo 4 :
Pi
which is positive.

From all this it appears, that if we construct a curve, the
ordinates of which shall represent the value of V correspond-
ing to the abscissa ἃ, that curve will pass through the origin,
and will for small values of ) lie beneath the abscissa, Its

OF STATISTICAL CONDITIONS 335

" convexity will, between the limits \ =o and A= ἢ be down-
1

wards, and at the extreme limit ἢ the curve will be above
1
the abscissa, its ordinate being positive. It follows from

this description, that it will intersect the abscissa once, and

only once, within the limits specified, viz., between the
I

values A=o, and A=—_.

1
The solution of the problem is, therefore, expressed by

(11), the value of ἃ being that root of the equation (10), which

lies within the limits o and—-, -,..—.
Ar Pa Ve
The constant ¢ is obviously the probability, that if the
events %,, %,-.%,; all happen, or all fail, they will all
happen.

This determination of the value of ἃ suffices for all pro-
blems in which the data are the same as in the one just con-
sidered. It is, as from previous discussions we are prepared
to expect, a determination independent of the form of the
function ¢.

[319] Let us, as another example, suppose

φ-- οὐ w=x (I -- 4) . . (I-X_) .. $%,(1-%)) . . (τ -- α,.-)-

This is equivalent to requiring the probability, that of the
events 4, %,..%, one, and only one, will happen. The
value of w will obviously be

Ww =5,(1 ~ ἡ: {{πΞ τὴ. $5n(1 — 51). «(1 - Spi) += -(1-,). (LS),

from which we should have
Prob. (οἱ ὦ - (1-%n) τ τα) (τ αν ἢ]

_ 5,(1 τὴ . - (te τ OE τ. Ok So ὺ
τ : ire gee {ΠΥ

Ο2ιλᾳ -- pod)» - ( - διλ) - - Ba Ek —prr) . . (1— ford)

λα τ) A(t - oe ry eee)
1 --2.λ 1 -- pA : Ὰ

336 THE LAWS OF FHOUGCAT

This solution serves well to illustrate the remarks made in
the introductory Chapter (I. 16). The essential difficulties
of the problem are founded in the nature of its data and not
in that of its gu@stta. The central equation by which ἃ is
determined, and the peculiar discussions connected there-
with, are equally pertinent to every form which that problem
can be made to assume, by varying the interpretation of the
arbitrary elements in its original statement.

[320] CHAPTER XX

PROBLEMS RELATING TO THE CONNEXION OF CAUSES
AND EFFECTS

1. SO to apprehend in all particular instances the relation
of cause and effect, as to connect the two extremes in thought
according to the order in which they are connected in nature
(for the modus operandi is, and must be, unknown to us),
is the final object of science. This treatise has shown, that
there is special reference to such an object in the constitution
of the intellectual faculties. There is a sphere of thought
which comprehends things only as coexistent parts of a
universe ; but there it also a sphere of thought (Chap. XI.)
in which they are apprehended as links of an unbroken, and,
to human appearance, an endless chain—as having their
place in an order connecting them both with that which has
gone before, and with that which shall follow after. In the
contemplation of such a series, it is impossible not to feel the
pre-eminence which is due, above all other relations, to the
relation of cause and effect.

Here I propose to consider, in their abstract form, some
problems in which the above relation is involved. There
exists among such problems, as might be anticipated from
the nature of the relation with which they are concerned, a
wide diversity. From the probabilities of causes assigned
@ priort, or given by experience, and their respective prob-
abilities of association with an effect contemplated, it may be
required to determine the probability of that effect ; and this
either, Ist, absolutely, or 2ndly, under given conditions.
To such an object some of the earlier of the following pro-
blems relate. On the other hand, it may be required to

determine the probability of a particular cause, or of some
3532 22

338 THE LAWS OF THOUGHT

particular connexion among a system of causes, from ob-
served effects, and the known tendencies of the said causes,
singly or in connexion, to the production of such effects.
This class of questions will be considered in a subsequent
portion of the [321] Chapter, and other forms of the general
inquiry will also be noticed. I would remark, that although
these examples are designed chiefly as illustrations of a
method, no regard has been paid to the question of ease or
convenience in the application of that method. On the con-
trary, they have been devised, with whatever success, as
types of the class of problems which might be expected to
arise from the study of the relation of cause and effect in the
more complex of its actual and visible manifestations.

2. PROBLEM I.—The probabilities of two causes A, and
A, are c, and c, respectively. The probability that if the
cause A, present itself, an event # will accompany it (whether
as a consequence of the cause A, or not) is ~,, and the prob-
ability that if the cause A, present itself, that event Ε will
accompany it, whether as a consequence of it or not, is Pp.
Moreover, the event & cannot appear in the absence of both
the causes A, and A,.* Required the probability of the
event 5

The solution of what this problem becomes in the case
in which the causes A,, A, are mutually exclusive, is well

known to be
Prob. ΞΟ ΩΣ:

* The mode in which such data as the above might be furnished by experience
is easily conceivable. Opposite the window of the room in which I write is a
field, liable to be overflowed from two causes, distinct, but capable of being com-

bined, viz., floods from the upper sources of the River Lee, and tides from the
ocean. Suppose that observations made on separate occasions have yielded the ©
following results : On 4 occasions the river was swollen by freshets, and on P of _
those occasions it was inundated, whether from this cause or not. On B occasions _
the river was swollen by the tide, and on Q of those occasions it was inundated, —
whether from ‘this cause or not. Supposing, then, that the field cannot be in- '
undated in the absence of 40/4 the causes above mentioned, let it be required to —

determine the total probability of its inundation.
Here the elements, a, ὁ, 2, φ of the general problem represent the ratios
ΟΡ cl ge:
Ne Δ 8,
or rather the values to which those ratios approach, as the value of Wis indefinitely
increased.

a ee

PROBLEMS: ON CAUSES 339

and it expresses a particular case of a fundamental and very
important principle in the received theory of probabilities.
Here it is proposed to solve the problem free from the re-
striction above stated.

[322] Let us represent

‘he .causec Ay bya.
The cause A, by y.
The effect Z by z.

Then we have the following numerical data :

Prob. x =¢,, Prob. y=¢,

Prob. xz=6),, Prob. yz Ξ ey

Again, it is provided that if the causes A,, A, are both
absent, the effect H does not occur ; whence we have the

logical equation
(1 -x)(1 —y) = (1 — 2).

Or, eliminating v, 2(1-x)(1 -y)=0. (2)

Now assume, CoS Ve (3)
Then, reducing these equations (VIII. 7), and connecting
the result with (2),

x2(t -- δ) Ἐσᾷ — xz) +y2(1 -- ὃ ἘΠῚ - 2) +2(1-x)(1-y)=0. (4)

From this equation, z must be determined as a developed

logical function of x, y, 5, and ¢, and its probability thence
deduced by means of the data,

Prob. «=¢,, Prob. y—¢, Prob.s=aqp,, Prob. f=. (5)

Now developing (4) with respect to 2, and putting # for
1—x, 7 for 1—y, and so on, we have

(x5 + si ty? + + XV)z + (s+ 2)2=0,
Ζ erat a an en ee es = τ απειηον
ae xs — sx — yt - ty -— xy
= stixy + * stxy + * sty + * sty
ο ο ο
τιν ἜΝ ΤΣ Ege
- —stxy + Stxy 4 —stxy + -ῖχγ
ο ο ο
ie τ: le
+ Stay 4+ —Stxy + Stxy 4+ —Stxy
ο ο fe)

+ ostxy + ostxy + ostxy + OStxy. (6)

340 THE LAWS OF THOUGHT

[323] From this result we find (XVII. 17),
V = stxy + sixy + Stxy + Stay + Stxy
+ Stxy + SUxy
= stxy + sixy + Stxy + δῇ
Whence, passing from Logic to Algebra, we have the follow-
ing system of equations, # standing for the probability sought: |
stay + sixy + ste _ stay + stay +5iy

(ΑἹ ζ

_ stay + stay _ stxy +stxy

OP) CoP»
_ stay + stay + Stxy _ stuy + sixy + 5tey + 5t Vv. (7)
u I :

from which we must eliminate s, 2 7, y, and V.
Now if we have any series of equal fractions, as

we know that

la+mb+ nc a
la +mb'+ne ν

And thus from the above system of equations we may deduce ©

τὰν stay Shae γ-
bap whoop, ee

whence we have, on equating the product of the three first
members to the cube of the last, ;

SSL x Υ _ ps 8 /
CE CeCe 8)

Again, from the system (7) we have

ΖΑ, sty ἣν stxy Ὁ

whence proceeding as before

Di) ΣΧ ΤῊ ὭΣ v3 (9) 4
(1 -G, +6) p, — Μ)(1 — 0 + 0g fy — U)(GP, + Copy — %)

[324] Equating the values of 7, in (8) and (9), we have

PROBLEMS ON CAUSES 341

(u -- α,2ι)ίω - efo)(1 - u)
= {1-4(1-p,)-u}{1- C(I -- 39) -—u}(Qp, + C5 py — u),

which may be more conveniently written in the form

(ὦ — Cpi)(u -- & fy) ay {1 = κ(ι -- 21) ~~} {1 -- κα -- 29) -- u} . (10)
Opi +f. -- Lh

From this equation the value of wz may be found. It re-
mains only to determine which of the roots must be taken
for this purpose.

3. It has been shown (XIX. 12) that the quantity w, in
order that it may represent the probability required in the
above case, must exceed each of the quantities c,f,, ¢.f,, and
fall short of each of the quantities 1 —c,(1 —f,), 1 τοι —p,),
and ¢,f, +, ; the condition among the constants, more-
over, being that the three last quantities must individually
exceed each of the two former ones. Now I shall show that
these conditions being satisfied, the final equation (10) has
but one root which falls within the limits assigned. That
root will therefore be the required value of z.

Let us represent the lower limits ¢,p,, ¢.f,, by a, 6 re-
spectively, and the upper limits 1 —c,(1—f,), I—c(1 —p,),
and ¢,f,+&/.,, by 2’, 3’, δ᾽ respectively. Then the general
equation may be expressed in the form

(uw — a)(u — 6)(1 —u) —(a’ -- μ)(δ' -- μ)(έ -- u) =0, (11)
or (1-a -- δ)μωξ -- [αὖ -- α'δ' - θ ι -- α' -- ὅγε }.6-Ὁ αὖ -- α΄ δ΄ ἐ΄ =o.

Representing the first member of the above equation by V,
we have
2
oan 2(1-a -- δ). (12)
Now let us suppose a the highest of the lower limits of w, a’
the lowest of its higher limits, and trace the progress of the
_ values of V between the limits =a and w=a’,
When wz=a, we see from the form of the first member of
(11) that V is negative, and when =a’ we see that V is
positive. [325] Between those limits V varies continuously

ery aye: .
without becoming infinite, and Tt β always of the same sign.

342 THE LAWS OF THOUGHT

Hence if # represent the abscissa and V the ordinate of a
plane curve, it is evident that the curve will pass from a point
below the axis of # corresponding to «=a, to a point above
the axis of # corresponding to =a’, the curve remaining
continuous, and having its concavity or convexity always
turned in the same direction. A little attention will show
that, under these circumstances, it must cut the axis of a
once, and only once.

Hence between the limits s=a, u=a’, there exists one
value of w, and only one, which satisfies the equation (11).
It will further appear, if in thought the curve be traced, that
the other value of z will be less than ὦ when the quantity
1—a’—0’ is positive and greater than any one of the quanti-
ties a’, δ΄, ε΄ when Ι -- α΄ -- δ΄ is negative. It hence follows
that in the solution of (11) the positive sign of the radical
must be taken. We thus find

αὖ -- α'δ' -- (τ -- α -- δ᾽ γέ +
a a Say ve (13)
where 0 - {αὖ -- α΄ δ΄ -- (τ -- α' -- ὅγΖ }3-Ξ 4(ι -- α' -0')(ab- «α΄ δ΄ 4).

24

A. s)he results ‘of this investigation may to some extent
be verified. Thus, it is evident that the probability of the
event / must in general exceed the probability of the con-
currence of the event # and the cause A, or A,. Hence we
must have, as the solution indicates,

uU>C pi, 1ῶ 9.

Again, it is clear that the probability of the effect E must
in general be less than it would be if the causes A,, A, were
mutually exclusive. Hence

“<li 21 +onpp.

Lastly, since the probability of the failure of the effect Εὶ
concurring with the presence of the cause A, must, in
general, be less than the absolute probability of the failure
of E, we have

q(1-p)<1-4,
“<1-¢(1-f,).

PROBLEMS ON CAUSES 343

[326] Similarly “<1 —Co(1 — fp).

And thus the conditions by which the general solution
was limited are confirmed.

Again, let £;=1, f,=1. This is to suppose that when
either of the causes A,, A, is present, the event & will
Eccl. Wwe Nave πιεπ — cy, δε at, 6-— 1c ee,
and substituting in (13) we get

uence aes amie Ξὰ + J { (G6 = 4 — & -- 1)? + 4(G% - ἢ - &)}
-2

=¢, + ¢, —¢,¢. on reduction
=1-(1-4)(1-4).

Now this is the known expression for the probability that
one cause at least will be present, which, under the circum-
stances, is evidently the probability of the event A.

Finally, let it be supposed that c, and c, are very small,
so that their product may be neglected ; then the expression
for w reduces to 2. - 22. Now the smaller the probability
of each cause, the smaller, in a much higher degree, is the
probability of a conjunction of causes. Ultimately, there-
fore, such reduction continuing, the probability of the event
E becomes the same as if the causes were mutually exclusive.

I have dwelt at greater length upon this solution, because
it serves in some respect as a model for those which follow,
some of which, being of a more complex character, might,
without such preparation, appear difficult.

5. PROBLEM II.—In place of the supposition adopted in
the previous problem, that the event & cannot happen when
both the causes A,, A, are absent, let it be assumed that the
causes A,, A, cannot both be absent, and let the other
circumstances remain as before. Required, then, the pro-
bability of the event £4.

Here, in place of the equation (2) of the previous solution,

we have the equation
(1 - x)(1-y)=0.

The developed logical expression of 2 is found to be

344 THE LAWS, OF THOUGHT
[327] 5 ΞΞ StXY + "stxp + sty + "step
Oo ο ο
+ * sty + S7XY + 1 stey + "step
ο ο ο
τ. Le ee
+ —Stxy 4+—Stxy + Stxy 4+ —Stxy
ο ο ο
+ ostxy + ostxy + ostxy + ast :
and the final solution is

Prob. Z=4;
the quantity «being determined by the solutionof the equation

(u-a)(u—6)_ (α’ -- μ)(δ' -- μὴ
a+b-u 4,--α' --δ +1

(1)

wherein
a=Of,, b=lypo, a@=1-¢(1-f,) F=1- C(I — 29).

The conditions of limitation are the following :—That value
of w must be chosen which exceeds each of the three quantities

a, 6, and a +06 —1,

and which at the same time falls short of each of the three
quantities

,

a’, 8, and até.

Exactly as in the solution of the previous problem, it may
be shown that the quadratic equation (1) will have one root,
and only one root, satisfying these conditions. The con-
ditions themselves were deduced by the same rule as before,
excepting that the minor limit a’ +6’ —1 was found by seek-
ing the major limit of 1 -- 2.

It may be added that the constants in the data, beside satis-
fying the conditions implied above, viz., that the quantities
a’, b’,anda+4é, must individually exceed a, 6, anda’ +6’—1,
must also satisfy the condition c,+c,>1. This also appears
from the application of the rule.

6. PROBLEM III.—The probabilities of two events 4 and B
are a and 6 respectively, the probability that if the event A
take place an event & will accompany it is 2, and the proba-
bility that [328] if the event B take place, the same event Z

PROBLEMS ON CAUSES 345

will accompany it is g. Required the probability that if the

event A take place the event B will take place, or vice versa,

the probability that if B take place, A will take place.

᾿ς Let us represent the event A by #, the event B by y, and
the event & by z. Then the data are—

Prob, x ὦ, Prob. y=d.
Prob. x«z=af, Prob. yz= dg.

Whence it is required to find
Prob. xy or Prob. xy

Prob.-x Probey
et XY=S, yo=t, xy=w.
Eliminating z, we have, on reduction,
δὰ + ty + SyE+ xt5 + xy + (τ — xy)w =o,

υ.. SET + SVE txts ἂν
= 2xy—1

= xyst + Txpst + "cyst Ἔ Txpst

ο ο ο
ται τορι ee ᾿ς τε Ὁ

+ —xyst + oxyst + —xyst + —xyst
ο ο ο
τ ΤΠ Ὁ rey vigoe

+ —xyst + —xyst + oxyst + —xy5st
ο ο ο

+ xyst + oxyst + oxysl + οαγτά (1)

Hence, passing from Logic to Algebra,

axyst + xyst
Teo hese

Prob. xy = ;

x, y, Ss, and ¢ being determined by the system of equations
xyst + xyst + xyst+ xyst xyst + xyst+ xyst + xyst

a b
_ xyst t+ xyst _ xyst + xyst
epee ee 07

= xyst + xyst + xyst + xyst + xyst + xyst + xysl= V.
[329] To reduce the above system to a more convenient form,
let every member be divided by Χ752, and in the result let

346 THE LAWS OF THOUGHT

We then find :
mm +m+nn'+n mm +m +nn'+n'
a ἯΙ b
mm +m_ mm +m |
= ap bg |
=mm+m+m +nn'+n+n' +1. |
mm +nn' |
Also, Prob. xy = ma

mm +m+m +nn +ntn +1

These equations may be reduced to the form

mm +m mm +m nan'+n nn'+n'

ap 2. αἷἱ 2): (xg)
=(m+1)(m' Ἔ 1) Ἡ (κα τα)ῷσ΄ -Ἐ 1) ---τ.

: mm + nn
(m+1)(m' +1)+(u+1)(“'+1)-1- |
Now assume |

Prob. xy =

͵ μ ’ V
+1)(m +1)= » (R+ I)" +1) > — τῷ 2) ἢ
(me nin’ ει) τ (nea. (a)
Then since |
ee et tl) mp .

m+1 (m+1)(v+p- 1)

and so on for the other numerators of the system, we find,
on substituting and multiplying each member of the system
by v+yu—1, the following results :

Mi TL ny : n'y =
(m+1)ap (m'+1)bg (n+ τ)α(ι -- 2) (κ΄ +1)b(1-g) ὁ
Prob. xy =(mm' +nn')(v -Ἑ μ -- 1). (3)

From the above system we have

1 ΞΡ: whence ma fo:
Si wt Salata p—ap ,
[330] Similarly |
- ἢ 4«(1-Ξ2) to) |
μ-- bg ν -α( -- 2) ν -- [((τ -- φ)
Hence eck = ν
ἴον Geran τ τι τ ΠῚ πὴ &c.

Substitute these values in (2) reduced to the form

PROBLEMS ON CAUSES 347

ds oil | Bo)

τ ιγδ ee ot 2 id
ae (m+1)(m' +1) (a+ 1)(#' + 1)
ἡ μος = SPM) {v—a(1 oes τ} eda)

Substitute also for m, m’, &c. their values in (3), and we have

es abpg αὔ(ι - p(t -- φ)
Prob. ay=| ayaa dg) * Eee.

_ pg , atx ape?) by (4).
μ ν

Now the first equation of the system (4) gives

et a ape ee (5)
μ

συ} 7... Ι- α2:- bg.
μ

Similarly,
αὐι1 — p)(1-g) _

Vv

p-1+a(1-p)+H(1 -9).

Adding these equations together, and observing that the first
member of the result becomes identical with the expression
just found for Prob. xy, we have
Prob, χγενε μεαό- 2.
Let us represent Prob. xy by w, and let a+0—2=m, then
p+tv=u-m, (6)
Again, from (5) we have
μν = abpg -- (ap +59 -- τ)μ. (7)
[331] Similarly from the first and third members of (4)
equated we have
pv = ab(x — p)(1 -- g) — {a(t ~ 2) + (1 -- Φ) -- 1}.
Let us represent af +0q—1 by 4, and al —p)+h(1-—g)-1
by Δ΄. We find on equating the above values of uy,
hp - ἔν =ab{ pg τ (ι --2)( -- 4)}
-- αὐ(2 -Ὁ 9 -- 1).
Let ab(p+q—1)=4, then
ἦἀμ-- ἔντε ἃ (8)

348 THE LAWS OF THOUGHT

Now from (6) and (8) we get

_h(u-—m)+e yp δίῶ - ) -

= m oe m
Substitute these values in (7) reduced to the form

μίν + h) = abpg,

and we have (hu—J/){h'(u-m)+ 4) =abpgm’, (9)
a quadratic equation, the solution of which determines 26, the
value of Prob. «y sought.

The solution may readily be put in the form

Ui + Whim = 1) [{λ’ ~ Adm - 1)} + 4hhabpgn]

΄ 2hh'

But if we further observe that
ih —h('m —1)=Kh+ 1) —hh'm=(t— hk')m,

since h=aptbg—-1, λ΄ -αί(ι -- 2) -Ἐόᾳ -- 4) -- τ,
whence ht+h'=a+b-2=m,
we find
Prob, ay H+ M sim —) £m ψ{{- 115: 4habpo} (4)

2hh'

It remains to determine which sign must be given to the
radical. We might ascertain this by the general method
exemplified in the last problem, but it is far easier, and it
fully suffices in the present instance, to determine the sign
by a comparison of the [332] above formula with the result
proper to some known case. For instance, if it were certain
that the event A is always, and the event B never, associated
with the event 4, then it is certain that the events 4 and B
are never conjoined. Hence if p=1, g=0, we ought to
have «=0, Now the assumptions p= 1, g=0, give

hkh=a-1, k=6-1, l=0, m=at6-2.
Substituting in (10) we have

(a-—1)(6-1)(@+6—2)+(a+d—2)(a—1)(6—1)
2(a — 1)(6—- 1)

and this expression vanishes when the lower sign is taken.

Prob. xy=

PROBLEMS ON CAUSES 349

Hence the final solution of the general problem will be ex-
pressed in the form

Prob. xy 4λ' + λ( τε -- ἢ -- κ΄ , {{{{--ὄ 12) 411. α24)
Prob. x _ 2ahh'

wherein A=apt+dg-1, Δ΄ --α( -- 2) Ἐ6(ι -- 4) --,
ἐξ αὐ(2 -Ἐφ-- τ), m=at+b-2,

As the terms in the final logical solution affected by the
coefficient = are the same as in the first problem of this

chapter, the conditions among the constants will be the
same, ViZ., =
apei1—b1-g) bg=1-a(i—f).

7. It is a confirmation of the correctness of the above
solution that the expression obtained is symmetrical with
respect to the two sets of quantities 2, g, and 1 --2, 1 -Φ,
2.6. that on changing p into 1 -- 2, and g into 1—g, the ex-
pression is unaltered. This is apparent from the equation

Prob. xy= αὖ ες Ἔ2.1: 2) }
μ ν

employed in deducing the final result. Now if there exist
probabilities ~, g of the event 4, as consequent upon a
knowledge of the occurrences of A and δ᾽, there exist pro-
babilities 1—f, 1 --φ of the contrary event, that is, of the
non-occurrence of £ under the same circumstances. As then
the data are unchanged in [333] form, whether we take
account in them of the occurrence or of the non-occurrence
of E, it is evident that the solution ought to be, as it is,
a symmetrical function of 2, g and --2, I—@.

Let us examine the particular case in which p=1, g=I.

We find
h=at+b-1, h#=-1, ΞΡ m=atb-2,

and substituting

Prob. xy —-—ab+(at+d- 1)(2 -a—-b-ab)-(a+b—2)(ab-a—b+ 1)
Probe x. —2a(a+b - τὴ

ὡς —2ab(at+b—-1)_ 5
Sonat -- τῈ

350 THE LAWS OF THOUGHT

It would appear, then, that in this case the events A and B
are virtually independent of each other. The supposition of

their invariable association with some other event Z, of the —
frequency of whose occurrence, except as it may be inferred —
from this particular connexion, absolutely nothing is known, —
does not establish any dependence between the events 4 and ©
Bthemselves. 1 apprehend that this conclusion is agreeable —

to reason, though particular examples may appear at first
sight to indicate a different result. For instance, if the

probabilities of the casting up, Ist, of a particular species ©
of weed, 2ndly, of a certain description of zoophytes upon —
the sea-shore, had been separately determined, and if it had ©
also been ascertained that neither of these events could happen ~
except during the agitation of the waves caused by a tempest, ἡ

it would, I think, justly be concluded that the events in
question were not independent. The picking up of a piece of
seaweed of the kind supposed would, it is presumed, render

more probable the discovery of the zoophytes than it would —
otherwise have been. But I apprehend that this fact is due —

to our knowledge of another circumstance not implied in the
actual conditions of the problem, viz., that the occurrence of
a tempest is but an occasional phenomenon, Let the range
of observation be confined to a sea always vexed with storm.
It would then, I suppose, be seen that the casting up of the
weeds and of the zoophytes ought to be regarded as inde-
pendent events. Now, to speak more generally, there are
conditions common to all phz-[334]nomena,—conditions
which, it is felt, do not affect their mutual independence.
I apprehend therefore that the solution indicates, that when
a particular condition has prevailed through the whole of our
recorded experience, it assumes the above character with re-
ference to the class of phenomena over which that experience
has extended.

8. PROBLEM IV.—To illustrate in some degree the above
observations, let there be given, in addition to the data of
the last problem, the absolute probability of the event Z,
the completed system of data being |

PROBLEMS ON CAUSES 251

Prob. «=a, Prob. γ- δ, Prob. z=<,
Prob. xz=ap, Prob. γε Ξε dg,

and let it be required to find Prob. ry.

Assuming, as before, +z=5, yz=¢, xy=w, the final logical
equation is

τὺ = xysts + xy Sts + O(xysis + xySts + xPZSE + Kysst + KYss7 + XZ )

: Ι
+ terms whose coefficients are —. (1)
ο

The algebraic system having been formed, the subsequent
eliminations may be simplified by the transformations adopted
in the previous problem. The final result is
Prob. xy ab | 29 ἘΞ {δ ι 2) } i (2)
c

1-¢
The conditions among the constants are
c>ap, ¢sbg, c=a1-ali-p) cHa1-db(1-4@).

Now if ΞΡ g=1, we find
Prob. yee ;
ἢ

Ε not admitting of any value less than a or ὁ, It follows
hence that if the event / is known to be an occaszonal one,
its invariable attendance on the events 4 and y zucreases the
probability of their conjunction in the inverse ratio of its
own frequency.

[335] The formula (2) may be verified in a large number
of cases. Asa particular instance, let g=c, we find

Prob. xy = ad. (3)

Now the assumption g = c involves, by Definition (Chap. XVI.),
the independence of the events B and Z. If then Band &
are independent, no relation which may exist between A
and ΑΕ can establish a relation between A and 4; wherefore
A and B are also independent, as the above equation (3)
implies.

It may readily be shown from (2) that the value of Prob. 2,
which renders Prob. «y a minimum, is

352 THE LAWS OF THOUGHT

aa ν (24) ᾿
oe aC ἢ

Prob. z=);

If p=g, this gives

a result, the correctness of which may be shown by the same
considerations which have been applied to (3).

PROBLEM V.—Given the probabilities of any three events,
and the probability of their conjunction ; required the pro-
bability of the conjunction of any two of them.

Suppose the data to be

Prob. x=, Prob. y=g, Prob.z=r7, Prob. xyz=™m,
and the gu@sttum to be Prob. xy.

Assuming zyz=s, xy=2, we find as the final logical
equation,

f= χγΖ5 + xyZ5 + O(xVS + XS) + : (sum of all other constituents) ;
ο

whence, finally, ;
Prob. ΟΣ — 4697? ~ apgrm)

27

wherein p=1-p,&e. 41--ὴῷ -- (2- Φ};.

This admits of verification when 2-ΞῚ, when g=1, when
r=0, and therefore m=o0, &c.

Had the condition, Prob. z=7, been omitted, the solution
would still have been definite. We should have had

[33] ρου.» 224 =m) + (1-2) - δ),

ME (3

and it may be added, as a final confirmation of their correct-
ness, that the above results become identical when m=pgr.

9. The following problem is a generalization of Problem I.,
and its solution, though necessarily more complex, is obtained
by a similar analysis.

PROBLEM VI.—If an event can only happen as a conse-
quence of one or more of certain causes A,, A,, .. A,, and
if generally c, represent the probability of the cause A,, and
2: the probability that if the cause A, exist, the event E will

a ΠΤ

PROBLEMS ON CAUSES 353

occur, then the series of values of c¢, and 2; being given, re-
quired the probability of the event Z,*

Let the causes 4,, A,,.. A, be represented by Wa Ue tee
and the event £ by 52.

Then we have generally,

Prob. x;=¢;,, Prob. x2=¢,f;.
Further, the condition that ΑΕ can only happen in connexion

with some one or more of the causes A,, A,,.. A, establishes
the logical condition,

2(1 -x,)(1-—x,) .. (1-—x,)=0. (1)
[337] Now let us assume generally
x,2= ἢ
which is reducible to the form
x2(1 -- 2) Ἐκ -- x3) =0,
forming the type of a system οἵ z equations which, together
with (1), express the logical conditions of the problem. Add-

ing all these equations together, as after the previous re-
duction we are permitted to do, we have

2{ x42(1 — 2) ἘΔ — x ,2)} + σά —%,)(1- x) . . (1-x,)=0, (2)
(the summation implied by © extending from 7=1 ἴο ΖΞ 3),

and this single and sufficient logical equation, together with
the 2 data, represented by the general equations

Prob. Δ; ΞΞ C55 Prob. L,=C; Pi, (3)

* It may be proper to remark, that the above problem was proposed to the
notice of mathematicians by the author in the Cambridge and Dublin Mathema-
tical Journal, Nov. 1851, accompanied by the subjoined observations :

‘‘The motives which have led me, after much consideration, to adopt, with
reference to this question, a course unusual in the present day, and not upon slight
grounds to be revived, are the following :—First, I propose the question as a test
of the sufficiency of received methods. Secondly, I anticipate that its discussion
will in some measure add to our knowledge of an important branch of pure analysis.
However, it is upon the former of these grounds alone that I desire to rest my
apology.

** While hoping that some may be found who, without departing from the line
of their previous studies, may deem this question worthy of their attention, I
wholly disclaim the notion of its being offered as a trial of personal skill or know-
ledge, but desire that it may be viewed solely with reference to those public and
scientific ends for the sake of which alone it is proposed.”

The author thinks it right to add, that the publication of the above problem led
to some interesting private correspondence, but did not elicit a solution.

23

354 THE LAWS OF THOUGHT

constitute the elements from which we are to determine
rons:

Let (2) be developed with respect to z. We have
[Z{a(1 -4) +4(1 — x;)} + (1 -- λα — χῳ) . . (1 — χ,}]5 + ΣΙ - z) =0,
whence
is Bis τὴν Ὁ δ (4)
BA, -— S{x,(1 -ἡ τᾷ — %)} — (τ τὴ) — Hq) . . (1- Hq)
Now any constituent in the expansion of the second member
of the above equation will consist of 22 factors, of which ἡ
are taken.out of the set. 4, 7.0, 4,51 ab 7 to
and #% out of the set ¢,, 7, .. ¢,, I—%,, I-—%,,.. 1—Z,, no
such combination as #,(1—4%,), 4,(1—4,), being admissible.
Let us consider first those constituents of which (1—4,),
(1—¢,)..(1—¢,) forms the ¢-factor, that is the factor de-
rived from the set z7,,.. 1—4,.

The coefficient of any such constituent will be found by
changing ¢,, ¢,,. . ¢, respectively into Oin the second member
of (4), and then assigning to ΤΊ, 4, . . %, their values as de-
pendent upon the nature of the x-factor of the constituent.
Now simply substituting for Ζι, 24, ..¢, the value o, the
second member becomes

3

ο
- Σα,-- ἃ -- ἡ) -- χ9) .. (αι - ἁ,)

[338] and this vanishes whatever values, 0, I, we subse-
quently assign to 4,, 4%, ..4%,. For if those values are not
all equal to ὁ; the term 2%, does not vanish, and if they are
all equal to o, the term —(1—+7%,)..(1—4%,) becomes —1,
so that in either case the denominator does not vanish, and
therefore the fraction does. Hence the coefficients of all
constituents of which (1—¢,).. (1 —¢,) is a factor will be o,
and as the sum of all possible x-constituents is unity, there
will be an aggregate term O(1—¢,). . (1 —¢,) in the develop-
ment of 2.

Consider, in the next place, any constituent of which the
¢-factor is ¢,7,.. ¢,(1—4,,,)..(1—4,), 7. being equal to or
greater than unity. Making in the second member of (4),
f,=1,..¢4,=1, ¢4,,,=0,.. 4,=0, we get the expression |

PROBLEMS ON CAUSES 355

Bie Oe Χο εχ oe) τ τ τ

Now the only admissible values of the symbols being o and
I, it is evident that the above expression will be equal to I
ποτ τ =. %—=1, 1%, =0, .. “,=0, and that for all
other combinations of value that expression will assume a
value greater than unity. Hence the coefficient 1 will be
applied to all constituents of the final development which are
of the form

My ALS Ae) (XA Ar 4) (τὴ),

the x-factor being similar to the ¢-factor, while other consti-
tuents included under the present case will have the virtual

coefficient ~. Also, it is manifest that this reasoning is in-
O

dependent of the particular arrangement and succession of
the individual symbols.
Hence the complete expansion of z will be of the form

=2XT)+0(1-4)(1-4) . . (1-4)

+ constituents whose coefficients are = (5)

‘

where 7 represents any ¢-constituent except (1 ae
(1—¢,), and X the corresponding or similar constituent of
“-- 6ἐ
[339] For instance, if z= 2, we shall have
D(AT) λα hat Βα Xel bo:
#1, %, δῖα, standing for 1—4,, 1—4,, &c. ; whence
Z= Ly Xoby ly + Uy Xoliby +X Xolile
+ OX Xol ile +X Xnh lo +X Noll + δὰ. ἢ (6)

. . I
+ constituents whose coefficients are = 3

This result agrees, difference of notation being allowed for,
with the developed form of z in Problem I. of this Chapter,
as it evidently ought to do.

10. To avoid complexity, | purpose to deduce from the
above equation (6) the necessary conditions for the deter-

356 THE LAWS OF THOUGHT

mination of Prob. ¢ for the particular case in which #= 2, in
such a form as may enable us, by pursuing in thought the
same line of investigation, to assign the corresponding con-
ditions for the more general case in which z possesses any

integral value whatever.
Supposing then = 2, we have
V = X,Xobjby + XyXalyby + Ly Xob ily + αγχρῆ ἧς, + HX Xgh ly + Ly Hobby + Hy Xol ilo.

OE LY ΚΣ ΞΕ re 7, owe ae Pe
ΠΝ ρα as

the conditions for the determination of 1, 4,, &c., being

Xy Nobby + Ky Xol ιν" 4X ally
Cy

Ζ
2

XX ybily Ἔ Δγλοίγΐ,, αχχοίγή, +A Koh ly _ V. ,

“Pi CoP2 |

Divide the members of this system of equations by —
%,%,/,f,, and the numerator and denominator of Prob. 2 by
the same quantity, and in the results assume !

xyb Xol x x. |
slam, 22=m, =m, τε». (7)
Xb, Kobo xy Xo
[340] we find
Prob. z= adie ABN ΤΗΣ
MMs + M+ My t+ MyMy+ Ny + 7. I

and MM + My + Myo tM _ MMe Ὁ My Ἔ MN t Ny
δ, τ a
M,M,+M, MM +m !
CP) CoP2 /
whence, if we assume, !
(m7, +1)(m,+1)=M, (4,4 1)(%.+1)=N, (9).
we have, after a slight reduction,
Prob. z= sas Mie, ;
M+N-1

(y+ 1) ποίη 1) my(My+1)_ my(m, +1) ΣΝ
q(t - 2) οἰ - 2.) A, “Po : ᾿

BROPLEALS ON CAUSES 357

or, m,M Mm oM n,1V
(m, + 1) p, (Mo + ΞΕ (2, + 1)¢,(1 - 2.)
nLV

=M+N-1

~ (tg an ἜΤ — po)

Now let a similar series of transformations and reductions

be performed in thought upon the final logical equation {6}.

We shall obtain for the determination of Prob. g the follow-
ing expression :

Prob. 2= toy : (10)
wherein M=(m,+1)(m,+1) .. (m,+1),
N= (7, +1)(tg+1) . . (%,+ 1),
M,,.. M,, δι n,, being given by the system of equations,
Pe MoM 2 My,
(m+ Da (mq + 1)eop, (Mtn + ΠΣ,
n,1V n,1V ρον. (11)

~ (1+ 1)¢(1 — py) π᾿ “(5 Ὲ 1 )¢,( 1 “2.

Still further to simplify the results, assume

[341] ΠΝ τ Με τ
M ft ͵ ΔΝ την
whence jo A τ 3, sae
μτεν--τ᾽ ἌΡΕΙ
We find
772) = Mo eS
(M,+1)o fp, (Mgt I)ly Po ~ (my HUE» Pn pe
ny a Ny ες Nn ine
(7, +1)e(1-—f,) (a+ 1)¢,(1 — po) (#,+1)e(1— Pa) Vv
whence m, = apy ye μη
μ- OP, CrP
and finally,
μ μ
22, τ Ὁ. ΞΞ “Ὁ eee Dae :
: μ-ῶ μ- &nPn

Vv

Vv
Ὁ ΣΑΣ ΚΣ Σ ΩΣ ὦ ἘΠ 22 te Ξ -
v-¢(1 —p,)’ ν-ο(τ — pn)

Substitute these values with those of 27 and WV in (9), and
we have

358 THE LAWS OF THOUGHT

μ' mated ee
(μ — & Py) (MH -- 29) (i eRe) μεν τ

The

uv
{v-a(1-p) Hy - αἱ =p,)} > -{v—e(1-p,)} ety =I
which may be reduced to the symmetrical form

(u— op). - (wr Cn Pn)

μήν --1-Ξ

VT se
Ὅν ΘΑ = Pisa poe A ~ Pn)} (12)
Vac
Finally,
M-1
ἔτους Sey grag τον (13)

Let us then assume I —y=#, we have then

Fe Coma al ες (μ - φιβη)
| Foote

_{1-4(t-A,) - 4} SAT GU Pa) τ}

(1 -«u)"""
[342] If we make for simplicity
οὔ αι Snbn=any 1-4(1-—p,) =4,, &e.,

the above equations may be written as follows:

pwn BO) A), 14)
wherein eg eee”) : “ἀμ (15)
(11)

This value of , substituted in (14) will give an equa-
tion involving only w, the solution of which will determine
Tropes 2, since: by (13) Etob..2—z, < It remains. to assign
the limits of z.

‘
es

11. Now the very same analysis by which the limits were —

determined in the particular case in which 2=2 (XIX. 12),
conducts us in the present case to the following result. The
quantity #, in order that it may represent the value of Prob. 2,
must have for its inferior limits the quantities ὅς, ν΄. @;
and for its superior limits the quantities 6,, 0,,.. δ
@,+a,..+a,. We may hence infer, a préord, that there
will always exist one root, and only one root, of the equa-

FROBLEMS ON CAUSES 359

tion (14) satisfying these conditions. I deem it sufficient,
for practical verification, to show that there will exist one,
and only one, root of the equation (14), a area the limits
Me, 4,5. a and δι, 0b

First, let us consider the nature of the changes to which
pw is subject in (15), as w varies from ὦ,, which we will
suppose the greatest of its minor limits, to 6,, which we will
suppose the least of its major limits. When w=a,, it is
evident that μ is positive and greater thana,. Whenuw=42,,
we have μεεὖό,, which is also positive. Between the limits
u=a,, u=b,, it may be shown that μ increases with a.
Thus we have :

n)

dp _ , _(4,-%) ve (ὁ, -- μὴ) (d,— μ)(ός -- μὴ) . . (2 Ξ
du {π| (1 —u)"" es
οὐ χ{7, -- @)(,= 4) «᾿. On—%) (16)
+ (2-1) Gaus :
Now let
Cina ὅ, πὸ ὦ
ΕΣ 1 τ τῇ n

Evidently +,, +, . . x,, will be proper fractions, and
343 γ 41, “2 prop

we have
ἄμ ᾿ :
i ~*~ Κλ Ἔν τα πα οι ee gees Oy,
τ πτν χιϊχεῖοι. τς σὲ eg) tg es ἀντ:
NE = hy) oe ἃ

Now the negative terms in the second member are (if we
may borrow the language of the logical developments) con-
stituents formed from the fractional quantities 7, %,.. *%,,

: : Tap νὰ
Their sum cannot therefore exceed unity; whence π, 5 Posi-

tive, and » increases with τε between the limits specified.
Now let (14) be written in the form

(w= ay) :- (ε΄ ts) _ (yu) m0, (17)

n—
and assume ~=a,, The first member becomes

(u-a,) | #— 4) “ΠΟ δε. 0 ᾿ (18)

n

μ

360 THE LAWS OF THOUGHT

and this expression is negative in value. For, making the
same assumption in (15), we find |

ae _ (4-4) τος (6,—4)
1 (1 - “yt

μ =a positive quantity.

At the same time we have

(4 -- ας) es (μ --α,) μ.-- ας ee
n—1 ne ae 2

μ μ μ

and since the factors of the second member are positive -
fractions, that member is less than unity, whence (18) is
negative. Wherefore the assumption u=a, makes the first
member of (17) negative. |

secondly, let «=0,, then by (15) n=“=6,, and the first
member of (17) becomes positive.

Lastly, between the limits #=a, and “=0,, the first
member of (17) continuously increases. For the first term —
of that expression written under the form

Be a On
μ “

(μ -- αι)

[344] increases, since » increases, and, with it, every factor
contained. Again, the negative term μ —z diminishes with
the increase of “4, as appears from its value deduced’ from
(15), viz.,
(Os -- ) e (ὁ, -- ἡ
(1 ae YM

Hence then, between the limits τέ = a,, «= ,, the first member
of (17) continuously increases, changing in so doing from a
negative toa positive value. Wherefore, between the limits ©
assigned, there exists one value of z, and only one, by which |
the said equation is satisfied.

12. Collecting these results together, we arrive at the
following solution of the general problem.

The probability of the event £ will be that value of w
deduced from the equation

eae) zu (μ - SrPn) | (19)

μ

PROBLEMS ON CAUSES 361

᾿..
Ww ee alacant —p,)-u}.. {1-61 melas

(1 -—x)""1

which (value) lies between the two sets of quantities,

Py: 220... .ὡξκ and 1-4(1-f,), 1-6,(1 -2) -. 1-6,(1 —f,),

_the former set being its inferior, the latter its superior, limits.

And it may further be inferred in the general case, as it
has been proved in the particular case of z= 2, that the value
of #, determined as above, will not exceed the quantity

Py το » + +onPn-

13. Particular verifications are subjoined.

Est. Jet 7, bs Pol. f= a ΤῊΣ is, torsupposesit
certain, that if any one of the events 4,, A,,.. A,, happen,
the event # will happen. In this case, then, the probability
of the occurrence of £ will simply be the probability that the
events or causes 4,, A,,.. A, do not all fail of occurring, and
its expression will therefore be 1—(1—c,)(I—¢,). . (I—©,).

Now the general solution (19) gives

[345] ey ea) ars)
wherein "ἢ πω
Ce (eae ἢ :
Hence, t-“%=(1-4).. (1-4),
“u=1—(1I-¢,) oss (1 — Cy),

equivalent to the α priorz determination above.
Onde Vet) ΞΟ 7,-0.7. Ξ-ο πο do) civics
ea adaael oo.
2,ΞΞ Ὸ.
as it evidently ought to be.
1 τεῦ, Cpe siball quantitics, so ἐπα} theit
squares and products may be neglected. Then developing

the second members of the equation (19),
τ (GP τον ἘΦ ΖΜ:
μ -—4=— n—-1
μ
Ξμ-(αζι 24. » +orPn)s

τεῦ, 21 τ 9 ere a One

362 THE LAWS OF THOUGHT

Now this is what the solution would be were the causes
A,, Ay, .. A, mutually exclusive. But the smaller the pro- |
babilities of those causes, the more do they approach the con-
dition of being mutually exclusive, since the smaller is the
probability of any concurrence among them. Hence the result
above obtained will undoubtedy be the limiting form of the
expression for the probability of 2.

4th. In the particular case of w=2, we may readily elimi-
nate » from the general solution. The result is

(u — ¢ Py)(4 -- C5 Do) a {1 —¢(1 —p,) — u} {1 — (1 — py) εν;
Pi top, u thi
which agrees with the particular solution before obtained for
this case, Problem I.

Though by the system (19), the solution is in general made
to depend upon the solution of an equation of a high order, its
[346] practical difficulty will not be great. For the con-
ditions relating to the limits enable us to select at once a near
value of z, and the forms of the system (19) are suitable for
the processes of successive approximation.

14. PROBLEM VII.—The data being the same as in the last
problem, required the probability, that if any definite and
given combination of the causes A,, A,, .. A,,
the event £ will be realized.

The cases ἡ A,,..A,, being represented as before by

present itself,

41, 4, 1 respectively, let the definite combination of
them, referred to in the statement of the problem, be repre-
sented by 4(4, +, . . #,) so that the actual occurrence of
that combination will be expressed by the logical equation,
ep oA ger i cod Pa
The data are

ΤῸ ec. 2 Probar, —c. (1)

Prob. ἘΞ Δ biol: το A;
and the object of investigation is

BIO: OC ea ee eel
P(X), Xp ) (2)

PLOUR οὐ σὸς aces

We shall first seek the value of the numerator.

PROBLEMS ON CAUSES 363

Let us assume,
HyZ=t,.. X=, (3)
OL yr ee (4)
Or, if for simplicity, we represent $(%,, %, . . 2.) by ¢, the
last equation will be
oz =w, (5)
to which must be added the equation

Hi ees ΞΟ (6)

Now any equation v,2=¢, of the system (3) may be reduced
to the form .
Xl, + t(1 — £,2) = Ὁ.
Similarly reducing (5), and adding the different results to-
gether, we obtain the logical equation

[347] δἰ {τ τα} +4, .. %,54+620+w(1-g2)=0, (7)

from which z being eliminated, w must be determined as a
Beve lo ped. ocical AnctiOns Olas τ 1

Now making successively z=1, ¢=0 in the above equa-
tion, and multiplying the results together, we have

{2(x,2,.4+4,t,) +4, .. &,+ φῶ - τῷ} x (2t,+w) =o.
Developing this equation with reference to w, and replac-
ing in the result S¢,+1 by 1, in accordance with Prop. I.
Chap. IX., we have
Lw+ E'(1 -w)=0;

wherein fea GA ea 59a το, OT gs
FE = ΣΡ ΣΙ ΣΟ + ea,) +0) στ Py
And hence ee 8
0 = rR (8)
The second member of this equation we must now develop
with respect to the double series of symbols z,, τῶ... . +,;
t,,%,..4,. In effecting this object, it will be most con-

venient to arrange the constituents of the resulting develop-
ment in three distinct classes, and to determine the coefficients
proper to those classes separately.

364 PHE LAWS OF ΟΝ

First, let us consider those constituents of which Z, . . Z, is

a factor. Making ¢,=0, . . 2,=0, we find

E=0, L=2x,+%,.. ἃς ἘΦ.
It is evident, that whatever values (0, 1) are given to the
a-symbols, Ὁ does not vanish. Hence the coefficients of
all constituents involving 4... 4, are o.

Consider secondly, those constituents which do not involve
the factor ὦ... ¢,, and which are symmetrical with reference
fo the, two.sets of symbols 4... 7 4 and 73.7: By syne
metrical constituents is here meant those which would re-
main unchanged if 74, were converted into ¢,, %, into ¢,, &c.,
aie gece cers tic constitucnts 2, ΤῊ ee τὰ ἢ
&c., are in this sense symmetrical.

[348] For all symmetrical constituents it is evident that

ΣΙ + t,X,)

vanishes. For those which do not involve 4 . . Z,, it is
further evident that 7, . . 7%, also vanishes, whence
E=¢ 2’ = ΣΙ, (φ),
ee),
Σή,(φ) -- φ

For those constituents of which the #-factor is found in ᾧ
the second member of the above equation becomes 1 ; for
those of which the +-factor is found in ᾧ it becomes o.
Hence the coefficcents of symmetrical constituents not involv-
ing t,.. t,, of which the x-factor ἐς found in p will be 1 ; of
those of which the x-factor ts not found in ¢ it will be o.

4

Consider lastly, those constituents which are unsymmetrical ~

with reference to the two sets of symbols, and which at the
same time do not involve Z,. . 7

n°

Here it is evident, that neither Ε nor Z’ can vanish, whence
the numerator of the fractional value of w in (8) must exceed
the denominator. That value cannot therefore be represented

O
νι 0; OF 5° It must then, in the logical development, be

represented by =. Such then will be the coefficient of this

class of constituents.

PROBLEMS ON CAUSES 365

15. Hence the final logical equation by which w is ex-
pressed as a developed logical function of 4,..4%,, 4,..¢
will be of the form

w=3,(X7)+0{3,(X7) +4 . . %}+ ὦ (sum of other con-
stituents),

n?

(9)

wherein 2,(X 7) represents the sum of all symmetrical con-
stituents of which the factor X is found in ¢, and Σ,(Χ 7),
the sum of all symmetrical constituents of which the factor
X is not found in ¢,—the constituent #,..7,Z,..7,, should
it appear, being in either case rejected.

Passing from Logic to Algebra, it may be observed, that
[349] here and in all similar instances, the function V, by
the aid of which the algebraic system of equations for the
Metermmination of the values of 4.4.4 γὲ 7, is formed,
is independent of the nature of any function ¢ involved, not
in the expression of the data, but in that of the questtum
of the problem proposed. Thus we have in the present

n

example,
Prob. w= ACSA) ;
KB
wherein VeS(XT)+3(XT)+4,..4,
SN ten oie (10)

Here S(X7) represents the sum of all symmetrical con-
stituents of the ἢ and ¢ symbols, except the constituent
Wea 7. ee, © is Valle Ole ale thessame as that νὴ:
tually employed in the solution of the preceding problem,
and hence we may avail ourselves of the results there
obtained.

If then, as in the solution referred to, we assume

xt, δ ἴῃ

ε--...----Ξ

τὰ,
a : —M,, ==", &e.,

we shall obtain a result which may be thus written :

M,
M+ WV a GD
M, being formed by rejecting from the function ¢ the con-
stituent +,.. Ζ,» if it is there found, dividing the result by

n?

Prob. w=

366 CHE LAWS OF. THOCGH I

a é ὌΝ
the same constituent Ζ . . Ζ,, and then changing — into m,,
og
1

*2 into m,, andsoon. The values of J and WM are the
a}
same as in the preceding problem. Reverting to these and
to the corresponding values of m,, m,, &c., we find
Prob. w= Ζη(μπν -- 1),

the general values of m,, 2, being

CP = (1 -2,) :
μ- ὦ» μ- οὐτ-- 2)
and uw and ν being given by the solution of the system of
‘equations.

[350] μὲν! _(H= οι): UH Grn) _ tv = 4 (1 - Ay)} Ξ ἘΠῚ = GAT = Dn)}

t=

n—1 =

The above value of Prob. w will be the numerator of the
fraction (2). It now remains to determine its denominator.
For this purpose assume

φ(α» Hg)» My) =Y,
or p=;
whence pi + vp =o.

Substituting the first member of this equation in (7) in place
of the corresponding form ¢zw-+w(I — 2) we obtain as the
primary logical equation,

2{x,2t,+4(1 —x,2)} +4, .. χε φῦ τ᾿ =0,
whence eliminating z, and reducing by Prop. II. Chap. IX.,

Hence oa b+ 3t.{2(x,F,+6,4,)+ 4, ..
2φ--ἰ

φῦ aM vd te A a Gee ἘΠ Ex} Bi Xy ον ὁ oe } ΞΞΟ.
oe)
’

and developing as before,
V=B(XT)+4 .. £3,(X)+0{2(XT)+4 . . 7,2,(X)}

+. (sum of other constituents). (12)

Here ΟΣ) indicates the, sum af all constituents found
in ¢, 2 (xX) the sum of all constituents not found in 9.

PROBLEMS ON CAUSES 367

The expressions are indeed used in place of ¢ and I—¢ to
preserve symmetry.

It follows hence that =\(X)+2,(X)=1, and that, as be-
fore, 2(X7)+2,(X7)=X(XT). Hence V will have the
same value as before, and we shall have

Prob pe a ὩΣ (Χ ye

Or transforming, as in the previous case,

M, +N,
ΕΟ gare ee
οὐ πὰ (13)

[351] wherein δῆ, is formed by dividing dy DY aa ee ane

ae ἘΠ: :
changing in the result “1 into 7), *2 into Nn, SLC,
ae 49
Now the final solution of the problem proposed will be
given by assigning their determined values to the terms of

the fraction
BEG: Olas Kn 2 ἢ Prob. τὺ

τ ν΄ Prob. Ὁ
Hence, therefore, by (11) and (13) we have
M,
: ἢ Ὁ “τ
Prob. soug M+ N,

A very slight attention to the mode of formation of the
functions 47, and JV, will show that the process may be
greatly simplified. We may, indeed, exhibit the solution of
the general problem in the form of a rule, as follows:

Reject from the function (41, %, .. %,) the constituent
4,.. 4%, 17 tt ts therein contained, suppress in all the remain-
ong constituents the factors 1, %,, &c., and change generally

in the result x, into se Call thts result M,.
—C,p,
Again, replace in the function (4,, %, ..%,) the con-

stituent <,.. 4%, tf 11 ἐς therein found, by unity ; suppress in

all the remaining constituents the factors %,, ἅς, δῶ, and
ε(τ -- 2.)

aot Nees)

Then the solution required will be expressed by the formula

change generally in the result x, into

368 THE LAWS OF THOUGHT

M,
M+,’

equations

( -- αΖι) “(ἃ aes)
"_fyaa(r=a)) -- ὕπο - 20}. G5)

ν

μ΄ ἘΡ--[ΞὉ

It may be added, that the limits of μαὶ and ν are the same
as in the previous problem. This might be inferred from
the general principle of continuity ; but conditions of limita-
tion, which are [352] probably sufficient, may also be estab-
lished by other considerations.

Thus from the demonstration of the general method in
probabilities, Chap. XVII. Prop. IV., it appears that the
Guanes τι το 47.2.5. in athe primary system, ol, al-
gebraic equations must be posztzve proper fractions. Now

χε (τ -- 2.)

πο Gay):

Hence generally 2, must be a positive quantity, and there-
fore we must have

vs¢,(1 -- 2,).

In like manner since we have

Xb, Se gs: C,Pr
eae) eo ope

we must have generally

μβωΐ,.

16. It is probable that the two classes of conditions thus
represented are together sufficient to determine generally
which of the roots of the equations determining y« and ν are
to be taken. Let us take in particular the case in which
i= δ... {τ we have

tees = ERP PY) = (ey + eghy) + SAPs,

Shiri ag: ὦ 1 -- (4-40) )ooP2

vV=1-6p),—%pot+ rf ᾿

(14)

μ and y being determined by the solution of the system of —

PROBLEMS ON CAUSES 369

Whence, since u>c,p, we have generally

V<I~-Cpy.
In like manner we have
ναὶ -hnpo, μετ —¢(1-p,), μξι- Co(I -- 29).
Now it has already been shown that there will exist but

one value of μ satisfying the whole of the above conditions
relative to that quantity, viz.

PSCPry pt -- ο( —f,),
whence the solution for this case, at least, is determinate.
- And I [353] apprehend that the same method is generally
applicable and sufficient. But this is a question upon which
a further degree of light is desirable.

To verify the above results, suppose $(4,. . ,)=1, which
is virtually the case considered in the previous problem. Now
the development of 1 gives all possible constituents of the
symbols 4, ..%,. Proceeding then according to the Rule,
we find

n

MM = B a) (ales τς aaa Og 6 ;
: (μ -- ὡ21) ὦ ἀν nea) πεν | γ (15)

n

LV, ae ᾿ -.. {t= ἘΠῚ ΣΝ os
=A) ae ee ee
Substituting in (14) we find

if

Prob. z=1-¥y,

which agrees with the previous solution.

Again, let τ. . #,)=%,, which after development, and
Puppression of the factotsz,.. 7, fives 7,(4,4+1).. @,,+1),
whence we find

Ἴ Ξ Opp ag ha are
1 (@—6,~p;) » ἀπ) tv ι y (15)

= al Sat ev Aa _4(1 -p;) :
sae cers (eS) eek (rs) ee

Substituting, we have

Probability that if the event 4, occur, Z will occur =).

And this result is verified by the data. Similar verifications

might easily be added.

370 THE LAWS OF THOUGHT

Let us examine the case in which
Dy. - X_)HHXy « + Het Xgh Xe. Hy s- FX ypXy > « Fn

Here we find
M eA δ ete ΩΣ ;
: bop, Μ - nPn
ose o(1 - 24) "ΠΝ C(I Pn) ;
: v— (1 -2.) v—¢,(1 — Pn)

whence we have the following result—

[354] Probability that if some one alone 5 Dr
Of the causes Ἢ... 4.1 μ -- Cpr
present itself, the event Z will sy obra y Cae aa,
follow μποΐ, ~v—6(1 -- 2,)

Let it be observed that this case is quite different from the
well-known one in which the mutually exclusive character of
the causes A, . . A, is one of the elements of the data, ex-
pressing a condition under which the very observations by
which the probabilities of A,, A,, &c. are supposed to have
been determined, were made.

Consider, lastly, the case in which $(4 .. ,)=4,4,. τὶ
Prete

Myx Sb» οὖς αδι-- ad
(μ τ αΖ1) Ὁ: (μ a CrPn) μ᾿ μ πα αὶ πὶ 1)
q(1-p;) «+ οἱ -2Ξ2.) ἘΠ 20,5. οἴ τ

N,= as
2 Waa =A)} fy =4(1- 2}} γον
Hence the following result—
Probability that if all 4
Causes το ΑΓ, con-| Doe Pv

spire, the event 25 a ἘΡ ey ee 012)...» τ as
follow

This expression assumes, as it ought to do, the value 1 when
any one of the quantities p, . . 2 is equal to 1.

17. PROBLEM VIII.—Certain causes A,, A,, . . A, being
so restricted that they cannot all fail, but still can only occur
in certain definite combinations denoted by the equation

$(A,, a ou © Aa,

PROBLEMS ON CAUSES 371

and there being given the separate probabilities Ge. Cer
the said causes, and the corresponding probabilities je ΤΑ
than an event £ will follow if those respective causes are
realized, required the probability of the event ἢ.

This problem differs from the one last considered in several
particulars, but chiefly in this, that the restriction denoted
by the equation $(4,.. A,)=1, forms one of the data,
and is supposed [355] to be furnished by or to be accordant
with the very experience from which the knowledge of the
numerical elements of the problem is derived.

Representing the events A,..A, by #,.., respect-
ively, and the event & by z, we have—

Prob: ἀξ, ΡΙΟΡ Yeo. (1)

Let us assume, generally,

ἌΞΕΙΝ

then combining the system of equations thus indicated with
the equations

ree = Ore, Oe ie OL a

furnished in the data, we ultimately find, as the developed
expression of g,
22x) Oni. ΣΙΝ (2)

where X represents in succession each constituent found in
¢@, and 7 a similar series of constituents of the symbols
t,..%,; X(X7) including only symmetrical constituents
with reference to the two sets of symbols.

The method of reduction to be employed in the present
case is so similar to the one already exemplified in former
problems, that I shall merely exhibit the results to which it
leads. We find

Prob. z= (3)

M
M+N’
with the relations

My Ὄπ δ Nn ey (4)
“Pi ΕΣ δία -2}) C(I = 2.)

372 HE LAWS On THROCGHT

Wherein J/ is formed by suppressing in ¢(7,. . %,) all the
factors 7, . . #,, and changing in the result x, into m,, x, into
m,, while WV is formed by substituting in 27, , for m,, &c. ;
moreover J/, consists of that portion of 27 of which m, is a
factor, MV, of that portion of VV of which 7, is a factor ; and
so on,

Let us take, in illustration, the particular case in which the

causes A,.. A, are mutually exclusive. Here we have
οί ἀπ Oe ee ey,
[356] Whence M=m,+m.,.. +Mn,
N=n, +m. τ +Mn;

M,=m,, N,=m, &c.
Substituting, we have

Me eae {πο τὸς Mn Ay ey.
OP; (ay Uf εἰ - 2.) rg — Pn)

Hence we find ἘΠ Εν ΠΝ A AT
Py + lah.» . τωι '

or M VN.
CP, a as Say Le

Hence, by (3), RIGO ΞΘ 07. Ἐπ}.

a known result.

There are other particular cases in which the system (4)
admits of ready solution. It is, however, obvious in most
instances it would lead to results of great complexity. Nor
does it seem probable that the existence of a functional rela-
tion among causes, such as is assumed in the data of the
general problem, will often be presented in actual experience ;
if we except only the particular cases above discussed.

Had the general problem been modified by the restriction
that the event ΖΦ cannot occur, all the causes A, . . A, being
absent, instead of the restriction that the said causes cannot
all fail, the remaining condition denoted by the equation
(A, . . A,)=1 being retained, we should have found for the
final logical equation

g=2(XT7)+03(X ),

——

PROBLEMS ON CAUSES 373

x(X) being, as before, equal to Olt. . #.), but 2(X T)
formed by rejecting from ᾧ the particular constituent
4,..4, if therein contained, and then multiplying each
#-constituent of the result by the corresponding ¢-constituent.
It is obvious that in the particular case in which the causes
are mutually exclusive the value of Prob. g hence deduced
will be the same as before.

18. PROBLEM IX.—Assuming the data of any of the pre-
[357]vious problems, let it be required to determine the pro-
bability that if the event Z present itself, it will be associated
with the particular cause 4,; in other words, to determine
the @ posterior? probability of the cause A, when the event
E has been observed to occur.

In this case we must seek the value of the fraction

Prob. x,2 Be ay ds

Prob. s ’ Prob. z’

by the data. (1)

As in the previous problems the value of Prob. z has been
assigned upon different hypotheses relative to the connexion
or want of connexion of the causes, it is evident that in all
those cases the present problem is susceptible of a determinate
solution by simply substituting in (1) the value of that element
thus determined.

If the a priort probabilities of the causes are equal, we
have c,=c,..=c,. Hence for the different causes the value
(1) will vary directly as the quantity 23,.. Wherefore what-
ever the nature of the connexion among the causes, the a pos-
tertort probability of each cause will be proportional to the
probability of the observed event # when that cause is known
toexist. The particular case of this theorem, which presents
itself when the causes are mutually exclusive, is well known.
We have then

Prous Crs Py

ΕΙΣ ΣΙ ee,

the values οὗ c, . . ς, being equal.
Although, for the demonstration of these and similar
theorems in the particular case in which the causes are

374 THE LAWS OF THOUGHT

mutually exclusive, it is not necessary to introduce the
functional symbol ¢, which is, indeed, to claim for ourselves
the choice of all possible and conceivable hypotheses of the
connexion of the causes, yet, under every form, the solution
by the method of this work of problems, in which the number
of the data is indefinitely great, must always partake of
a somewhat complex character. Whether the systematic
evolution which it presents, first, of the logical, secondly,
of the numerical relations of a problem, furnishes any com-
pensation for the length and occasional tediousness of its
[358] processes, I do not presume to inquire. Its chief
value undoubtedly consists in its power,—in the mastery
which it gives us over questions which would apparently
baffle the unassisted strength of human reason. For this
cause it has not been deemed superfluous to exhibit in this
chapter its application to problems, some of which may
possibly be regarded as repulsive, from their difficulty,
without being recommended by any prospect of immediate
utility. Of the ulterior value of such speculations it is, |
conceive, impossible for us, at present, to form any decided
judgment.

19. The following problem is of a much easier description
than the previous ones, |

PROBLEM X.— The probability of the occurrence of a certain
watural phenomenon under given circumstances ts p. Obser-
vation has also recorded a probability a of the existence of a
permanent cause of that phenomenon, t.e. of a cause which
would always produce the event under the circumstances
supposed. What is the probability that if the phenomenon
7s observed to occur n times in succession under the given
circumstances, tt will occur the n+" time? What also ts
the probability, after such observation, of the existence of the
permanent cause referred to ?

FIRST CASE. —Let ¢ represent the existence of a permanent
cause, and %, +,,.. %,,, the successive occurrences of the
natural phznomenon.

PROBLEMS ON CAUSES 375

If the permanent cause exist, the events Ay, 495... Vy. ἈΓδ
necessary consequences, Hence

f=0x,, f=vx,, &e,
and eliminating the indefinite symbols,
3 ‘(1 —2,)=0, ¢(1—x,)=0, 41 —«,,,)=0.
Now we are to seek the probability that if the combination

Ἔν, %, happen, the event x,,, will happen, z.e. we are to
seek the value of the fraction

Prob. aX. ἄμμι,
PIOD) ivy an

We witefrst.seck the value of Prob:-4,7, 3:

[359] Represent the combination 4,2, . . x, by w, then
we have the following logical equations :
{i —x,)=0, #1—-x,)=0 .. f1—%,)=0,
ee ee
Reducing the last to the form
εὐ τς αι εξ 0) Fw ie,
and adding it to the former ones, we have
Die x) ee ταιτ γε a, α) Ξο τ
wherein > extends to all values of z from 1 to , for the one
logical equation of the data. With this we must connect
the numerical conditions,
Prob. x,= Prob. x, .. =Prob. x,=, Prob. f=a;
and our object is to find Prob. w.
From (1) we have

wy a EAE = σὴ + 1% ere

Bi vl
Σίι = Hi) +H Hy - . Hug Kye . . Hn es (2)
χα ee ΣῈ ike AE ᾿

on developing with respect to 5 This result must further

be developed with respect to 4%, %,.. %,,.
Now if we make 4,-—1, %=1,.. 4%, =1, the coefficients
both of ¢ and of 1—¢ become 1. If we give to the same

376 TARE LAWS CFL, THOUGHT,

symbols any other set of values formed by the interchange
of o and 1, it is evident that the coefficient of ¢ will become
negative, while that of 1—¢ will become 0. Hence the full
development (2) will be

Ὁ χα. ἀπ αι, a Mall Ξε Ol Aan ee 2)
Ὶ . I : I
+ constituents whose coefficients are — , or equivalent to — .
° ο
Here we have

Vee Lt a AE oP) 33 eo ce, τὶ}: ἢ
ΞΕ Xo. - ἌΣ τ ἢ»

whence, passing from Logic to Algebra,

[360] “115 - - f+ αχ( - ξ ἢ 2%.» Xb+H—(1 - δ) -

POO oe ΣΧ ee Ζ Be Χο Be Ot 5
es er nt + Xf 4)". 172 XX.» ΧΙΓΠῚ
p a
Prob. zw = 1. Ὁ Ὡ :
Χο . . x,t+1-—¢

From the forms of the above equations it is evident that
we have 7,=4,..=4%,. Replace then each of these quantities
by x, and the system becomes

αι ποῖ 2" py y τ
p a
μεθ, τ
ΧΙ J

from which we readily deduce

2 ᾿ς a. n—1
Prob. w= Prob. #,%, . . %,»=a+(p— a)( ) :

1 ΞΩ
If in this result we change z into z+1, we get
Φιαν
Γτόῦ, eWeek, eat pa) =)
I-—@
Hence we find—

“εῳ- 254"

ΒτΟθ 2 2, I-a

angie,
EIOD. 4X,

era a

as the expression of the probability that if the phanomenon

(3)

PROBLEMS ON CAUSES 377

be z times repeated, it will also present itself the 2+ 1” time.
_ By the method of Chapter XIX. it is found that a cannot
exceed 2 in value.

The following verifications are obvious :—

ist. If @=o, the expression reduces to Ὁ, as it ought
to do. For when it is certain that no permanent cause
exists, the successive occurrences of the phenomenon are
independent.

2nd. If p=1, the expression becomes 1, as it ought to do.

rd. If 7 —a, the expression becomes 1, unless a—o, “If
the probability of a phenomenon is equal to the probability
that there [361] exists a cause which under given circum-
stances would always produce it, then the fact that that
phenomenon has ever been noticed under those circum-
stances, renders certain its re-appearance under the same. Ἐ

4th. As z increases, the expression approaches in value to
unity. This indicates that the probability of the recurrence
of the event increases with the frequency of its successive
appearances,—a result agreeable to the natural laws of
expectation.

SECOND CASE.—We are now to seek the probability
a postertort of the existence of a permanent cause of the
phenomenon. This requires that we ascertain the value

of the fraction
LODE IX i Σ.
b]

Prob. να:

the denominator of which has already been determined.
To determine the numerator assume

Ζχηλο . - Xp =D,

* As wecan neither re-enter nor recall the state of infancy, we are unable to say
how far such results as the above serve to explain the confidence with which young
children connect events whose association they have once perceived. But we
may conjecture, generally, that the strength of their expectations is due to the
necessity of inferring (as a part of their rational nature), and the narrow but
impressive experience upon which the faculty is exercised. Hence the reference
of every kind of sequence to that of cause and effect. A little friend of the
author’s, on being put to bed, was heard to ask his brother the pertinent question,
—‘* Why does going to sleep at night make it light in the morning?” The
brother, who was a year older, was able to reply, that it would be light in the
morning even if little boys did not go to sleep at night.

378 THE LAWS OF THOUGHT

then proceeding as before, we obtain for the logical
development,
W=1X,Xy . . αι Ἔσ( — 2).
Whence, passing from Logic to Algebra, we have at once
Prob, Ζ ὦ,

a result which might have been anticipated. Substituting
then for the numerator and denominator of the above frac-
tion their values, we have for the ὦ posterior? probability of

a permanent cause, the expression
[362] a
ie n—1
pol)

I

It is obvious that the value of this expression increases with
the value of x. |

I am indebted to a learned correspondent,* whose original
contributions to the theory of probabilities have already been
referred to, for the following verification of the first of the
above results (3).

‘‘ The whole a priort probability of the event (under the
circumstances) being 2, and the probability of some cause C
which would necessarily produce it, a, let x be the proba-
bility that it will happen if no such cause as C exist. Then
we have the equation

p=at(t—a)x,
whence 2 τ
I= @
Now the phenomemon observed is the occurrence of the
event # times. The a priori probability of this would be—
I supposing C to exist,
#£" supposing C not to exist ;

whence the a posterior? probability that C exists is

a
a +(1—a)x” :
that C does not exist is
(1 —a)x"
a+(t—a)a"

* Professor Donkin.

PROBLEMS ON CAUSES 379

- Consequently the probability of another occurrence is

a (1 -- a)x”
xX 1+
a+(t-a)x” a+(1—a)x" Ὁ
or idols Sc a
a+(1—a)x” Ὁ

~ [363] which, on replacing x by its value2—%, will be found

I—a’
to agree with 3.”
Similar verifications might, it is probable, also be found
_ for the following results, obtained by the direct application
of the general method.
The probability, under the same circumstances, that if,
out of #z occasions, the event happen 7. times, and fail x—r

times, it will happen on the z+ 1” time is

ἘΠῚ
α- (2 -- Da

Breet 4

wherein m2" 2h) 33a Ht ΤῊ ae
2. cere 7c 72

The probability of a permanent cause (7 being less than 7)
is O. This is easily verified.

If p be the probability of an event, and c the probability
that if it occur it will be due to a permanent cause ; the pro-
bability after 2 successive observed occurrences that it will
recur on the (x+1)” similar occasion is

c+(1 -0)x"
Cha oa
wherein 3 TAG τον
τ 72

20. It is remarkable that the solutions of the previous pro-
blems are void of any arbitrary element. We should scarcely,
from the appearance of the data, have anticipated such a
circumstance, It is, however, to be observed, that in all
- those problems the probabilities of the causes involved are
supposed to be known a priort. In the absence of this as-

380 THE LAWS OF THOUGHT

sumed element of knowledge, it seems probable that arbitrary
constants would xecessarily appear in the final solution. Some
confirmation of this remark is afforded by aclass of problems
to which considerable attention has been directed, and which,
in conclusion, I shall briefly consider.

[364] It has been observed that there exists in the heavens
a large number of double stars of extreme closeness. Either
these apparent instances of connexion have some physical
ground or they have not. If they have not, we may regard
the phenomenon of a double star as the accidental result of a
‘random distribution ” of stars over the celestial vault, z.e.
of a distribution which would render it just as probable that
either member of the binary system should appear in one spot
as in another. If this hypothesis be assumed, and if the
number of stars of a requisite brightness be known, we can
determine what is the probability that two of them should
be found within such limits of mutual distance as to con-
stitute the observed phenomenon. Thus Mitchell,* esti-
mating that there are 230 stars in the heavens equal in
brightness to 8 Capricorni, determines that it is 80 to 1
against such a combination being presented were those stars
distributed at random. The probability, when such a com-
bination has been observed, that there exists between its
members a physical ground of connexion, is then required.

Again, the sum of the inclinations of the orbits of the ten
known planets to the plane of the ecliptic in the year 1801
was Ο15 4187, according to the French measures. Were all
inclinations equally probable, Laplace + determines, that
there would be only the excessively small probability
‘OOOO000I 1235 that the mean of the inclinations should fall
within the limit thus assigned. And he hence concludes,
that there is a very high probability in favour of a disposing
cause, by which the inclinations of the planetary orbits have
been confined within such narrow bounds. Professor De
Morgan, taking the sum of the inclinations at 92°, gives to

* Phil. Transactions, An. 1767.
+ Lhéorte Analytigue des Probabilités, p. 276.
+ Encyclopedia Metropolitana, art. ““ Probabilities.”

nae Gag

PROBLEMS ON CAUSES 381

_ the above probability the value ‘oooo0012, and infers that

“1 is I : 00000012, that there was a necessary cause in the

_ formation of the solar system for the inclinations being what

they are.” An equally determinate conclusion has been

drawn from observed coincidences between the direction of

[3657 circular polarization in rock-crystal, and that of certain

oblique faces in its crystalline structure. *
These problems are all of a similar character. A certain

hypothesis is framed, of the various possible consequences

of which we are able to assign the probabilities with perfect

_rigour. Now some actual result of observation being found

among those consequences, and its hypothetical probability
being therefore known, it is required thence to determine
the probability of the hypothesis assumed, or its contrary.
In Mitchell’s problem, the hypothesis is that of a ‘‘random

distribution of the stars,”—the possible and observed con-

_ sequence, the appearance of a close double star. The very

small probability of such a result is held to imply that the

probability of the hypothesis is equally small, or, at least,

of the same order of smallness. And hence the high and,
and as some think, determinate probability of a disposing

cause in the stellar arrangements is inferred. Similar re-

marks apply to the other examples adduced.

21. The general problem, in whatsoever form it may be
presented, admits only of an zudefinite solution. Let # re-
present the proposed hypothesis, y a phanomenon which
might occur as one of its possible consequences, and whose

calculated probability, on the assumption of the truth of the

hypothesis, is 23, and let it be required to determine the
probability that if the phenomenon y is observed, the hypo-
thesis x is true. The very data of this problem cannot be
expressed without the introduction of an arbitrary element.

_ We can only write

Prob. x=a, Prob. xy=ap; (1)
a being perfectly arbitrary, except that it must fall within

* Edinburgh Review, No. 185, p. 32. This article, though not entirely free
from error, is well worthy of attention.

382 THE LAWS ‘OF THOUGHT

the limits o and 1 inclusive. If then P represent the con-
ditional probability sought, we have

ἔτοσς 2) ae 88 (2)

~ Prob. y Prob. γ᾽

It remains then to determine Prob, γ.
[366] Let xy =z, then

ee mtx +21 - 2) +0(1 = Ax + τα ~ 2)(1 ~ αν). (3)

t

x

Hence observing that Prob. =a, Prob. ¢=ap, and passing

from Logic to Algebra, we have

7 te τ κα - t)x
τεῦ - 2

Prob. y=

3

with the relations

τειν ἐν
a

=fx+1-f,

Hence we readily find |
Prob. y=ap+c(1-a). (4)

Now recurring to (3), we find that ¢ is the probability,
that if the event (1 —¢)(1—~) occur, the event y will occur.
But

(1 -¢)(1-x)=(1-xy)(I-x)=1-4%,

Hence ¢ is the probability that tf the event x do not occur,
the event y will occur.

Substituting the value of Prob. y in (2), we have the
following theorem :

The calculated probability of any phenomenon y, upon an
assumed physical hypothesis x, being p, the a posteriori pro-
bability P of the physical hypothesis, when the phenomenon
has been observed, ts expressed by the equation

ἜΘΟΣ aes
apt+c(1—a)’ (5)
where a and c are arbitrary constants, the former representing
the a priori probability of the hypothesis, the latter the pro-

bability that tf the hypothesis were false, the event y would
present itself.

PROBLEMS ON CAUSES 383

The principal conclusion deducible from the above theorem
is that, other things being the same, the value of P in-
creases and diminishes simultaneously with that of . Hence
the greater or less the probability of the phanomenon when
the hypothesis is assumed, the greater or less is the pro-
bability of the hypothesis when the phenomenon has been
observed. When 2. is very small, then generally P also is
small, unless either ὦ is large or c small. [367] Hence,
secondly, if the probability of the phznomenon is very
small when the hypothesis is assumed, the probability of
the hypothesis is very small when the phenomenon is ob-
served, unless either the a 2220γὲ probability a of the hypo-
thesis is large, or the probability of the phanomenon upon
any other hypothesis small.

The formula (5) admits of exact verification in various
cases, as when c=O, or a@=1, ora=O. But it is evident that
it does not, unless there be means for determining the values
of aand ¢, yield a definite value of P. Any solutions which
profess to accomplish this object, either are erroneous in
principle, or involve a tacit assumption respecting the above
arbitrary elements. Mr. De Morgan’s solution of Laplace’s
problem concerning the existence of adetermining cause of the
narrow limits within which the inclinations of the planetary
orbits to the plane of the ecliptic are confined, appears to
me to be of the latter description. Having found a proba-
bility 23 = ‘00000012, that the sum of the inclinations would
be less than 92° were all degrees of inclination equally pro-
bable in each orbit, this able writer remarks: ‘‘If there be
a reason for the inclinations being as described, the proba-
bility of the event is 1. Consequently, it is I : ὉΟΟΟΟΟΙΖ
(2.6. 1: 2), that there was a necessary cause in the formation
of the solar system for the inclinations being what they are.”
Now this result is what the equation (5) would really give, if,

I
assigning to 23 the above value, we should assume c= 1, a= >

For we should thus find

ea ee

384 THE LAWS OF THOUGAT
I
ἘΠ
δοὺς ΤΡ»
gee
To fk op. (6)

But P representing the probability, @ posteriori, that all
inclinations are equally probable,1—VP is the probability, @
posteriori, that such is not the case, or, adopting Mr.

De Morgan’s alternative, that a determining cause exists.

The equation (6), therefore, agrees with Mr. De Morgan’s
result.

22. Are we, however, justified in assigning to a and ¢ par-
ticular values? I am strongly disposed to think that we are
not. [368] The question is of less importance in the special
instance than in its ulterior bearings. In the received applica-

tions of the theory of probabilities, arbitrary constants do not |
explicitly appear ; but in the above, and in many other in-

stances sanctioned by the highest authorities, some virtual
determination of them has been attempted. And this circum-
stance has given to the results of the theory, especially in
reference to questions of causation, a character of definite
precision, which, while on the one hand it has seemed to
exalt the dominion and extend the province of numbers, even
beyond.the measure of their ancient claim to rule the world ; ἢ
on the other hand has called forth vigorous protests against
their intrusion into realms in which conjecture is the only
basis of inference. The very fact of the appearance of
arbitrary constants in the solutions of problems like the above,
treated by the method of this work, seems to imply, that
definite solution is impossible, and to mark the point where
inquiry ought to stop. We possess indeed the means of in-
terpreting those constants, but the experience which is thus
indicated is as much beyond our reach as the experience which
would preclude the necessity of any attempt at solution
whatever. +

* Mundum regunt numeri.

+ See an interesting paper by Prof. Forbes in the Philosophical Magazine, Dec.
1850 ; also Mill’s Zogzc, chap, xviii.

PROBLEMS ON CAUSES 385

Another difficulty attendant upon these questions, and
inherent, perhaps, in the very constitution of our faculties, is
that of precisely defining what is meant by Order. The
manifestations of that principle, except in very complex in-
stances, we have no difficulty in detecting, nor do we hesitate
to impute to it an almost necessary foundation in causes
operating under Law. But to assign to it a standard of
numerical value would be a vain, not to say a presumptuous,
endeavour. Yet must the attempt be made, before we can
aspire to weigh with accuracy the probabilities of different

constitutions of the universe, so as to determine the elements

upon which alone a definite solution of the problems in
question can be established.

23. The most usual mode of endeavouring to evade the
necessary arbitrarinesss of the solution of problems in the
theory of [369] probabilities which rest upon insufficient
data, is to assign to some element whose real probability
is unknown all possible degrees of probability ; to suppose
that these degrees of probability are themselves equally
probable ; and, regarding them as so many distinct causes
of the phznomenon observed, to apply the theorems

which represent the case of an effect due to some one

of a number of equally probable but mutually exclusive
causes (Problem IX.). For instance, the rising of the sun
after a certain interval of darkness having been observed
m times in succession, the probability of its again rising
under the same circumstances is determined, on received
principles, in the following manner. Let fp be any unknown
probability between o and 1, and ς (infinitesimal and
constant) the probability, that the probability of the
sun’s rising after an interval of darkness lies between
the limits and p+dp. Then the probability that the

sun will rise # times in succession is
1
cf Pap
0

and the probability that he will do this, and will rise again,
25

386 THE. LAWS'‘OF THOUGHT

A : . . . . ° 3
or, which is the same thing, that he will rise #2+ 1 times in ~
succession, 15

C i ods,

Hence the probability that if he rise # times in succession,
he will rise the (#+1)” time, is

1
[Pe m4 I
“| "Δ m+2’
0
the known and generally received solution.

The above solution is usually founded upon a supposed
analogy of the problem with that of the drawing of balls
from an urn containing a mixture of black and white balls,
between which all possible numerical ratios are assumed to
be equally probable. And it is remarkable, that there are
two or three distinct hypotheses which lead to the same final —
result. For instance, if the balls are finite in number, and
those which are drawn are not [370] replaced, or if they
are infinite in number, whether those drawn are replaced or
not, then, supposing that 7 successive drawings have yielded
only white balls, the probability of the issue of a white ball
at the (m+1)” drawing is

M+ κα
m+ 2

It has been said, that the principle involved in the above
and in similar applications is that of the equal distribution
of our knowledge, or rather of our ignorance—the assigning
to different states of things of which we know nothing, and
upon the very ground that we know nothing, equal degrees
of probability. 1 apprehend, however, that this is an arbi-
trary method of procedure. Instances may occur, and one
such has been adduced, in which different hypotheses lead
to the same final conclusion. But those instances are ex-
ceptional. With reference to the particular problem in
question, it is shown in the memoir cited, that there is one

* See a memoir by Bishop Terrot, Zainburgh Trans., vol. xx. part iv.

PROBLEMS ON CAUSES 387

hypothesis, viz., when the balls are finite in number and not
replaced, which leads to a different conclusion, and it is easy
to see that there are other hypotheses, as strictly involving
the principle of the ‘‘equal distribution of knowledge or
ignorance,” which would also conduct to conflicting results.

24. For instance, let the case of sunrise be represented
by the drawing of a white ball from a bag containing an
infinite number of balls, which are all either black or white,
and let the assumed principle be, that αὐ possible constttu-
tions of the system of balls are equally probable. By a con-
stitution of the system, | mean an arrangement which assigns
to every ball in the system a determinate colour, either black
or white. Let us thence seek the probability, that if 7
white balls are drawn in m drawings, a white ball will be
drawn in the (#+1)” drawing.

First, suppose the number of the balls to be μ, and let the
symbols 11, 22». . 4, be appropriated to them in the follow-
ing manner. Let x, denote that event which consists in the
7. ball of the system being white, the proposition declaratory
of such a state of things being #,=1. In like manner the
compound [371] symbol 1 -- ας will represent the circumstance
of the z” ball being black. It is evident that the several
constituents formed of the entire set of symbols 4,, 4%, . - %,
will represent in like manner the several possible constitu-
tions of the system of balls with respect to blackness and
whiteness, and the number of such constitutions being 2",
the probability of each will, in accordance with the hypothesis,

I cee .
be re This is the value which we should find if we sub-
stituted in the expression of any constituent for each of the

I
που 47. 2... τ the value = Hence, then, the pro-

bability of any event which can be expressed as a series of
constituents of the above description, will be found by sub-

ane : : I
stituting in such expression the value 5 for each of the above

symbols,

388 THE LAWS OF THOUGHT

Now the larger mw is, the less probable it is that any ball |
which has been drawn and replaced will be drawn again. As ©

μ᾽ approaches to infinity, this probability approaches to 0. —

And this being the case, the state of the balls actually drawn |

5

can be expressed as a logical function of m of the symbols —
a. te, τι λα therciolc, by development, as a series of |
constituents of the said » symbols. Hence, therefore, its |
probability will be found by substituting for each of the |
symbols, whether in the undeveloped or the developed form,

the value ὦ. But this is the very substitution which it would
2

be necessary, and which it would suffice, to make, if the pro-
bability of a white ball at each drawing were known, a prior,

I
to be -.
2

It follows, therefore, that if the number of balls be infinite, ~
and all constitutions of the system equally probable, the

probability of drawing m white balls in succession will be se a

“=

and the probability of drawing # + 1 white balls in succession
oe whence the probability that after # white balls have
been drawn, the next drawing will furnish a white one, will
be 2 In other [372] words, past experience does not int

«“

this case affect future expectation.

25. It may be satisfactory to verify this result by ordinary

methods. To accomplish this, we shall seek—

First : The probability of drawing 7 white balls, and 2 —7
black balls, in 2) trials, out of a bag containing yu balls, every
ball being replaced after drawing, and all constitutions of
the systems being equally probable, ὦ priorz.

Secondly : The value which this probability assumes when
pw becomes infinite.

Thirdly : The probability hence derived, that if #z white

balls are drawn in succession, the #+1” ball drawn will be
white also. |;

Sf
ἐξ.

PROBLEM S°OW CAUSES 389

The probability that ~ white balls and p—,r black ones
will be drawn in 2. trials out of an urn containing μ balls,
each ball being replaced after trial, and all constitutions of:
the system as above defined being equally probable, is equal
to the sum of the probabilities of the same result upon the
separate hypotheses of there being no white balls, 1 white
ball,-—lastly « white balls inthe urn. Therefore, it is the
sum of the probabilities of this result on the hypothesis of
there being x white balls, 2 varying from 0 to μ.

Now supposing that there are z white balls, the probability

of drawing a white ball in a single drawing is “and the
a

probability of drawing 7 white balls and 2 —7 black ones in
a particular order in 2 drawings, is

πὰ

But there being as many such orders as there are combina-
tions of x things in p things, the total probability of drawing

-y white balls in 2 drawings out of the system of u& balls of

which z are white, is

pI po) τ ay : =\i (1)

Again, the number of constitutions of the system of μ balls,
which admit of exactly balls being white, is

[373] μίμ - 1)... (ᾳ--π τ)

Li Τὰ 70

and the number of possible constitutions of the system is 2".
Hence the probability that exactly 7 balls are white is
Lace Vso et ica)

1. 2 9 27 ΝΣ
Multiplying (1) by this expression, and taking the sum of the

products from z=o to m=, We have

Heat paregrylenD nA BA 2 9
μ

τ.,2 Ὁ. 2-0 To eee 2

for the expression of the total probability, that out of a

390 THE LAWS ΟΣ ΤΟΣ

system of « balls of which all constitutions are equally pro-
bable, γ white balls will issue in drawings. Now

grew) ss (want Hee -")"
n=0 | Pay ae Ὁ: 25 μ πε
ER ae oan ae
_ yt 3). + (eo ei. Ve é? .. (8=0)
n=0 1 25. 0 222 μ ἰ

“(1 ἜΣ -ὴς (un t1) ag

μ n=0 1.2,

-22\(1- 2)" sey, οἷ

μ ad :

so that P(D)e9=p(~)e". Βυῖ

D standing for the symbol a

by a known theorem,

2m 3.m
f= 1 4 Aor + eal 1}. cath t(¢—1)(¢- 2).
LZ 1.7. 1

D1 + &)u={1+Ao

—1)+&c.}(1 + 6).

In the second member let ¢°=%, then
a ρος δὲ

D"(1 + εθ)μ = (1 + Ao™x ee x? + &c.)(1 + x)¥
dx v ae
»since : a
DD ὃ De ay iar
( 1) (D-it+1)=x ᾿

[374] In the second member of the above equation, perform-
ing the differentiations and making x= 1 (since 9=0), we get

D"(1 + 8)" = p(Ao") 2-1 ME (Aon ee + &e.
te

The last term of the second member of this equation will be |

Mw—t). . (u—-m+1)A™”™
1 2. ΠΣ

2u-™ μίβ τα) μ᾿ ἘΈ ΤΠ ΖΡ,

since A”o"=1.2.. m. Μπέη μ is a large quantity this |
term exceeds all the others in value, and as μ᾿ approaches to
infinity tends to become infinitely great in comparison with
them. And as moreover it assumes the form ,”2""", we
have, on passing to the limit,

7 Dey pe εθ)μ = oh ae (£) 2h,
2

PROBLEMS ON CAUSES 391

Hence if @(D) represent any function of the symbol D, which
is capable of being expanded in a series of ascending powers
of D, we have

D 0)u = ΕῚ
H(D)(x + δ). = φίδ):», (4)
if@=oand n= oo. Strictly speaking, this implies that the
ratio of the two members of the above equation approaches
a state of equality, as u increases towards infinity, θ being
equal to o.

By means of this theorem, the last member of (3) reduces
to the torm
Aa) ahs a)
Εν τὸ Dba oe, 2é— | — .
2ΜΧᾺ2 2 2

P(p-1).. or

ΤΟΣ ee OA 2

Hence (2) gives

as the expression for the probability that from an urn con-
taining an infinite number of black and white balls, all con-
stitutions of the system being equally probable, 7 white balls
will issue in 2 drawings.

Hence, making p=m, r=m, the probability that in m

drawings all the balls will be white is G) , and the proba-

bility that this [375] will be the case, and that moreover the
m+1

(m+ 1)" drawing will yield a white ball is (3: , whence the

probability, that if the first # drawings yield white balls only,
the (m +1)" drawing will also yield a white ball, is

De ΓΝ L.
() +() ae
and generally, any proposed result will have the same pro-
bability as if it were an even chance whether each particular
drawing yielded a white or a black ball. This agrees with
the conclusion before obtained.

26. These results only illustrate the fact, that when the
defect of data is supplied by hypothesis, the solutions will,

392 LHE LA WS OF THOUGHT

in general, vary with the nature of the hypotheses assumed ;
so that the question still remains, only more definite in form,
whether the principles of the theory of probabilities serve to
guide us in the election of such hypotheses. I have already
expressed my conviction that they do not—a conviction
strengthened by other reasons than those above stated.
Thus, a definite solution of a problem having been found by
the method of this work, an equally definite solution is
sometimes attainable by the same method when one of the
data, suppose Prob. +=f,, is omitted. But I have not been
able to discover any mode of deducing the second solution
from the first by zxztegratzon, with respect to 2 supposed
variable within limits determined by Chap. XIX. This de-
duction would, however, I conceive, be possible, were the
principle adverted to in Art. 23 valid. »-Still it is with
diffidence that I express my dissent on these points from
mathematicians generally, and more especially from one
who, of English writers, has most fully entered into the
spirit and the methods of Laplace ; and I venture to hope,
that a question, second to none other in the Theory of
Probabilities in importance, will receive the careful attention
which it deserves.

cg ee ee ee ee ee ee oe

[376] CHART ER Ox]

PARTICULAR APPLICATION OF THE PREVIOUS GENERAL
METHOD TO THE QUESTION OF THE PROBABILITY
OF JUDGMENTS

I. ON the presumption that the general method of this
treatise for the solution of questions in the theory of
probabilities, has been sufficiently elucidated in the previous
chapters, it is proposed here to enter upon one of its practical
applications selected out of the wide field of social statistics,
viz., the estimation of the probability of judgments. Per-
haps this application, if weighed by its immediate results,
is not the best that could have been chosen. One of the
first conclusions to which it leads is that of the necessary
insufficiency of any data that experience alone can furnish,
for the accomplishment of the most important object of the
inquiry. But in setting clearly before us the xecesszty of
hypotheses as supplementary to the data of experience, and
in enabling us to deduce with rigour the consequences of
any hypothesis which may be assumed, the method accom-
plishes all that properly lies within its scope. And it may
be remarked, that in questions which relate to the conduct
of our own species, hypotheses are more justifiable than in
questions such as those referred to in the concluding sections
of the previous chapter. Our general experience of human
nature comes in aid of the scantiness and imperfection of
statistical records.

2. The elements involved in problems relating to criminal
assize are the following :—

ist. The probability that a particular member of the jury
will form a correct opinion upon the case.

2nd. The probability that the accused party is guilty.
393

394 THE LAWS OF THOUGAT

3rd. The probability that he will be condemned, or that
he will be acquitted.

4th. The probability that his condemnation or acquittal
will be just.

5th. The constitution of the jury.

[377] 6th. The data furnished by experience, such as the
relative numbers of cases in which unanimous decisions have
been arrived at, or particular majorities obtained; the
number of cases in which decisions have been reversed by
superior courts, &c.

Again, the class of questions under consideration may be
regarded as either direct or inverse. The direct questions
of probability are those in which the probability of correct
decision for each member of the tribunal, or of guilt for the
accused party, are supposed to be known a priorz, and in
which the probability of a decision of a particular kind, or
with a definite majority, is sought. Inverse problems are
those in which, from the data furnished by experience, it is
required to determine some element which, though it stand
to those data in the relation of cause to effect, cannot
directly be made the subject of observation ; as when from
the records of the decisions of courts it is required to deter-
mine the probability that a member of a court will judge
correctly. To this species of problems, the most difficult
and the most important of the whole series, attention will
chiefly be directed here.

3. There is no difficulty in solving the direct problems
referred to in the above enumeration. Suppose there is
but one juryman. Let & be the probability that the accused
person is guilty ; x the probability that the jurymen will
form a correct opinion ; X the probability that the accused
person will be condemned : then—

kx = probability that the accused party is guilty, and that
the juryman judges him to be guilty.

(1 —&)(1 —*)= probability that the accused person is inno-
cent, and that the juryman pronounces him guilty.

ΑΝ eS ee Ee | ee eS ee ere

PROBABILITY OF JUDGMENTS 395

Now these being the only cases in which a verdict of
condemnation can be given, and being moreover mutually
exclusive, we have

X=hx + (1 - &)(1 - 2), (1)

In like manner, if there be x jurymen whose separate
probabilities of correct judgment are z,, 4, .. #,, the pro-
bability of an unanimous verdict of condemnation will be

X= ha y% . . X+(1- kU -2,)(1 τα) (1 2...

[378] Whence, if the several probabilities a, ty 2 ale
equal, and are each represented by x, we have

X=hx" + (1 -&)(1— x)", (2)

The probability in the latter case, that the accused person
is guilty, will be
i
kx” + (1-21 - 4)"

All these results assume, that the events whose proba-
bilites are denoted by 4, x,, 7, &c., are independent, an
assumption which, however, so far as we are concerned, is
involved in the fact that those events are the only ones of
which the probabilities are given.

The probability of condemnation by a given number of
voices may be found on the same principles. If a jury is
composed of three persons, whose several probabilities of
correct decision are +, +’, x’, the probability that. the
accused person will be declared guilty by two of them will be

ΧΑ, Ξε ξίχα ( -- x") + xx"(1 - x’) + ἀχ (i x)}
τὰ - λα = a)(1~ a')a" τα τ λα 2")a + (1 αγα - ἀμ}
which if x=2’=7” reduces to
φάχξᾳι — x) + 3(ι-- α)χᾷ -- x)?
And by the same mode of reasoning, it will appear that if

X;, represent the probability that the accused person will be
declared guilty by z voices out of a jury consisting of ἡ

396 THE LAWS OF FHOUGHT

persons, whose separate probabilities of correct judgment
are equal, and represented by +, then

πὸ πι a eee ee — xt 4 (1 A) — α)}. (3)

1.2.
If the probability of condemnation by a determinate
majority ὦ is required, we have simply
t-a=n-1,

whence Pa +a
2

3

[379] which must be substituted in the above formula. Of
course a admits only of such values as make z an integer.
lf wis even, those values: are-O, 2.4, &c?; 1f odd, 1,3, 5,
&c., as is otherwise obvious.

The probability of a condemnation by a majority of at
least a given number of voices m, will be found by adding
together the following several probabilities determined as
above, Viz. :

ist. The probability of a condemnation by an exact
majority m ;

2nd. The probability of condemnation by the next
greater majority m+2;

and so on; the last element of the series being the prob-
ability of unanimous condemnation. Thus the probability of
condemnation by a majority of 4 at least out of 12 jurors,
would be

PG oe a τ
the values of the above terms being given by (3) after making
(herein ai 15.

4. When, instead of a jury, we are considering the case
of a simple deliberative assembly consisting of persons,
whose separate probabilities of correct judgment are denoted
by x, the above formule are replaced by others, made some-
what more simple by the omission of the quantity &.

The probability of unanimous decision is

X= x" + (1 -- x)",

PROBABILITY OF JUDGMENTS 397

The probability of an agreement of z voices out of the
whole number is |

x ue) εν (2-t+1)
ae Lees

(“Ὁ -- αὐτί τ αὐτὰ τα). (4)

Of this class of investigations it is unnecessary to give
any further account. They have been pursued to a con-
siderable extent by Condorcet, Laplace, Poisson, and other
writers, who have investigated in particular the modes of
calculation and reduction which are necessary to be employed
when # andz are large numbers. It is apparent that the
whole inquiry is of a very speculative character. The values
of x and & cannot be deter-[380]mined by direct observation.
We can only presume that they must both in general exceed

I
the value Ξ that the former, x, must increase with the

progress of public intelligence ; while the latter, %, must
depend much upon those preliminary steps in the adminis-
tration of the law by which persons suspected of crime are
brought before the tribunal of their country. It has been
remarked by Poisson, that in periods of revolution, as during
the Reign of Terror in France, the value of & may fall, if

account be taken of political offences, far below the limit

I : ;
+ The history of Europe in days nearer to our own would

probably confirm this observation, and would show that it
is not from the wild license of democracy alone, that the
accusation of innocence is to be apprehended.

Laplace makes the assumption, that all values of x from

x= ἢ to X— 1,

NIM

are equally probable. He thus excludes the supposition that
a juryman is more likely to be deceived than not, but
assumes that within the limits to which the probabilities of
individual correctness of judgment are confined, we have no
reason to give preference to one value of x over another.
This hypothesis is entirely arbitrary, and it would be un-
availing here to examine into its consequences.

398 THE LAWS OF THOOCGHT

Poisson seems first to have endeavoured to deduce the
values of x and &, inferentially, from experience. In the
six years from 1825 to 1830 inclusively, the number of in-
dividuals accused of crimes against the person before the
tribunals of France was 11016, and the number of persons con-
demned was 5286. The juries consisted each of 12 persons,
and the decision was pronounced by a simple majority.
Assuming the above numbers to be sufficiently large for the
estimation of probabilities, there would therefore be a prob-

5286
11016’
accused person would be condemned by a simple majority.

We should have the equation
X, +X... +X 1.=°4782, (5)

[381] the general expression for α΄, being given by (3) after
making therein z=12. In the year 1831 the law, having
received alteration, required a majority of at least four
persons for condemnation, and the number of persons tried
for crimes against the person during that year being 2046,
and the number condemned 743, the probability of the con-

743,
2046’

ability measured by the fraction or ‘4782, that an

demnation of an individual by the above majority was

or *3631. Hence we should have
Met Ag te ty = 3038: (6)
Assuming that the values of & and x were the same for
the year 1831 as for the previous six years, the two equa-
tions (5) and (6) enable us to determine approximately their
values. Poisson thus found,

k='5354, x= 6786.
For crimes against property during the same periods, he
found by a similar analysis,
k="6744, X= "7771.
The solution of the system (5) (6) conducts in each case
to two values of £, and to two values of x, the one value in

; I
each pair being greater, and the other less, than = It was

oo

PROBABILITY OF JUDGMENTS 399

assumed, that in each case the larger value should be pre-
ferred, it being conceived more probable that a party
accused should be guilty than innocent, and more probable

that a juryman should form a correct than an erroneous

opinion upon the evidence.

5. The data employed by Poisson, especially those which
were furnished by the year 1831, are evidently too im-
perfect to permit us to attach much confidence to the above
determinations of + and £; and it is chiefly for the sake of
the method that they are here introduced. It would have
been possible to record during the six years, 1825-30, or
during any similar period, the number of condemnations
pronounced with each possible majority of voices. The
values of the several elements Y,, X,, .. X,., were there
no reasons of policy to forbid, might have been accurately
ascertained. Here then the conception of the general
problem, of which Poisson’s is a particular case, arises.
How shall we, from [382] this apparently supernumerary
system of data, determine the values of x and &? If the
hypothesis, adopted by Poisson and all other writers on the
subject, of the absolute independence of the events whose

_ probabilities are denoted by x and & be retained, we should

be led to form a system of five equations of the type (3),
and either select from these that particular pair of equations
which might appear to be most advantageous, or combine
together the equations of the system by the method of least
squares, There might exist a doubt as to whether the latter
method would be strictly applicable in such cases, especially
if the values of x and & afforded by different selected pairs
of the given equations were very different from each other.
M. Cournot has considered a somewhat similar problem, in
which, from the records of individual votes in a court con-
sisting of four judges, it is proposed to investigate the
separate probabilities of a correct verdict from each judge.
For the determination of the elements 7, 2’, x”, τὺ he
obtains eight equations, which he divides into two sets of
four equations, and he remarks, that should any consider-

400 THE LAWS OF THOUGHT

able discrepancy exist between the values of 4, 2’, x”, x’,
determined from those sets, it might be regarded as an
indication that the hypothesis of the independence of the
opinions of the judges was, in the particular case, unten-
able. The principle of this mode of investigation has been
adverted to in (XVIII. 4).

6. I proceed to apply to the class of problems above indi-
cated, the method of this treatise, and shall inquire, first,
whether the records of courts and deliberative assemblies,
alone, can furnish any information respecting the proba-
bilities of correct judgment for their individual members,
and, it appearing that they cannot, secondly, what kind
and amount of necessary hypothesis will best comport with
the actual data.

PROPOSITION |

From the mere records of the dectsions of a court or delibera-
tive assembly, it ts not possible to deduce any definite conclu-
ston respecting the correctness of the individual judgments of
ets members.

Though this Proposition may appear to express but the
conviction of unassisted good sense, it will not be without
interest to show that it admits of rigorous demonstration.

[383] Let us suppose the case of a deliberative assembly
consisting of 2. members, no hypothesis whatever being
made respecting the dependence or independence of their
judgments. Let the logical symbols αὶ, #, .. +, be em-
ployed according to the following definition, viz.: Let the
generic symbol #, denote that event which consists in the
uttering of a correct opinion by the z” member, A,, of the
court. We shall consider the values of Prob. ΖΦ, Prob.
ae τοῦ: τ as the gucstfa of a proplem. the. ex-
pression of whose possible data we must in the next place
investigate,

Now those data are the probabilities of events capable of
being expressed by definite logical functions of the symbols

ee δ μὰ

Pe Eee ΩΝ ee ee ee ee ““

PROBABILITY OF JUDGMENTS 401

Εν, %,,.-.%,. Let X,, X,,.. X,, represent the functions
in question, and let the actual system of data be

Prob. X;=a,, Prob. X,=a,, Prob. X,,=am.-

Then from the very nature of the case it may be shown that
X,, X,.. X, are functions which remain unchanged if
@,, %,,-. #, are therein changed into I—z,, I—4,,.. 1—2,
respectively. Thus, if it were recorded that in a certain
proportion of instances the votes given were unanimous,
the event whose probability, supposing the instances suf-
ficiently numerous, is thence determined, is expressed by
the logical function

χὰ oO Oe op) Crater) Eas mad @ ear 20)
a function which satisfies the above condition. Again, let
it be recorded, that in a certain proportion of instances, the
vote of an individual, suppose A,, differs from that of all
the other members of the court. The event, whose proba-
bility is thus given, will be expressed by the function

BRE ey) eM eel Ly ae Os

also satisfying the above conditions. Thus, as agreement

Γ᾽ in opinion may be an agreement in either truth or error ;

eG

and as, when opinions are divided, either party may be
right or wrong; it is manifest that the expression of any
particular state, whether of agreement or difference of senti-
ment in the assembly, will depend upon a logical function
of the symbols x,, %, . . %,, [384] which similarly involves
the privative symbols 1—+,, 1-43, .. 1—%,. But in the
records of assemblies, it is not presumed to declare which
set of opinions is right or wrong. Hence the functions
X,, X,, .. X,, must be solely of the kind above described.

7. Now in proceeding, according to the general method,
to determine the value of Prob. x,, we should first equate
the functions X,, .. X,, to a new set of symbols 4, . . Z,,.
From the equations

X,=4; Xo = boy etd Xm = bn

thus formed, we should eliminate the symbols %, 43, . .ὄ ¥,,
26

402 THE LAWS OF THOUGHT

and then determine +, as a developed logical function of the

symbols ¢,, ¢,, . . Z,, expressive of events whose probabilities

are given. Let the result of the above elimination be
Ex,+ £1 -x,)=0; Gy

anda ΑΕ being functions of-2,, 2; .. 722. then
E' |

if eee ν 2
if Fee 3 ( )
Now the functions Y,, X,,.. X, are symmetrical with
reference to the symbols 4, .. 4, and 1—4,,..1-%,. It

is evident, therefore, that in the equation 4’ must be identical
Wit &. Hence.(2) gives

and it is evident, that the only coefficients which can appear
in the development of the second member of the above

: fe) I
equation are 5 and = The former will present itself when-

ever the valiles assioned to 4), .. 2.,.in. détermining the
coefficient of a constituent, are such as to make E=o0, the
latter, or an equivalent result, in every other case. Hence
we may represent the development under the form

τς}, (3)

Cand D being constituents, or aggregates of constituents,
OC the Πρ 2 |
[385] Passing then from Logic to Algebra, we have

Ῥτοῦ: x, = m6
the function V of the general Rule (XVII. 17) reducing in
the present case to C. The value of Prob. ἢ is therefore
wholly arbitrary, if we except the condition that it must not
transcend the limitso andi. The individual values of Prob.
%,,.. Prob. x,, are in like manner arbitrary. It does not
hence follow, that these arbitrary values are not connected
with each other by necessary conditions dependent upon

PROBABILITY OF JUDGMENTS 403

the data. The investigation of such conditions would, how-
ever, properly fall under the methods of Chapter XIX.

If, reverting to the final logical equation, we seek the in-
terpretation of c, we obtain but a restatement of the original
problem. For since Cand D together include all possible
constituents of 7,, 4,,. . Z,,, we have

Cpt;
: : . ες :
and since D is affected by the coefficient a: it is evident that

On substituting therein for ¢,, 7,,... ¢,, their expressions in
eric Ol 2. 4,2. 4, we should have D—o; Hencethe
same substitution would give C=1. Now by the rule, c is
the probability that if the event denoted by C take place,
the event τὸ will take place... Hence.C being equal to. 1,
and, therefore, embracing all possible contingencies, c must
be interpreted as the absolute probability of the occurrence
of the event σι.

It may be interesting to determine in a particular case
the actual form of the final logical equation. Suppose, then,
that the elements from which the data are derived are the
records of events distinct and mutually exclusive. For in-
stance, let the numerical data @,, a,,..4@,, be the respec-
tive probabilities of distinct and definite majorities. Then
the logical functions X,, X,,.. X,, being mutually ex-
clusive, must satisfy the conditions

Ae Mo = Oyu An Oh, Ne Os OLE.

Whence we have,
tly=0, tytn=0o, &c.

[386] Under these circumstances it may easily be shown,
that the developed logical value of x, will be

wy = (Az pa een eee)
: Ι
+ constituents whose coefficients are =

In the above equation ἢ, stands for 1—¢,, &c.
These investigations are equally applicable to the case in

404 THE LAWS OF THOUGAT

which the probabilities of the verdicts of a jury, so far as
agreement and disagreement of opinion are concerned, form
the data of a problem. Let the logical symbol w denote
that event or state of things which consists in the guilt of
the accused person. Then the functions X,, X,,.. X,, of
the present problem are such, that no change would therein
ensue from simultaneously converting w, 4,, 4%, ..%, into
W, t,, 4, . . #, respectively. Hence the final logical value of
w, as well as those of +,, 2%, . . %, will be exhibited under
the same form (3), and a like general conclusion thence
deduced.

It is therefore established, that from mere statistical
documents nothing can be inferred respecting either the in-
dividual correctness of opinion of a judge or counsellor, the
guilt of an individual, or the merits of a disputed question.
If the determination of such elements as the above can be
reduced within the province of science at all, it must be by
virtue either of some assumed criterion of truth furnishing
us with new data, or of some hypothesis relative to the
connexion or the independence of individual judgments,
which may warrant a new form of the investigation. In the
examination of the results of different hypotheses, the
following general Proposition will be of importance.

PROPOSITION II

8. Given the probabilities of the n simple events x1, . . Δ,» 2,»
725. -—
ΓΕΘ XG bLODs 4, τ POD 7s (1)

also the probabilities of the m—1 compound events X,, Xo,
- Ay, UES. τ.

Prob αὐ Rob, ν᾽ --ἰἰ περ paar; (2)

[387] the latter events X,, .. X,,, being distinct and
mutually exclusive ; required the probability of any other com-
pound event X.

In this Proposition it is supposed, that X,, X,, . . X.

m—1)

as well as X, are functions of the symbols 21, 4%, . - %, alone.

PROBABILITY OF JUDGMENTS 405
Moreover, the events X,, τ΄ Α΄ being mutually ex-
clusive, we have

AX X_=0, . . XXy1=0, X,X,=0, &c.; (3)

the product of any two members of the system vanishing.
Now assume

X,=h, Xn—1=t -y X=, (4)
Then ¢ must be determined as a logical function of x,, . . ,,
Pee ee
Now by (3),
bi,=0, Lbn1=0, ff,=0, &c.; (5)
811 binary products οἵ ¢,, . . Ζ,.., vanishing. The developed

expression for ¢ can, therefore, only involve in the list of
constituents which have 1, 0, or - for their coefficients, such

as contain some one of the following factors, viz. :—

ra Ou ms ee, ae 55 ΖΞ oo A yous PS Ε1 (6)

ἢ, standing for 1—Z,, &c. It remains to assign that portion
of each constituent which involves the symbols 7, .. %,; to-
gether with the corresponding coefficients.

Since X,=7, (2 being any integer between 1 and m-—1I in-
clusive), it is evident that

Xf, or Zn—1 = 9,

from the very constitution of the functions. Any constituent
included in the first member of the above equation would,

I :
therefore, have a for its coefficient.

Now let
Xm=i-X, Ἔτι τ om—1 > (7)

and it is evident that such constituents as involve ὦ... Z,_,,
O
as a factor, and yet have coefficients of the form 1, 0, or =

must be [388] included in the expression

ΧΟ er.

Now X,, may be resolved into two portions, viz., XX, and

406 THE LAWS OF THOUGHT

(1—X)YX,,, the former being the sum of those constituents
of X,, which are found in XY, the latter of those which are
not found in X. ΤῈ is evident that in the developed expres-
sion of ¢, which is equivalent to Y, the coefficients of the
constituents in the former portion XY, will be 1, while those
of the latter portion (1 —Y)X,, will beo. Hence the elements
we have now considered will contribute to the development
of 2 the terms

DOG te τον Xe Fe

Again, since X,=2,, while X,¢,=2,t,=0, &c., it is evi-

dent that the only constituents involving. το ¢.., asa

: : O :
factor which have coefficients of the form tO, OF a will be

included in the expression

GUNES Meta i mg

and reasoning as before, we see that this will contribute to
the development of ¢ the terms

2G. GF AE ee lm ΕΟ ΕΘ Δ ΧΙ; Chey 2

m—1°*

Proceeding thus with the remaining terms of (6), we de-
duce for the final expression of Ζ,

ee οὐχ τ te pe
τοι AGN Fy ο( me) Ale te re. (8)
+ terms whose coefficients are ὦ.
O°

In this expression it is to be noted that ΧΑ, denotes the
sum of those constituents which are common to Y and De Ἰἴῶολ
that sum being actually given by multiplying Y and X,, to-
gether, according to the rules of the calculus of Logic.

In passing from Logic to Algebra, we shall represent by
(X-X,,,) what the above product becomes, when, after effecting
the multiplication, or selecting the common constituents, we
give to the symbols 4, .. #,, a quantitative meaning.

[389] With this understanding we shall have, by the
general Rule (XVII. 17),

ΡΣ ἘΠ ΤΝ

PROBABILITY OF JUDGMENTS 407
Prob. ¢
COC το 10 6 on ae ee
a » (9)
ee et NGL ey eer ee ee tig)
mnence-the relations determining τὶ 24/2)... 4.0 will

be of the following type (ὦ varying from 1 to 2),

(x: Xn)A, 0 fee ea Fe, Gp a os bet 5.288 + (4; Xm—1)2, ἐς ote aa
C;
_ hh rae pe es Χο νὴ iris ἐπ τεῦ Ἐν V. (41)

a, Am -1

From the above system we shall next eliminate the sym-

DOs hk... foe.
We have

δ: ay, V z a An-1 V

hea . . gs Age OG 3 Ly oe SS 3 “1. = ΣῈ ; e (12)
m—

Substituting these values in (10), we find
ΖΞ ΧΕ oe En—1+a,V ir) + An V.

Pience i Fe pi Sol ΕΞ ἤν
Now let a= 1-4, < —ay1, (13)
then we have
at ina (14)

Now reducing, by means of (12) and (14), the equation
(9), and the equation formed by equating the first line of (11)
to the symbol VY; writing also Prob, X for Prob. 2, we have

a (XX) | a(XX9) | : μα XXm) | Ι
Prob. X = τ X, (15)
a,(x;X)) a(x; Xp) : 4 lH Xm) _ o (16)
pay ππτ Xn

wherein X,, and a,, are given by (7) and (13).

[390] These equations involve the direct solution of the
problem under consideration. In (16) we have the type of
m equations (formed by giving to Zine yalles 1, 2... 9.0%
successively), from which the Wallies Olt, 4a, εν

408 THE ΕΝ OF Tf HOCGH YE.

be found, and those values substituted in (15) give the value
of Prob. X as a function of the constants ὧς, εἰ, &c.

One conclusion deserving of notice, which is deducible from
the above solution, is, that if the probabilities of the com-
pound events X,, . . X,,_,, are the same as they would be
were the events 7,, . . τὸ entirely independent, and with
given probabilities c,, . . c,, then the probability of the
event X will be the same as if calculated upon the same
hypothesis of the absolute independence of the events ἜΝ

. ¥,, For upon the hypothesis supposed, the assumption
4,=¢, %,=C¢,, inthe quantitative system would give AE Ξεαι,
X,=4,, Whence (15) and (16) would give

Brob: = (Υχλ 4 (0X) x.) (17)
(x;,X,)+(4Xq) ... + (0: Xp) = eG. (18)

But since. Y,+X,.. +_X,,=1,it isevident that thesecond
member of (17) will be formed by taking all the constituents
that are contained in Α΄, and giving them an algebraic signi-
ficance. And a similar remark applies to (18). Whence
those equations respectively give

Prob. X (logical) =X (algebraic),
Xx, = ¢;-
Wherefore, if Y=¢(%,, %, . . x,), we have
ΕΟ οὐ ee),

which is the result in question.

Hence too it would follow, that if the Quantities 9.0
were indeterminate, and no hypothesis were made as to the
possession of a mean common value, the system (15) (16)
would be satisfied by giving to those quantities any such
values, 41, %, . . #,, as would satisfy the equations

X, =a, er ae PG as SR Aa,
supposing the value of the element a, like the values of
a, . . @,_,, to be given by experience.

[391] 9. Before applying the general solution (15). (16);.t0
the question of the probability of judgments, it will be con-

a
eat
ἕξ

PROBABILITY OF JUDGMENTS 409

venient to make the following transformation. Let the
data be
Ny =O... By =ey,

Γοῦ <i 4, 2. Prob. αὶ . a

Ans 3

and let it be required to determine Prob. X,,., the unknown
value of which we will represent by @,,,. Then in (15) and
(16) we must change

x mito X45. Prob. <x into...
Nay into ues; Am—1 into an—2
Χο into 2 m—1 a AG an into Am—1 Ἢ Qn >

with these transformations, and observing that (τ, OX) =0:

εἰ except ‘when r=m— I, and that it is then equal to _Y may tHe

equations (15) (16) give
ine 1+ Gn) Xin ik

am — 1s Dear Exe ? (19)
af a,(x;,X}) ee Ans (ὦ: a 2) 4 (Gm lay τ Am); ae tal a x Xm) (20)
Oe Χο τος ἀν a De "ΟΣ

Now from (19) we find
Ae 1 eo EEX. Sta

’
am 1 Qn an 1 τς LP

by virtue of which the last term of (20) may be reduced to
the form

Ay XAG 4) 4 GA.) :
Χο v.68

With these reductions the system (17) and (18) may be re-
placed by the following symmetrical one, viz. :

Gar ae Xin

Am-1 an : oe
ας (Gala XG) Am Xi Xm) _
τ a as gee Cie (22)

These equations, in connexion with (7) and (13), enable us
ΠΕ 102] dcterimine 2) as a function οἱ ἐν δ αἰ δ.
the numerical data supposed to be furnished by experience.
We now proceed to their application.

410 THE LAWS OF THOUGHT

PROPOSITION III

10. Given any system of probabilities drawn from recorded
instances of unanimity, or of assigned numerical majority in
the decisions of a deliberate assembly ; required, upon a certain
determinate hypothesis, the mean probability of correct judg-
ment for a member of the assembly.

In what way the probabilities of unanimous decision and of
specific numerical majorities may be determined from ex-
perience, has been intimated in a former part of this chapter.
Adopting the notation of Prop. I. we shall represent the events
whose probabilities are given by the functions. X,, X,, . . Χο
It has appeared from the very nature of the case that these
events are mutually exclusive, and that the functions by which
they are represented are symmetrical with reference to the
symbols 4,, #, . . %,. Those symbols we continue to use in
the same sense as in Prop. I., viz., by 2, we understand that
event which consists in the formation of a correct opinion by
the z’ member of the assembly.

Now the immediate data of experience are—

Prob. X,=a,, Prob. X,=a,, .. Prob. Xn_3=@n_a; (1)
ΡΟ i s 4. (2)

Ay. ot, , being functions of the locical symbols 4,, . τ to
the probabilities of the events denoted by which we shall
assign the intermediate value c. Thus we shall have

Prob. x, = Prob. x, . . =Prob. x,=<¢. (3)

Now it has been seen, Prop. I., that the immediate data
(1) (2), unassisted by any hypothesis, merely conduct us to
a restatement of the problem. On the other hand, it is
manifest that if, adopting the methods of Laplace and
Poisson, we employ the system (3) alone as the data for the
application of the method of this work, finally comparing
the results obtained with the experimental system (1) (2), we
are relying wholly upon a doubtful hypothesis,—the inde-
pendence of individual judgments. But [393] though we

PROBABILITY OF JUDGMENTS 411

ought not wholly to rely upon this hypothesis, we cannot
wholly dispense with it, or with some equivalent substitute.
Let us then examine the consequences of a 4mzted indepen-
dence of the individual judgments ; the conditions of limita-
tion being furnished by the apparently superfluous data.
From the system (1) (3) let us, by the method of this work,
determine Prob. X,,_,, and, comparing the result with (2),
determine c. Even here an arbitrary power of selection is
claimed. But it is manifest from Prop. I. that something
of this kind is unavoidable, if we would obtain a definite
solution at all. As tothe principle of selection, I apprehend
that the equation (2) reserved for final comparison should be
that which, from the magnitude of its numerical element
@,-1, is esteemed the most important of the primary series
furnished by experience.

Now, from the mutually exclusive character of the events
denoted by the functions X,, X,,.. X,_1, the concluding
equations of the previous proposition become applicable. On
account of the symmetry of the same functions, and the re-
duction of the system of values denoted by ¢; to a single
value c, the equations represented by (22) become identical,
the values of +,, 25» . - 2, become equal, and may be replaced
by a single value x, and we have simply,

care ῳ
πα, ee ie + Slee) we, (5)
The following is the nature of the solution thus indicated :
The functions X,, .. X,,-1, and the values a, .. @,_,
being given in the data, we have first,
Ne LX) ag ee
Ga. κα

From each of the functions X,, X,,.-. X,, thus given or
determined, we must select those constituents which contain
a particular symbol, as +4, for a factor. This will determine
the functions (#X,), (¢X,), &c., and then in all the functions

412 THE LAWS OF THOUGHT

we must change 7,, 22», - - 2, individually tox. Or we may
regard any [394] algebraic function X,in the system (4)
(5) as expressing the probability of the event denoted by
the logical function .Y,, on the supposition that the logical
symbols #,, %,, . . #, denote independent events whose
common probability is % On the same supposition (#X;,)
would denote the probability of the concurrence of any
Marticuiar event of the series 2,, 4,,° . 2% with 4; πὸ
forms of X,, («X,), &c. being determined, the equation (4)
gives the value of x, and this, substituted in (5), determines
the value of the element c required. Of the two values
which its solution will offer, one being greater, and the
other less, than 4, the greater one must be chosen, when-
soever, upon general considerations, it is thought more
probable that a member of the assembly will judge correctly,
than that he will judge incorrectly.

Here then, upon the assumed principle that the largest of
the values a,,_, shall be reserved for final comparison in the
equation (2), we possess a definite solution of the problem
proposed, And the same form of solution remains applicable
should any other equation of the system, upon any other
ground, as that of superior accuracy, be similarly reserved
in the place of:(2),

11. Let us examine to what extent the above reserva-
tion has influenced the final solution. It is evident that the
equation (5) is quite independent of the choice in question.
So is likewise the second member of (4). Had we reserved
the function .Y,, instead of X the equation for the de-

m—-1)

termination of « would have been

Ay δες

Die Gu

(6)

but the value of # thence determined would still have to be
substituted in the same final equation (5). We know that
were the events %,, 4%, . Ὁ. really independent, the equa-
tions (4) (6), and all others of which they are types, would
prove equivalent, and that the value of x furnished by any

ee ee ee Ὁ

i ie

CASES

PROBABILITY OF JUDGMENTS 413

one of them would be the true value of c. This affords a
means of verifying (5). For if that equation be correct, it
ought, under the above circumstances, to be satisfied by the
assumption c=. In other words, the equation

ΠΗ ΦΧ a(xX,) ἀκ τ
a ae Ps an ὁ fe es ams ἘΞ (7)

[395] ought, on solution, to give the same value of x as the
equation (4) or (6). Now this will be the case. For since,
by hypothesis,

ἜΣ Xe

a, a, a

?

we have, by a known theorem,

X,_X; eg te ee

: : τ
ay Ay rs i Ages) te On

Hence (7) becomes on substituting a, for X,, &c.
(xX) +(#X,) .. τ =x

a mere identity.

Whenever, therefore, the events +,, +, . . %, are really
independent, the system (4) (5) is a correct one, and is in-
dependent of the arbitrariness of the first step of the process
by which it was obtained. When the said events are not in-
dependent, the final system of equations will possess, leaving
in abeyance the principle of selection above stated, an arbi-
trary element. But from the persistent form of the equation
(5) it may be inferred that the solution is arbitrary in a less
degree than the solutions to which the hypothesis of the
absolute independence of the individual judgments would con-
duct us. The discussion of the limits of the value of c, as
dependent upon the limits of the value of 4, would determine
such points.

These considerations suggest to us the question whether
the equation (7), which is symmetrical with reference to the
functions Y,, Y,, . . X,,, free from any arbitrary elements,
and rigorously exact when the events +,,%,, . . ¥,are really
independent, might not be accepted as a mean general solu-

414 THE LAWS: OF THOUGHT

tion of the problem. The proper mode of determining this
point would, I conceive, be to ascertain whether the value
of « which it would afford would, in general, fall within the
limits of the value of c, as determined by the systems of
equations of which the system (4) (5) presents the type.
It seems probable that under ordinary circumstances this
would be the case. Independently of such considerations,
however, we may regard (7) as itself the expression of a
certain principle of solution, viz., that regarding X,, X,,

. X,, as exclusive causes of the event whose probability is
x, we accept the [396] probabilities of those causes a, . . a,,
from experience, but form the conditional probabilities of
the event as dependent upon such causes,

ee, or &ec. (XVII. Prop. I.)
on the hypothesis of the independence of individual judg-
ments, and so deduce the equation (7). I conceive this,
however, to be a less rigorous, though possibly, in practice
a more convenient mode of procedure than that adopted in
the general solution.

12. It now only remains to assign the particular forms
which the algebraic functions X,, (~X,), &c. in the above
equations assume when the logical function Α΄; represents
that event which consists in 7, members of the assembly voting
one way, and z—yv members the other way. It is evident
that in this case the algebraic function Y, expresses what the
probability of the supposed event would be were the events
41, %,, . . x, independent, and their common probability
measured by x. Hence we should have, by Art. 3,

Ἐπ OE rns

Under the same circumstances (1 Δ ὃ would represent the pro-
bability of the compound event, which consists ina particular
member of the assembly forming a correct judgment, con-
jointly with the general state of voting recorded above. It
would, therefore, be the probability that a particular member

PROBABILITY OF JUDGMENTS 415

votes correctly, while of the remaining 2—1 members, r—1
᾿ vote correctly ; or that the same member votes correctly,
while of the remaining »—1 members 7, vote incorrectly.
Hence

ἐλ). Ξ τ)(γι -- 2)... τω ὡν ΤΕ λα Dee) (7-7) enor

PROPOSITION IV

13. Grven any system of probabilities drawn from recorded
instances of unanimity, or of assigned numerical majority in
the decisions of a criminal court of justice, required upon
hypotheses stmtlar to those of the last proposition, the mean
probability c of [397] correct judgment for a member of the
court, and the general probability k of guilt in an accused
person.

The solution of this problem differs in but a slight degree
from that of the last, and may be referred to the same
general formule, (4) and (5), or (7). It is to be observed,
that as there are two elements, c and ᾧ, to be determined,
tL is Necessary. to reserve two ol the functions. x, τ, Πὰς
let us suppose Y, and YX,,_,, for final comparison, employing
either the remaining 772 — 3 functions in the expression of the
ata or tiie: two “respective. Seto: A) ον τ and
as πα΄ in either case it 45 supposea that there
must be at least two original independent data. If the
equation (7) be alone employed, it would in the present
instance furnish two equations, which may thus be written :

Gi we “ας 3 fe Wan) ΕΣ
"ἢ + Sa eae x; (1)
a(AX,) | ao(kX 9) a An(AXm) _ 2. 2
ἘΠ an ae (2)

These equations are to be employed in the following manner :
—Let x,, 4, .. 4%, represent those events which consist in
the formation of a correct opinion by the members of the
court respectively. Let also w represent that event which

416 THE LAWS OF THOUGHT

consists in the guilt of the accused member. By the aid
of these symbols we can logically express the functions
Αἱ, X,,.. X,,,, whose probabilities are given, as also the
function Y,,. Then from the function , select those con-
stituents which contain, as a factor, any particular symbol
of the set +,, *%,..%,, and also those constituents which
contain as a factor w. In both results change 4,, 4%, .. %,
severally into x, and w into k. The above results will
give (7X,) and (£X,). Effecting the same transformations
throughout, the system (1) (2) will, upon the particular
hypothesis involved, determine + and &.

14. We may collect from the above investigations the
following facts and conclusions :

_ Ist. That from the mere records of agreement and dis-
agreement in the opinions of any body of men, no definite
numerical conclusions can be drawn respecting either the
probability of cor-[398]rect judgment in an individual
member of the body, or the merit of the questions submitted
to its consideration.

2nd. That such conclusions may be drawn upon various
distinct hypotheses, as—rst, Upon the usual hypothesis of
the absolute independence of individual judgments ; 2ndly,
upon certain definite modifications of that hypothesis war-
ranted by the actual data; 3rdly, upon a distinct principle
of solution suggested by the appearance of a common form
in the solutions obtained by the modifications above ad-
verted to. |

Lastly. That whatever of doubt may attach to the final
results, rests not upon the imperfection of the method, which
adapts itself equally to all hypotheses, but upon the un-
certainty of the hypotheses themselves.

It seems, however, probable that with even the widest
limits of hypothesis, consistent with the taking into account
of afl the data of experience, the deviation of the results
obtained would be but slight, and that their mean values
might be determined with great confidence by the methods
of Prop. III. Of those methods I should be disposed to

PROBABILITY OF JUDGMENTS 417

give the preference to the first. Such a principle of mean
solution having been agreed upon, other considerations seem
to indicate that the values of ¢ and & for tribunals and
assemblies possessing a definite constitution, and governed
in their deliberations by fixed rules, would remain nearly
constant, subject, however, to a small secular variation,
dependent upon the progress of knowledge and of justice
among mankind, There exist at present few, if any, data
proper for their determination.

27

[399] CHAPTER XXII

ON THE NATURE OF SCIENCE, AND THE CONSTITUTION OF
THE INTELLECT

1. WHAT I mean by the constitution of a system is the
aggregate of those causes and tendencies which produce its
observed character, when operating, without interference,
under those conditions to which the system is conceived to
be adapted. Our judgment of such adaptation must be
founded upon a study of the circumstances in which the
system attains its freest action, produces its most harmonious
results, or fulfils in some other way the apparent design of
its construction. There are cases in which we know dis-
tinctly the causes upon which the operation of a system
depends, as well as its conditions and its end. This is the
most perfect kind of knowledge relatively to the subject
under consideration. There are also cases in which we know
only imperfectly or partially the causes which are at work,
but are able, nevertheless, to determine to some extent the
laws of their action, and, beyond this, to discover general
tendencies, and to infer ulterior purpose. It has thus, I
think rightly, been concluded that there is a moral faculty
in our nature, not because we can understand the special in-
struments by which it works, as we connect the organ with
the faculty of sight, nor upon the ground that men agree
in the adoption of universal rules of conduct ; but because,
while, in some form or other, the sentiment of moral appro-
bation or disapprobation manifests itself in all, it tends,
wherever human progress is observable, wherever society is
not either stationary or hastening to decay, to attach itself
to certain classes of actions, consentaneously, and after a

manner indicative both of permanency and of law. Always
418

CONSTITUTION OF THE INTELLECT 10

and everywhere the manifestation of Order affords a pre-
sumption, not measurable indeed, but real (XX. 20)" ot
the fulfilment of an end or purpose, and the existence of a
ground of orderly causation.

[400] 2. The particular question of the constitution of the
intellect has, it is almost needless to say, attracted the
efforts of speculative ingenuity in every age. For it not
only addresses itself to that desire of knowledge which the
greatest masters of ancient thought believed to be innate in
our species, but it adds to the ordinary strength of this
motive the inducement of a human and personal interest.
A genuine devotion to truth is, indeed, seldom partial in its
aims, but while it prompts to expatiate over the fair fields of
outward observation, forbids to neglect the study of our own
faculties. Even in ages the most devoted to material in-
terests, some portion of the current of thought has been
reflected inwards, and the desire to comprehend that by
which all else is comprehended has only been baffled in
order to be renewed.

It is probable that this pertenacity of effort would not have
been maintained among sincere inquirers after truth, had
the conviction been general that such speculations are hope-
lessly barren. We may conceive that it has been felt that
if something of error and uncertainty, always incidental to
a state of partial information, must ever be attached to the
results of such inquiries, a residue of positive knowledge
may yet remain; that the contradictions which are met
with are more often verbal than real; above all, that even
probable conclusions derive here an interest and a value from
their subject, which render them not unworthy to claim
regard beside the more definite and more splendid results of
physical science. Such considerations seem to be perfectly
legitimate. Insoluble as many of the problems connected
with the inquiry into the nature and constitution of the
mind must be presumed to be, there are not wanting others
upon which a limited but not doubtful knowledge, others
upon which the conclusions of a highly probable analogy,

420 THE LAWS. OF THOUGH!

are attainable. As the realms of day and night are not
strictly conterminous, but are separated by a crepuscular
zone, through which the light of the one fades gradually off
into the darkness of the other, so it may be said that
every region of positive knowledge lies surrounded by a
debateable and speculative territory, over which it in some
degree extends its influence and its light. Thus there may
be questions relating to the constitution of the intellect
which, though they do not admit, in the present state of
know-[40I]ledge, of an absolute decision, may receive so
much of reflected information as to render their probable
solution not difficult ; and there may also be questions re-
lating to the nature of science, and even to particular truths
and doctrines of science, upon which they who accept the
general principles of this work cannot but be led to entertain
positive opinions, differing, it may be, from those which
are usually received in the present day.* In what follows
I shall recapitulate some of the more definite conclusions
established in the former parts of this treatise, and shall
then indicate one or two trains of thought, connected with
the general objects above adverted to, which they seem to
me calculated to suggest.

3. Among those conclusions, relating to the intellectual
constitution, which may be considered as belonging to the
realm of positive knowledge, we may reckon the scientific
laws of thought and reasoning, which have formed the basis
of the general methods of this treatise, together with the
principles, Chap. V., by which their application has been
determined. The resolution of the domain of thought into
two spheres, distinct but coexistent (IV. XI.); the sub-
jection of the intellectual operations within those spheres

* The following illustration may suffice :—

It is maintained by some of the highest modern authorities in grammar that
conjunctions connect propositions only. Now, without inquiring directly whether
this opinion is sound or not, it is obvious that it cannot consistently be held by
any who admit the scientific principles of this treatise ; for to such it would seem
to involve a denial, either, Ist, of the possibility of performing, or 2ndly, of the
possibility of express¢ng, a mental operation, the laws of which, viewed in both

these relations, have been investigated and applied in the present work.—
(Latham on the English Language ; Sir John Stoddart’s Unzversal Grammar, &c.)

CONSTITUTION OF THE INTELLECT 421

to a common system of laws (XI.); the general mathe-
matical character of those laws, and their actual expression
(iI. I1I.); the extent of their affinity with the laws of
thought in the domain of number, and the point of their
divergence therefrom ; the dominant character of the two
limiting conceptions of universe and eternity among all the
subjects of thought with which Logic is concerned ; the
relation of those conceptions to the fundamental conception
of unity in the science of number,—these, with many
similar results, are not to be ranked as merely [402] pro-
bable or analogical conclusions, but are entitled to be re-
garded as truths of science. Whether they be termed meta-
physical or not, is a matter of indifference. The nature of
the evidence upon which they rest, though in kind distinct,
is not inferior in value to any which can be adduced in
support of the general truths of physical science.

Again, it is agreed that there is a certain order observable
in the progress of all the exacter forms of knowledge. The
study of every department of physical science begins with
observation, it advances by the collation of facts to a pre-
sumptive acquaintance with their connecting law, the validity
of such presumption it tests by new experiments so devised

as to augment, if the presumption be well founded, its

probability indefinitely ; and finally, the law of the pheno-
menon having been with sufficient confidence determined,
the investigation of causes, conducted by the due mixture
of hypothesis and deduction, crowns the inquiry. In this
advancing order of knowledge, the particular faculties and
laws whose nature has been considered in this work bear
their part. It is evident, therefore, that if we would im-
partially investigate either the nature of science, or the in-
tellectual constitution in its relation to science, no part of
the two series above presented ought to be regarded as
isolated. More especially ought those truths which stand in
any kind of supplemental relation to each other to be con-
sidered in their mutual bearing and connexion.

4. Thus the necessity of an experimental basis for all

422 THE LAWS OF THOUGHT

positive knowledge, viewed in connexion with.the existence
and the peculiar character of that system of mental laws,
and principles, and operations, to which attention has been
directed, tends to throw light upon some important questions
by which the world of speculative thought is still in a great
measure divided. How, from the particular facts which
experience presents, do we arrive at the general propositions
of science? What is the nature of these propositions? Are
they solely the collections of experience, or does the mind
supply some connecting principle of its own? In a word,
what is the nature of scientific truth, and what are the
grounds of that confidence with which it claims to be
received ?

[403] That to such questions as the above, no single and
general answer can be given, must be evident. There are
cases in which they do not even need discussion. Instances
are familiar, in which general propositions merely express
per enumerationem simplicem, a fact established by actual
observation in all the cases to which the proposition applies.
The astronomer asserts upon this ground, that all the known
planets move from west to east round the sun. But there
are also cases in which general propositions are assumed
from observation of their truth in particular instances, and
extension of that truth to instances unobserved. No prin-
ciple of merely deductive reasoning can warrant such a pro-
cedure. When from a large number of observations on the
planet Mars, Kepler inferred that it revolved in an ellipse,
the conclusion was larger than his premises, or indeed than
_any premises which mere observation could give. What
other element, then, is necessary to give even a prospective
validity to such generalizations as this? It is the ability
inherent in our nature to appreciate Order, and the con-
current presumption, however founded, that the phenomena
of Nature are connected by a principle of Order. Without
these, the general truths of physical science could never have
been ascertained. Grant that the procedure thus established
can only conduct us to probable or to approximate results;

Se ee ee! eae a oe

CONSTITUTION OF THE INTELLECT _ 423

it only follows, that the larger number of the generalizations
of physical science possess but a probable or approximate
truth. The security of the tenure of knowledge consists in
this, that wheresoever such conclusions do truly represent
the constitution of Nature, our confidence in their truth
receives indefinite confirmation, and soon becomes undis-
tinguishable from certainty. The existence of that principle
above represented as the basis of inductive reasoning enables
us to solve the much disputed question as to the necessity
of general propositions in reasoning. The logician affirms,
that it is impossible to deduce any conclusion from particular
premises. Modern writers of high repute have contended,
that all reasoning is from particular to particular truths.
They instance, that in concluding from the possession of a
property by certain members of a class, its possession by
some other member, it is not necessary to establish the
intermediate general conclu-[404]sion which affirms its
possession by aé/ the members of the class incommon. Now
whether it is so or not, that principle of order or analogy
upon which the reasoning is conducted must either be stated
or apprehended as a general truth, to give validity to the
final conclusion. In this form, at least, the necessity of
general propositions as the basis of inference is confirmed,
—a necessity which, however, I conceive to be involved in
the very existence, and still more in the peculiar nature, of
those faculties whose laws have been investigated in this
work, For if the process of reasoning be carefully analyzed,
it will appear that abstraction is made of all peculiarities of
the individual to which the conclusion refers, and the atten-
tion confined to those properties by which its membership
of the class is defined.

5. But besides the general propositions which are derived
by induction from the collated facts of experience, there
exist others belonging to the domain of what is termed
necessary truth. Such are the general propositions of Arith-
metic, as well as the propositions expressing the laws of
thought upon which the general methods of this treatise are

424 THE LAWS OF THOUGHT

founded; and these propositions are not only capable of
being rigorously verified in particular instances, but are made
manifest in all their generality from the study of particular
instances. Again, there exist general propositions expressive
of necessary truths, but incapable, from the imperfection of
the senses, of being exactly verified. Some, if not all, of
the propositions of Geometry are of this nature ; but it is
not in the region of Geometry alone that such propositions
are found. The question concerning their nature and origin
is a very ancient one, and as it is more intimately connected
with the inquiry into the constitution of the intellect than
any other to which allusion has been made, it will not be
irrelevant to consider it here. Among the opinions which
have most widely prevailed upon the subject are the follow-
ing. It has been maintained, that propositions of the class
referred to exist in the mind independently of experience,
and that those conceptions which are the subjects of them
are the imprints of eternal archetypes. With such arche-
types, conceived, however, to possess a reality of which all
the objects of sense are but a faint shadow or dim suggestion,
Plato furnished his ideal world. It [405] has, on the other
hand, been variously contended, that the subjects of such
propositions are copies of individual objects of experience ;
that they are mere names; that they are individual objects
of experience themselves ; and that the propositions which
relate to them are, on account of the imperfection of those
objects, but partially true ; lastly, that they are intellectual
products formed by abstraction from the sensible perceptions
- of individual things, but so formed as to become, what the
individual things never can be, subjects of science, ze. sub-
jects concerning which exact and general propositions may
be affirmed. And there exist, perhaps, yet other views, in
some of which the sensible, in others the intellectual or ideal,
element predominates.

Now if the last of the views above adverted to be taken
(for it is not proposed to consider either the purely ideal or
the purely nominalist view) and if it be inquired what, in

Pe eS eT τ -

CONSTITUTION OF THE INTELLECT 425

the sense above stated, are the proper objects of science,
objects in relation to which its propositions are true without
any mixture of error, it is conceived that but one answer
can be given. It is, that neither do individual objects of
experience, nor with all probability do the mental images
which they suggest, possess any strict claim to this title.
It seems to be certain, that neither in nature nor in art do
we meet with anything absolutely agreeing with the geo-
metrical definition of a straight line, or of a triangle, or of
a circle, though the deviation therefrom may be inappreciable
by sense; and it may be conceived as at least doubtful,
whether we can form a perfect mental image, or conception,
with which the agreement shall be more exact. But it is
not doubtful that such conceptions, however imperfect, do
point to something beyond themselves, in the gradual ap-

proach towards which all imperfection tends to disappear.

Although the perfect triangle, or square, or circle, exists
not in nature, eludes all our powers of representative con-
ception, and is presented to us in thought only, as the limit
of an indefinite process of abstraction, yet, by a wonderful
faculty of the understanding, it may be made the subject of
propositions which are absolutely true. The domain of reason
is thus revealed to us as larger than that of imagination.
Should any, indeed, think that we are able to picture to
ourselves, with rigid accuracy, the scientific elements of form,
direction, mag-[406]nitude, &c., these things, as actually
conceived, will, in the view of such persons, be the proper
objects of science. But if, as seems to me the more just
opinion, an incurable imperfection attaches to all our attempts
to realize with precision these elements, then we can only
affirm, that the more external objects do approach in reality,
or the conceptions of fancy by abstraction, to certain /zmzt-
ing states, never, it may be, actually attained, the more do
the general propositions of science concerning those things
or conceptions approach to absolute truth, the actual devia-
tion therefrom tending to disappear. To some extent, the
same observations are applicable also to the physical sciences.

=

426 THE LAWS OF THOUGHT

What have been termed the ‘‘ fundamental ideas” of those
sciences as force, polarity, crystallization, &c.,* are neither,
as I conceive, intellectual products independent of experience,
nor mere copies of external things; but while, on the one
hand, they have a necessary antecedent in experience, on
the other hand they require for their formation the exercise
of the power of abstraction, in obedience to some general
faculty or disposition of our nature, which ever prompts us
to the research, and qualifies us for the appreciation, of
order.t Thus we study approximately the effects of gravi-
tation on the motions of the heavenly bodies, by a refer-
ence to the Aimzting supposition, that the planets are perfect
[407]spheres or spheroids. We determine approximately the
path of a ray of light through the atmosphere, by a process
in which abstraction is made of all disturbing influences of
temperature. And such is the order of procedure in all the
higher walks of human knowledge. Now what is remark-
able in connexion with these processes of the intellect is the
disposition, and the corresponding ability, to ascend from
the imperfect representations of sense and the diversities of
individual experience, to the perception of general, and it
may be of immutable truths. Wherever this disposition
and this ability unite, each series of connected facts in nature
may furnish the intimations of an order more exact than
that which it directly manifests. For it may serve as ground
and occasion for the exercise of those powers, whose office

* Whewell’s Philosophy of the Inductive Sciences, pp. 71, 775 213.

+ Of the idea of order it has been profoundly said, that it carries within itself its
own justification or its own control, the very trustworthiness of our faculties being
/ judged by the conformity of their results to an order which satisfies the reason.
““Lridée de Vordre a cela de singulier et d’eminent, qu’elle porte en elle méme sa
justification ou son contréle. Pour trouver si nos autres facultés nous trompent
ou nous ne trompent pas, nous examinons si les notions qu’elles nous donnent
s’enchainent on ne s’enchainent pas suivant un ordre qui satisfasse la raison,” —
Cournot, Zssaz sur les fondements de nos Connaissances, Admitting this principle
as the guide of those powers of abstraction which we undoubtedly possess, it seems
unphilosophical to assume that the fundamental ideas of the sciences are not deriv-
able from experience. Doubtless the capacities which have been given to us
for the comprehension of the actual world would avail us in a differently con-
stituted scene, if in some form or other the dominion of order was still maintained.
It is conceivable that in such a new theatre of speculation, the laws of the in-

tellectual procedure remaining the same, the fundamental ideas of the sciences
might be wholly different from those with which we are at present acquainted.

ah ita ae ier

EE OR ee ee ee Ee ee, ae

CONSTITUTION OF THE INTELLECT 427

=

it is to apprehend the general truths which are indeed ex-
emplified, but never with perfect fidelity, in a world of
changeful phanomena.

6. The truth that the ultimate laws of thought are mathe-
matical in their form, viewed in connexion with the fact of
the possibility of error, establishes a ground for some re-
markable conclusions. If we directed our attention to the
scientific truth alone, we might be led to infer an almost
exact parallelism between the intellectual operations and the
movements of external nature. Suppose any one conversant
with physical science, but unaccustomed to reflect upon the
nature of his own faculties, to have been informed, that it
had been proved, that the laws of those faculties were mathe-
matical; it is probable that after the first feelings of in-
credulity had subsided, the impression would arise, that the
order of thought must, therefore, be as necessary as that of
the material universe. We know that in the realm of
natural science, the absolute connexion between the initial
and final elements of a problem, exhibited in the mathe-
matical form, fitly symbolizes that physical necessity which
binds together effect and cause. The necessary sequence
of states and conditions in the inorganic world, and the
necessary connexion of premises and conclusion in the pro-
cesses of exact demonstration thereto applied, seem to be
co-ordinate. It may possibly be a question, to which of the
two series the primary application of the term ‘‘necessary ”
is due ; whether to the observed constancy of [408] Nature,
or to the indissoluble connexion of propositions in all valid
reasoning upon her works. Historically we should perhaps
give the preference to the former, philosophically to the
latter view. But the fact of the connexion is indisputable,
and the analogy to which it points is obvious.

Were, then, the laws of valid reasoning uniformly obeyed,
a very close parallelism would exist between the operations
of the intellect and those of external Nature. Subjection to
laws mathematical in their form and expression, even the
subjection of an absolute obedience, would stamp upon the

428 ΤΗΣ LAWS: OF THOUGHT

two series one common character. The reign of necessity
over the intellectual and the physical world would be alike
complete and universal.

But while the observation of external Nature testifies with
ever-strengthening evidence to the fact, that uniformity of
operation and unvarying obedience to appointed laws pre-
vail throughout her entire domain, the slightest attention to
the processes of the intellectual world reveals to us another
state of things. The mathematical laws of reasoning are,
properly speaking, the laws of rvzght reasoning only, and
their actual transgression is a perpetually recurring pheeno-
menon. Error, which has no place in the material system,
occupies a large one here. We must accept this as one of
those ultimate facts, the origin of which it lies beyond the
province of science to determine. We must admit that there
exist laws which even the rigour of their mathematical forms
does not preserve from violation. We must ascribe to them
an authority the essence of which does not consist in power,
a supremacy which the analogy of the inviolable order of
the natural world in no way assists us to comprehend.

As the distinction thus pointed out is veal, it remains un-
affected by any peculiarity in our views respecting other
portions of the mental constitution. If we regard the in-
tellect as free, and this is apparently the view most in
accordance with the general spirit of these speculations, its
freedom must be viewed as opposed to the dominion of
necessity, not to the existence of a certain just supremacy
of truth. The laws of correct inference may be violated, but
they do not the less truly erzs¢ on this account. Equally
do they remain unaffected in character and authority if the
hypothesis of necessity in its extreme form be [409] adopted.
Let it be granted that the laws of valid reasoning, such as
they are determined to be in this work, or, to speak more
generally, such as they would finally appear in the con-
clusions of an exhaustive analysis, form but a part of the
system of laws by which the actual processes of reasoning,
whether right or wrong, are governed. Let it be granted

Page ΠΤ ee ee

ae Se ee ee a ee

CONSTITUTION OF THE INTELLEC? 429

that if that system were known to us in its completeness,
we should perceive that the whole intellectual procedure was
necessary, even as the movements of the inorganic world are
necessary. And let it finally, as a consequence of this
hypothesis, be granted that the phenomena of incorrect
reasoning or error, wheresoever presented, are due to the
interference of other laws with those laws of which right
reasoning is the product. Still it would remain that there
exist among the intellectual laws a number marked out from
the rest by this special character, viz., that every movement
of the intellectual system which is accomplished solely under
their direction is v¢ght, that every interference therewith by
other laws is not interference only, but vzolation. It cannot
but be felt that this circumstance would give to the laws in
question a character of distinction and of predominance.
They would but the more evidently seem to indicate a final
purpose which is not always fulfilled, to possess an authority
inherent and just, but not always commanding obedience.
Now a little consideration will show that there is nothing
analogous to this in the government of the world by natural
law. The realm of inorganic Nature admits neither of pre-
ference nor of distinctions. We cannot separate any portion
of her laws from the rest, and pronounce them alone worthy
of obedience,—alone charged with the fulfilment of her
highest purpose. On the contrary, all her laws seem to
stand co-ordinate, and the larger our acquaintance with them,
the more necessary does their united action seem to the
harmony, and, so far as we can comprehend it, to the general
design of the system. How often the most signal departures
from apparent order in the inorranic world, such as the
perturbations of the planetary system, the interruption of the
process of crystallization by the intrusion of a foreign force,
and others of a like nature, either merge into the conception of
some more exalted scheme of order, or lose toa [410] more
attentive and instructed gaze their abnormal aspect, it is
needless to remark. One explanation only of these facts
can be given, viz., that the distinction between ¢vve and

430 THE LAWS OF THOUGHT

false, between correct and incorrect, exists in the processes
of the intellect, but not in the region of a physical necessity.
As we advance from the lower stages of organic being to the
higher grade of conscious intelligence, this contrast gradually
dawns upon us. Wherever the phenomena of life are mani-
fested, the dominion of rigid law in some degree yields to
that mysterious principle of activity. Thus, although the
structure of the animal tribes is conformable to certain
general types, yet are those types sometimes, perhaps, in
relation to the highest standards of beauty and proportion,
always, imperfectly realized. The two alternatives, between
which Art in the present day fluctuates, are the exact imita-
tion of individual forms, and the endeavour, by abstraction
from all such, to arrive at the conception of an ideal grace
and expression, never, it may be, perfectly manifested in
forms of earthly mould. Again, those teleological adapta-
tions by which, without the organic type being sacrificed,
species become fitted to new conditions or abodes, are but
slowly accomplished,—accomplished, however, not, appar-
ently, by the fateful power of external circumstances, but
by the calling forth of an energy from within. Life in all
its forms may thus be contrasted with the passive fixity of
inorganic nature. But inasmuch as the perfection of the
types in which it is corporeally manifested is in some measure
of an ideal character, inasmuch as we cannot precisely define
the highest suggested excellency of form and of adaptation,
the contrast is less marked here than that which exists
between the intellectual processes and those of the purely
_material world. For the definite and technical character of
the mathematical laws by which both are governed, places
in stronger light the fundamental difference between the kind
of authority which, in their capacity of government, they
respectively exercise.

7. There is yet another instance connected with the general
objects of this chapter, in which the collation of truths or
facts, drawn from different sources, suggests an instructive
train of reflection. It consists in the comparison of the laws

CONSTITUTION OF THE INTELLECT 431

of thought, in their scientific expression, with the actual
forms which physical [411] speculation in early ages, and
metaphysical speculation in allages, have tended to assume.
There are two illustrations of this remark, to which, in
particular, 1 wish to direct attention here.

Ist. It has been shown (III. 13) that there is a scientific
connexion between the conceptions of unity in Number, and
the universe in Logic. They occupy in their respective
systems the same relative place, and are subject to the same
formal laws. Now to the Greek mind, in that early stage
of activity,—a stage not less marked, perhaps not less
necessary, in the progression of the human intellect, than the
era of Bacon or of Newton,—when the creat problems of
Nature began to unfold themselves, while the means of
observation were as yet wanting, and its necessity not
understood, the terms ‘‘ Universe” and ‘*The One” seem
to have been regarded as almost identical. To assign the
nature of that unity of which all existence was thought to be
a manifestation, was the first aim of philosophy.* Thales
sought for this fundamental unity in water. Anaximenes
and Diogenes conceived it to be air. Hippasus of
Metapontum, and Heraclitus the Ephesian, pronounced that
it was fire. Less definite or less confident in his views,
Parmenides simply declared that all existing things were
One; Melissus that the Universe was infinite, unsusceptible
of change or motion, One, like to itself, and that motion
was not, but seemed to be.t In a spirit, which, to the
reflective mind of Aristotle, appeared sober when contrasted
with the rashness of previous speculation, Anaxagoras of
Clazomene, following, perhaps, the steps of his fellow-
citizen, Hermotimus, sought in Intelligence the cause of the
world and of its order.t The pantheistic tendency which

* See various passages in Aristotle’s Meéaphyszcs, Book I.

Τ᾿ Ἐδόκει δὲ αὐτῷ τὸ πᾶν ἄπειρον εἶναι, καὶ ἀναλλοίωτον, καὶ ἀκίνητον, καὶ ἕν, ὅμοιον
ἑαυτῷ καὶ πλῆρες. κινησίν τε μὴ εἶναι δοκεῖν δὲ elyou.—Diog. Laert. ix. cap. 4.

t Νοῦν δή τις εἰπὼν ἐνεῖναι, καθάπερ ἐν τοῖς ζῴοις, καὶ ἐν τῇ φύσει, τὸν αἴτιον
τοῦ κόσμον καὶ τῆς τάξεως πάσης οἷον νήφων ἐφάνη παρ᾽ εἰκῇ λέγοντας τοὺς

πρότερον. Φανερῶς μὲν οὖν ᾿Αναξαγόραν ἴσμεν ἁψάμενον τούτων τῶν λόγων, αἰτίαν
δ᾽ ἔχει πρότερον Ἑρμότιμος ὁ Κλαζομένιος εἰπεῖν. ---Αὐἴδί, 27ε1., I. 3.

432 THE LAWS OF THOUGHT

pervaded many of these speculations is manifest in the
language of Xenophanes, the founder of the Eleatic school,
who, ‘‘surveying the expanse of [412] heaven, declared
that the One was God.” * Perhaps there are few, if any,
of the forms in which unity can be conceived, in the abstract
as numerical or rational, in the concrete as a passive sub-
stance, or a central and living principle, of which we do not
meet with applications in these ancient doctrines. The
writings of Aristotle, to which 1 have chiefly referred,
abound with allusions of this nature, though of the
larger number of those who once addicted themselves to
such speculations, it is probable that the very names have
perished. Strange, but suggestive truth, that while Nature
in all but the aspect of the heavens must have appeared as
little else than a scene of unexplained disorder, while the
popular belief was distracted amid the multiplicity of its
gods,—the conception of a primal unity, if only ina rude,
material form, should have struck deepest root ; surviving in
many a thoughtful breast the chills of a lifelong disappoint-
ment, and an endless search ! +

2ndly. In equally intimate alliance with that law of
thought which is expressed by an equation of the second
degree, and which has been termed in this treatise the law
of duality, stands the tendency of ancient thought to those
forms of philosophical speculation which are known under
the name of dualism. The theory of Empedocles,t which

* Ἐενοφάνης δὲ... els τὸν ὅλον οὐρανὸν ἀποβλέψας, τὸ ἕν εἶναι φησι τὸν
θεόν. --- 70.

+ The following lines, preserved by Sextus Empiricus, and ascribed to Timon
the Sillograph, are not devoid of pathos :—

ὡς Kal ἐγὼν ὄφελον πυκινοῦ νόου ἀντιβολῆσαι
ἀμφοτερόβλεπτος (δολίῃ δ᾽ ὁδῷ ἐξεπατήθην,
πρεσβυγενὴς ἔ ἔτ ἐὼν) καὶ ἀναμφήριστος ἁ ἁπάσης
σκεπτοσύνης᾽ ὅππη γὰρ ἐμὸν νόον εἰρύσαιμι,
εἰς ἕν 7 αὐτὸ τε πᾶν ἀνέλνετο.

I quote them from Ritter, and venture to give the following version :

Be mine, to partial views no more confin’d
Or sceptic doubts, the truth illumin’d mind !
For, long deceiv’d, yet still on Truth intent,
Life’s waning years in wand’rings wild are spent.
Still restless thought the same high quest essays,
And still the One, the All, eludes my gaze.

~ Arist. AZed, I. 4, 6.

SS ee περλωδ ξὺν

ge eS ee Te ee ΡΨ

Ὶ
4
4
;

CONSTITUTION OF THE INTELLECT 433

explained the apparent contradictions of nature by referring
them to the two opposing principles [413] of ‘strife’ and
‘friendship ” ; and the theory of Leucippus, * which resolved
all existence into the elements of a plenum and a vacuum, are
of this nature. The famous comparison of the universe to a
lyre or a bow,t its ‘‘ recurrent harmony ” being the product
of opposite states of tension, betrays the same origin. In
the system of Pythagoras, which seems to have been a com-
bination of dualism with other elements derived from the
study of numbers, and of their relations, ten fundamental
antitheses are recognised : finite and infinite, even and odd,
unity and multitude, right and left, male and female, rest
and motion, straight and curved, light and darkness, good
and evil, the square and the oblong. In that of Alemzon
the same fundamental dualism is accepted, but without the
definite and numerical limitation with which it is connected
in the Pythagorean system. The grand development of
this idea is, however, met with in that ancient Manichzan
doctrine, which not only formed the basis of the religious
system of Persia, but spread widely through other regions
of the East, and became memorable in the history of the
Christian Church. The origin of dualism as a speculative
opinion, not yet connected with the personification of the
Evil Principle, but naturally succeeding those doctrines
which had assumed the primal unity of Nature, is thus
stated by Aristotle :—‘‘ Since there manifestly existed in
Nature things opposite to the good, and not only order and
beauty, but also disorder and deformity ; and since the evil
things did manifestly preponderate in number over the good,
and the deformed over the beautiful, some one else at length
introduced strife and friendship as the respective causes of
these diverse phznomena.”+ And in Greece, indeed, it

* Arist. 7722. I. 4, 9. : ΤΣ

+ παλίντροπος ἁρμονίη ὅκως περ τόξου καὶ Adpns.—Heraclitus, quoted in Orzgenzs
Philosophumena, 1X. 9. Also Plutarch, De /szde et Ostride. :

+ "Emel δὲ καὶ τἀναντία τοῖς ἀγαθοῖς ἐνόντα ἐφαίνετο ἐν τῇ φύσει, καὶ οὐ μόνον
τάξις καὶ τὸ καλὸν ἀλλὰ καὶ ἀταξία καὶ τὸ αἰσχρόν, καὶ πλείω τὰ κακὰ τῶν ἀγαθῶν
καὶ τὰ φαῦλα τῶν καλῶν, οὕτως ἄλλος τις φιλίαν εἰσήνεγκε καὶ νεῖκος ἑκάτερον
ἑκατέρων αἴτιον τούτων. ---Ατἱϑδί. Metaphysica, 1. 4. ᾿

2

434 THE LAWS OF THOUGHT

seems to have been chiefly as a philosophical opinion, or as
an adjunct to philosophical speculation, that the dualistic
theory obtained ground.* The moral application of the
doctrine most in [414] accordance with the Greek mind is
preserved in the great Platonic antithesis of ‘‘ being and
non-being,”—the connexion of the former with whatsoever
is good and true, with the eternal ideas, and the archetypal
world: of the latter with evil, witherror, with the perishable
phenomena of the present scene. The two forms of specu-
lation which we have considered were here blended together ;
nor was it during the youth and maturity of Greek philo-
sophy alone that the tendencies of thought above described
were manifested. Ages of imitation caught up and adopted
as their own the same spirit. Especially wherever the
genius of Plato exercised sway was this influence felt. The
unity of all real being, its identity with truth and goodness
considered as to their essence ; the illusion, the profound
unreality, of all merely phanomenal existence ; such were
the views,—such the dispositions of thought, which it chiefly
tended to foster. Hence that strong tendency to mysticism
which, when the days of renown, whether on the field of
intellectual or on that of social enterprise, had ended in
Greece, became prevalent in her schools of philosophy, and
reached their culminating point among the Alexandrian
Platonists. The supposititious treatises of Dionysius the
Areopagite served to convey the same influence, much
modified by its contact with Aristotelian doctrines, to the
scholastic disputants of the middle ages. It can furnish no
just ground of controversy to say, that the tone of thought
_ thus encouraged was as little consistent with genuine de-
votion as with a sober philosophy. That kindly influence of
human affections, that homely intercourse with the common
things of life, which form so large a part of the true, because
intended, discipline of our nature, would be ill replaced by

* Witness Aristotle’s well-known derivation of the elements from the qualities
warm,” and ‘‘ dry,” and their contraries. It is characteristic that Plato connects
their generation with mathematical principles. — 77maeus, cap. xi.

[1

SS a pe δὰ λυ eee Oe υνὰ —" ———— poi ale

CONSTITUTION OF THE INTELLECT 435

the contemplation even of the highest object of thought,
viewed by an excessive abstraction as something concerning
which not a single intelligible proposition could either be
affirmed or denied.* I would but slightly allude to those
connected speculations on the Divine Nature which ascribed
[415] to it the perfect union of opposite qualities, + or to the
remarkable treatises of Anselm, designed to establish a
theory of the universe upon the analogies of thought and
being.{ The primal unity is there represented as having its
abode in the one eternal Truth. The conformity of Nature
to her laws, the obedience of moral agents to the dictates of
rectitude, are the same Truth seen in action ; the world itself
being but an expression of the self-reflecting thought of its
Author.§ Still more marked was the revival of the older
forms of speculation during the sixteenth and seventeenth

centuries. The friends and associates of Lorenzo the

Magnificent, the recluses known in England as the Cam-
bridge Platonists, together with many meditative spirits
scattered through Europe, devoted themselves anew, either
to the task of solving the ancient problem, De Uno, Vero,
Bono, or to that of proving that all such inquiries are futile
and vain.|| The logical elements which underlie all these
speculations, and from which they appear to borrow at least

* Αὐτὸς καὶ ὑπὲρ θέσιν ἐστὶ καὶ dpaipeoww.—Dion. Areop. De Divinis Nomini-
bus, cap. li.

t+ See especially the lofty strain of Hildebert beginning ‘‘ Alpha et 2 magne
Deus.” (Trench’s Sacred Latin Poetry.) The principle upon which all these
speculations rest is thus stated in the treatise referred to in the last note.
Οὐδὲν οὖν ἄτοπον, ἐξ ἀμυδρῶν εἰκόνων ἐπὶ τὸ πάντων αἴτιον ἀναβάντας, ὕπερ-
κοσμιοῖς ὀφθαλμοῖς θεωρῆσαι πάντα ἐν τῷ παντῶν αἰτίῳ, καὶ τὰ ἀλλήλοις ἐναντία
μονοειδῶς καὶ ἡνωμένως.---72ὲ Divinis Nominibus, cap. v. And the kind οἵ
knowledge which it is thus sought to attain is described as a ‘‘darkness beyond
light,” ὑπερφῶτος γνόφος. (De Mystica Theologia, cap. i.) Milton has a similar
thought-—

κε Dark with excessive bright Thy skirts appear.”
Par, Lost, Book III.

Contrast with these the nobler simplicity of 1 Johni. 5.

1 Monologium, Prosologium, and De Verztate.

§ ‘‘Idcirco cum ipse summus spiritus dicit seipsum dicit omnia quz facta sunt.”
— Monolog. cap. xxiii.

| See dissertations in Spinoza, Picus of Mirandula, H. More, ἄς, Modern
discussions of this nature are chiefly in connexion with esthetics, the ground of
the application being contained in the formula of Augustine: ‘‘ Omnis porro
pulchritudinis forma, unitas est.”

436 THE LAWS OF THOUGHT

their form, it would be easy to trace in the outlines of more
modern systems,—more especially in that association of the
doctrine of the absolute unity with the distinction of the ego
and the zon-ego as the type of Nature, which forms the
basis of philosophy of Hegel. The attempts of specula-
tive minds to ascend to some high pinnacle of truth, from
which they might survey the entire framework and con-
[416]nexion of things zx the order of deductive thought,
have differed less in the forms of theory which they have pro-
duced, than through the nature of the interpretations which
have been assigned to those forms.* And herein lies the
real question as to the influence of philosophical systems
upon the disposition and the life. For though it is of slight
moment that men should agree in tracing back all the forms
and conditions of being to a primal unity, it is otherwise as
concerns their conceptions of what that unity is, and what
are the kinds of relation, beside that of mere causality,
which it sustains to themselves. Herein too may be felt
the powerlessness of mere Logic, the insufficiency of the
profoundest knowledge of the laws of the understanding, to
resolve those problems which lie nearer to our hearts, as
progressive years strip away from our life the illusions of its
golden dawn.

8. If the extremely arbitrary character of human opinion
be considered, it will not be expected, nor is it here main-
tained, that the above are the only forms in which speculative
men have shaped their conjectural solutions of the problem
of existence. Under particular influences other forms of
_ doctrine have arisen, not unfrequently, however, masking

* For instance, the learned mysticism of Gioberti, widely as it differs in its
spirit and its conclusions from the pantheism of Hegel (both being, perhaps,
equally remote from truth), resembles it in applying both to ¢hought and to
being the principles of unity and duality. It is asked :—‘‘ Or non ὃ egli chiaro
che ogni discorso si riduce in fine alle idee di Dio, del mondo, e della creazione,
P ultima delle quali ἃ il legame delle due prime?” And this question being
affirmatively answered in the formula, ‘1’ Ente crea le esistenze,” it is said of that
formula, —‘‘ Essa abbraccia la realta universale nella dualita del necessario e del
contingente, esprime il vincolo di questi due ordini, e collocandolo nella creazion
sostanziale, riduce la dualita reale a un principio unico, all unita primordiale dell’

Ente non astratto, complessivo, e generico,g{ma concreto, individuato, assoluto,
e creatore,”—De/ Bello e del Buono, pp. 30, 31.

ee ὅδε

ee eee ee ΤΣ

CONSTITUTION OF THE INTELLECT 437

those portrayed above. * [417] But the wide prevalence of
the particular theories which we have considered, together
with their manifest analogy with the expressed laws of
thought, may justly be conceived to indicate a connexion
between the two systems. As all other mental acts and
procedures are beset by their peculiar fallacies, so the Opera-
tion of that law of thought termed in this work the law of
duality may have its own peculiar tendency to error, exalt-
ing mere want of agreement into contrariety, and thus form
a world which we necessarily view as formed of parts supple-
mental to each other, framing the conception of a world
fundamentally divided by opposing powers. Such, with
some large but hasty inductions from phenomena, may
have been the origin of dualism,—independently of the
question whether dualism is in any form a true theory or
not. Here, however, it is of more importance to consider
in detail the bearing of these ancient forms of speculation,
as revived in the present day, upon the progress of real
knowledge ; and upon this point I desire, in pursuance of
what has been said in the previous section, to add the follow-
ing remarks :

Ist. All sound philosophy gives its verdict against such
speculations, if regarded as a means of determining the
actual constitution of things. It may be that the progress
of natural knowledge tends towards the recognition of some
central Unity in Nature. Of such unity as consists in the
mutual relation of the parts of a system there can be little
doubt, and able men have speculated, not without grounds,
on a more intimate correlation of physical forces than the
mere idea of a system would lead us to conjecture. Further,
it may be that in the bosom of that supposed unity are

* Evidence in support of this statement will be found in the remarkable
treatise recently published under the title (the correctness of which seems doubt-
ful) of Origents Philosophumena. The early corruptions of Christianity of which
it contains the record, though many of them, as is evident from their Ophite
character, derived from the very dregs of paganism, manifest certain persistent
forms of philosophical speculation. For the most part they either belong to the

dualistic scheme, or recognise three principles, primary or derived, between two
of which the dualistic relation may be traced.—Orvzg. Phil, pp. 135, 139, 150,

235, 253, 264.

4:8. THE LAWS OF THOUGHT

involved some general principles of division and re-union,
the sources, under the Supreme Will, of much of the related
variety of Nature. The instances of sex and polarity have
been adduced in support of such a view. As a supposition,
I will venture to add, that it is not very improbable that, in
some such way as this, the constitution of things without
may correspond to that of the mind within. But such corre-
spondence, if it shall ever be proved to exist, will appear as
the last induction from human knowledge, not as the first
principle of scientific inquiry. The natural order of dis-
covery is from the particular to the universal, and it may
confidently be affirmed [418] that we have not yet advanced
sufficiently far on this track to enable us to determine what
are the ultimate forms into which all the special differences
of Nature shall merge, and from which they shall receive
their explanation.

2ndly. Were this correspondence between the forms of
thought and the actual constitution of Nature proved to
exist, whatsoever connexion or relation it might be supposed
to establish between the two systems, it would in no degree
affect the question of their mutual independence. It would
in no sense lead to the consequence that the one system is
the mere product of the other. A too great addiction to
metaphysical speculations seems, in some instances, to have
produced a tendency toward this species of illusion. Thus,
among the many attempts which have been made to explain
the existence of evil, it has been sought to assign to the
fact a merely relatzve character,—to found it upon a species
_ of logical opposition to the equally relative element of good.
It suffices to say, that the assumption is purely gratuitous.
What evil may be in the eyes of Infinite wisdom and purity,
we can at the best but dimly conjecture ; but to us, in all
its forms, whether of pain or defect, or moral transgression,
or retributory woe, it can wear but one aspect,—that of a
sad and stern reality, against which, upon somewhat more
than the highest order of prudential considerations, the whole .
preventive force of our nature may be exerted. Now what

CONSTITUTION OF THE INTELLECT 439

has been said upon the particular question just considered,
is equally applicable to many other of the debated points of
philosophy ; such, for instance, as the external reality of
space and time. We have no warrant for resolving these
into mere forms of the understanding, though they un-
questionably determine the present sphere of our knowledge.
And, to speak more generally, there is no warrant for the
extremely subjective tendency of much modern speculation.
Whenever, in the view of the intellect, different hypotheses
are equally consistent with an observed fact, the instinctive
testimony of consciousness as to their relative value must be
allowed to possess authority.

3rdly. If the study of the laws of thought avails us neither
to determine the actual constitution of things, nor to explain
the facts involved in that constitution which have perplexed
the wise [419] and saddened the thoughtful in all ages, —
still less does it enable us to rise above the present con-
ditions of our being, or lend its sanction to the doctrine
which affirms the possibility of an zztuztzve knowledge of the
infinite, and the unconditioned,—whether such knowledge
be sought for in the realm of Nature, or above that realm.
‘We can never be said to comprehend that which is repre-
sented to thought as the limit of an indefinite process of
abstraction. A progression ad infinitum is impossible to
finite powers. But though we cannot comprehend the in-
finite, there may be even scientific grounds for believing
that human nature is constituted in some relation to the in-
finite. We cannot perfectly express the laws of thought, or
establish in the most general sense the methods of which
they form the basis, without at least the implication of ele-
ments which ordinary language expresses by the terms
“Universe” and ‘‘Eternity.” As in the pure abstractions
of Geometry, so in the domain of Logic it is seen, that the
empire of Truth is, in a certain sense, larger than that of
Imagination. And as there are many special departments
of knowledge which can only be completely surveyed from
an external point, so the theory of the intellectual processes,

440 THE LAWS OF THOUGHT

as applied only to finite objects, seems to involve the recog-
nition of a sphere of thought from which all limits are with-
drawn. If then, on the one hand, we cannot discover in the
laws of thought and their analogies a sufficient basis of proof
for the conclusions of a too daring mysticism ; on the other
hand we should err in regarding them as wholly unsuggestive.
As parts of our intellectual nature, it seems not improbable
that they should manifest their presence otherwise than by
merely prescribing the conditions of formal inference. What-
ever grounds we have for connecting them with the peculiar
tendencies of physical speculation among the Ionian and
Italic philosophers, the same grounds exist for associating
them with a disposition of thought at once more common
and more legitimate. To no casual influences, at least,
ought we to attribute that meditative spirit which then most
delights to commune with the external magnificence of
Nature, when most impressed with the consciousness of
sempiternal verities,—which reads in the nocturnal heavens
a bright manifestation of order; or feels in some wild scene
among the [420] hills, the intimations of more than that
abstract eternity which had rolled away ere yet their dark
foundations were laid. *

9. Kefraining from the further prosecution of a train of
thought which to some may appear to be of too speculative
a character, let us briefly review the positive results to
which we have been led. It has appeared that there exist
in our nature faculties which enable us to ascend from the
particular facts of experience to the general propositions
which form the basis of Science ; as well as faculties whose
office it is to deduce from general propositions accepted as
true the particular conclusions which they involve. It has
been seen, that those faculties are subject in their operations
to laws capable of precise scientific expression, but invested
with an authority which, as contrasted with the authority of
the laws of nature, is distinct, suz generts, and underived.
Further, there has appeared to be a manifest fitness between

* Psalm xc. 2.

CONSTITUTION OF THE INTELLECT 4a

the intellectual procedure thus made known to us, and the
conditions of that system of things by which we are sur-
rounded,—such conditions, I mean, as the existence of
species connected by general resemblances, of facts associated
under general laws ; together with that union of permanency
with order, which while it gives stability to acquired know-
ledge, lays a foundation for the hope of indefinite progression.
Human nature, quite independently of its observed or mani-
fested tendencies, is seen to be constituted in a certain relation
to Truth; and this relation, considered as a subject of
speculative knowledge, is as capable of being studied in its
details, is, moreover, as worthy of being so studied, as are
the several departments of physical science, considered in
the same aspect. I would especially direct attention to
that view of the constitution of the intellect which represents
it as subject to laws determinate in their character, but not
operating by the power of necessity ; which exhibits it as
redeemed from the dominion of fate, without being aban-
doned to the lawlessness of chance. We cannot embrace
this view without accepting at least as probable the intima-
tions which, upon the principle of analogy, it seems to
furnish respecting another and a higher aspect of our nature,
—its subjection in the sphere of duty as well as in that of
knowledge to [421] fixed laws whose authority does not
consist in power,—its constitution with reference to an
ideal standard and a final purpose. It has been thought,
indeed, that scientific pursuits foster a disposition either to
overlook the specific differences between the moral and the
material world, or to regard the former as in no proper sense
a subject for exact knowledge. Doubtless all exclusive
pursuits tend to produce partial views, and it may be, that
a mind long and deeply immersed in the contemplation of
scenes over which the dominion of a physical necessity is
unquestioned and supreme, may admit with difficulty the
possibility of another order of things. But it is because of
the exclusiveness of this devotion to a particular sphere of
knowledge, that the prejudice in question takes possession,

442 THE LAWS OF THOUGHT

if at all, of the mind. The application of scientific methods
to the study of the intellectual phanomena, conducted in an
impartial spirit of inquiry, and without overlooking those
elements of error and disturbance which must be accepted
as facts, though they cannot be regarded as Jaws in the
constitution of our nature, seems to furnish the materials of
a juster analogy.

10. If it be asked to what practzcal end such inquiries as
the above point, it may be replied, that there exist various
objects, in relation to which the courses of men’s actions
are mainly determined by their speculative views of human
nature. Education, considered in its largest sense, is one
of those objects. The ultimate ground of all inquiry into
its nature and its methods must be laid in some previous
theory of what man is, what are the ends for which his
several faculties were designed, what are the motives which
have power to influence them to sustained action, and to
elicit their most perfect and most stable results. It may be
doubted, whether these questions have ever been considered
fully, and at the same time impartially, in the relations
here suggested. The highest cultivation of taste by the
study of the pure models of antiquity, the largest acquaint-
ance with the facts and theories of modern physical science,
viewed from this larger aspect of our nature, can only appear
as parts of a perfect intellectual discipline. Looking from
the same point of view upon the means to _be employed, we
might be led to inquire, whether that all but exclusive appeal
which is made in [422] the present day to the spirit of
emulation or cupidity, does not tend to weaken the influence
of those more enduring motives which seem to have been
implanted in our nature for the immediate end in view.
Upon these, and upon many other questions, the just limits
of authority, the reconciliation of freedom of thought with
discipline of feelings, habits, manners, and upon the whole
moral aspect of the question,—what unfixedness of opinion,
what diversity of practice, do we meet with! Yet, in the
sober view of reason, there is no object within the compass

CONSTITUTION OF THE INTELLECT 443

of human endeavours which is of more weight and moment
than this, considered, as I have said, in its largest meaning.
Now, whatsoever tends to make more exact and definite our
view of human nature, in any of its real aspects, tends, in
the same proportion, to reduce these questions into narrower
compass, and restrict the limits of their possible solution.
Thus may even speculative inquiries prove fruitful of the
most important principles of action.

11. Perhaps the most obviously legitimate bearing of such
speculations would be upon the question of the place of
Mathematics in the system of human knowledge, and the
nature and office of mathematical studies, as a means of in-
tellectual discipline. No one who has attended to the course
of recent discussions can think this question an unimportant
one. Those who have maintained that the position of Mathe-
matics is in both respects a fundamental one, have drawn one
of their strongest arguments frem the actual constitution of
things. The material frame is subject in all its parts to the
relations of number. All dynamical, chemical, electrical,
thermal, actions, seem not only to be miserable in them-
selves, but to be connected with each other, even to the
extent of mutual convertibility, by numerical relations of a
perfectly definite kind. But the opinion in question seems
to me to rest upon a deeper basis than this. The laws of
thought, in all its processes of conception and of reasoning,
in all those operations of which language is the expression
or the instument, are of the same kind as are the laws of the
acknowledged processes of Mathematics. It is not con-
tended that it is necessary for us to acquaint ourselves with
those laws in order to think coherently, or, in the ordinary
sense of [423] the terms, to reason well. Men draw in-
ferences without any consciousness of those elements upon
which the entire procedure depends. Still less is it desired
to exalt the reasoning faculty over the faculties of observa-
tion, of reflection, and of judgment. But upon the very
ground that human thought, traced to its ultimate elements,
reveals itself in mathematical forms, we have a presumption

444 THE LAWS OF THOUGHT

that the mathematical sciences occupy, by the constitution
of our nature, a fundamental place in human knowledge, and
that no system of mental culture can be complete or funda-
mental, which altogether neglects them.

But the very same class of considerations shows with
equal force the error of those who regard the study of Mathe-
matics, and of their applications, as a sufficient basis either
of knowledge or of discipline. If the constitution of the
material frame is mathematical, it is not merely so. If the
mind, in its capacity of formal reasoning, obeys, whether
consciously or unconsciously, mathematical laws, it claims
through its other capacities of sentiment and action, through
its perceptions of beauty and of moral fitness, through its
deep springs of emotion and affection, to hold relation to a
different order of things. There is, moreover, a breadth of
intellectual vision, a power of sympathy with truth in all its
forms and manifestations, which is not measured by the force
and subtlety of the dialectic faculty. Even the revelation of
the material universe in its boundless magnitude, and per-
vading order, and constancy of law, is not necessarily the
most fully apprehended by him who has traced with minutest
accuracy the steps of the great demonstration. And if we
embrace in our survey the interests and duties of life, how
little do any processes of mere ratiocination enable us to
comprehend the weightier questions which they present !
As truly, therefore, as the cultivation of the mathematical
or deductive faculty is a part of intellectual discipline, so
truly is it only a part. The prejudice which would either
-banish or make supreme any one department of knowledge
or faculty of mind, betrays not only error of judgment, but
a defect of that intellectual modesty which is inseparable
from a pure devotion to truth. It assumes the office of
criticising a constitution of things which no human ap-
pointment has established, or can annul. It sets aside the
[424] ancient and just conception of truth as one though
manifold. Much of this error, as actually existent among us,
seems due to the special and isolated character of scientific

Sas Oe eee

CONSTITUTION OF THE INTELLECT 4as

teaching—which character it, in its turn, tends to foster,
The study of philosophy, notwithstanding a few marked in-
stances of exception, has failed to keep pace with the advance
of the several departments of knowledge, whose mutual rela-
tions it is its province to determine. It is impossible, how-
ever, not to contemplate the particular evil in question as part
of a larger system, and connect it with the too prevalent view
of knowledge as a merely secular thing, and with the undue
predominance, already adverted to, of those motives, legiti-
mate within their proper limits, which are founded upon a
regard to its secular advantages. In the extreme case it is
not difficult to see that the continued operation of such
motives, uncontrolled by any higher principles of action, un-
corrected by the personal influence of superior minds, must
tend to lower the standard of thought in reference to the
objects of knowledge, and to render void and _ ineffectual
whatsoever elements of a noble faith may still survive. And
ever in proportion as these conditions are realized must the
same effects follow. Hence, perhaps, it is that we some-
times find juster conceptions of the unity, the vital con-
nexion, and the subordination to a moral purpose, of the
different parts of Truth, among those who acknowledge
nothing higher than the changing aspect of collective
humanity, than among those who profess an intellectual
allegiance to the Father of Lights. But these are questions
which cannot further be pursued here. To some they will
appear foreign to the professed design of this work. But
the consideration of them has arisen naturally, either out
of the speculations which that design involved, or in the
course of reading and reflection which seemed necessary to

its accomplishment.

INDEX

Abbreviation, methods of, 139 ff.

Abelard, 2.

Addition, logical, 36, 60, 72.

Adimantus, 193.

Alcmezeon, 433.

Alexandrian Platonists, 434.

Anaxagoras, 431.

Anaximenes, 431.

Anselm, 2, 435.

Aristotelian logic and its modern exten-
sions, 239 ff.

Aristotle, 1, 4, ΤΊ, 54, 143, 146, 147,
173, 206, 239, 254, 431, 432, 433,
434-

Astronomical observations,
of. 21,

Augustine, 435.

reduction

Bacon, Francis (Lord), 2, 431.
Boole, Mrs. Mary Everest, v.
Butler, Joseph (Bishop), 212, 213, 229.

Ceesar, 38.

Calculus of logic, 6.

Calculus of 0 and 1, 41.

Cause and effect, relation of, 337 ff.

Chrysippus, 191.

Cicero, 190, 191.

Clarke, Samuel, 197 ff., 232.

Class term, 60.

Combination, law of, 39, 54.

Condorcet, Marie Jean Antoine Nicolas
Caritat, 397.

Constituents of an expansion, 82.

Contradiction, principle of, 53.

Conversion, logical, 113.

Copula, 38.

opiate Antoine Augustine, 297, 399,
426.

De Morgan, Augustus, xii, 240, 315,
380, 383, 384.

Descartes, René, 2.

Development, 77 ff.

Dichotomy, 55.

446

Dickson,
ΧΙ,

Diogenes, 431.

Dionysius the Areopagite, 434, 435.

‘* Divisible,” 140.

Division, logical, 78, 94.

Donkin, William, 266, 278.

Duality, law of (see Law of duality),

George Stephens (Rev.),

Effect, relation of cause and, 337 ff.
Elimination, problem of, 9, 107 ff.
Empedocles, 432. :
Expansion, 77 ff.

Fabius, 190, 101} 192.
Faugere, Prosper, 257.
Fermat, Pierre de, 257.
Forbes, James David, 385.
Fractions, logical, 81, 94.

| Free-agency, human, 22.

Function, logical, 78, 87, 90, 109.

General method in logic, 7.
General truths, 5.

Gioberti, Vincenzo, 436.
Graves, Charles (Rev.), 259.

- Hamilton, Sir William, 240.
_ Hegel, Georg Wilhelm Friedrich, 436.

Heraclitus, 431.
Hermotimus, 431.
Hildebert, 435.
Hincks, Edward, 259.
Hippasus, 431.

Identity, sign of, 29.

Indefinite class, 96.

Independent events, 270.

Independent system of propositions,
163.

Intellect, constitution of the, 418 ff.

Interpretability, 73.

Interpretability, condition of, ror.

Interpretation of logical equations,
87 ff.

INDEX

John (Evangelist), 435.
Judgments, probability of, 393.

Kant, Immanuel, 253.
Kepler, Johann, 5, 422.

Language, 26 ff., 46.

Laplace, Pierre Simon, 2, 22, 263,
380, 383, 392, 397, 410.

Latham, 420.

ae commutative, of logical addition,
36.

Law, commutative, of logical multipli-
cation, 31.

Law, distributive, 36.

Law of duality, 34, 50, 56, 62, 71, 85,
108, 132, 139, 160, 162, 164, 176,
195, 196, 373.

Laws of Nature, 4.

Laws of the mind, 4.

Layard, Austen Henry (Sir), 259.

Leibnitz, Gottfried Wilhelm, 254.

Leucippus, 433.

Locke, John, 2, 196.

Logic, Aristotelian, and its modern
extensions, 2309 ff.

Logic, definition of, 5.

Logic, general method in, 7.

Lorenzo the Magnificent, 435.

Lubbock, Sir John, xii.

Manichzean doctrine, 433.

Melissus, 431.

Méré, Chevalier de, 257.

Method, conditions of a perfect, 159 ff.

Method of reduction, 232 ff.

Methods of abbreviation, 130 ff.

Mill, John Stuart, 253, 385.

Milton, John, 33, 435

Mind, derivation of logical laws from
laws of, 43.

Mind, nature and constitution of, 24.

Mitchell, J. (Rev.), 380, 381.

More, Henry, 435.

Multiplication, logical, 31.

Multipliers, arbitrary constant, 124.

Newton, Isaac (Sir), 208, 431.
Nineveh, 259.
Nothing, 52, 96.

Ogham inscriptions, 259.
Operation, signs of, 29.
Origen, 437.

Pallas, 253.
Parmenides, 431.
Particular affirmative propositions, 66.

447

Particular negative propositions, 69.

Pascal, Blaise, 2, 257.

Picus of Mirandula, 435.

Plato, 173, 192, 424, 434.

Platonists, Alexandrian, 434.

Platonists, Cambridge, 435.

Plutarch, 433.

Poisson, Siméon Dénis, 22, 258, 397,
398, 399, 410.

Porphyry, 2.

asia! propositions, 57 ff.

Probabilities, theory of, 14, 155, 257 fh,
ees y OF, 14, 155, 257

Probability of Judgments, 393 ff.

Proclus, 2.

Propositions, primary, 57.

Propositions, secondary, 57, 168.

Propositions, universal negative, 68

Quality, 246.

Quantitative symbols, 282, 287.
(Quantity, 246.

Quetelet, Lambert Adolphe Jacques,

X11,

Ramus, Petrus, 2.

Reduction, method of, 232 ff.

Reduction of systems of propositions,
123 ff.

Relations among things and relations
among facts, 8.

Ritter, Heinrich, 432.

Ryall, John, ix.

Sainte Beuve, Charles Augustine de,
257.

Science, nature of, 418 ff,

Secondary propositions, 57, 168 ff.

Senior, Nassau William, 65, 114.

Sextus Empiricus, 432.

Sign for copula, 38.

Signs, 26 ff. ὯΝ
Signs, appellative or descriptive, 30.
| Social system, movements of the

22.
‘Sones 64,07 ΤΙ Ὶ τῆ. 102.
Space, notion of, in theory of secondary
propositions, 185.
Spinoza, Benedict de, 197 ff., 435.
Statistical conditions, 311 ff.
Stoddart, John (Sir), 420.
Subtraction, logical, 37, 53, 60.
Syllogism, 107, 244 ff.
Symbolical reasoning, 72 ff.
Symbols, literal, 29.

Taylor, Brook, 79.
Terrot (Bishop), 386.

448 THE LAWS OF THOGGAT

Thales, 431.

Thomson, William (Archbishop) xi,
240,

Time, secondary propositions and idea
of, 171 ff.

Timon the Sillograph, 432.

Transposition, law of, 39, 54.

Trench, Richard Chenevix (Arch-
bishop), 435.

Truths, general, 5.

Universal affirmative proposition, 64.
Universal negative propositions, 68.
Universe of discourse, 46, 52, 176.

Whately, Richard (Archbishop), xi,
11. 7252:
Whewell, William (Rev. ), 426.

Xenophanes, 433.

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