The Mathematical Analysis of Logic

The Mathematical

Analysis of Logic By

GEORGE BOOLE

I.I

NIVER51TY OF TORONTO

UNiV

OF TORONTO

^

THE MATHEMATICAL ANALYSIS OF LOGIC

THE MATHEMATICAL ANALYSIS OF LOGIC BEING AN ESSAY TOWARDS A CALCULUS OF DEDUCTIVE REASONING

By

GEORGE BOOLE

PHILOSOPHICAL LIBRARY

NEW YORK

Published in the United States of America 1948, by the Philosophical Library, Inc., 15 East 40th Street, New York, N.Y.

THE MATHEMATICAL ANALYSIS

OF LOGIC,

BEING AN ESSAY TOWARDS A CALCULUS OF DEDUCTIVE REASONING.

BY GEORGE BOOLE.

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ARISTOTLE, Anal.

CAMBRIDGE

Posf., lib.

:

MACMILLAN, BARCLAY, & MACMILLAN LONDON: GEORGE BELL. 1847

i.

;

cap. xi.

o

/

PRINTED IN ENGLAND BY

HENDERSON & SPALDING LONDON. W.I

PREFACE. IN presenting

this

Work

deem

public notice, I

to

irrelevant to observe, that speculations similar to those it

at different periods,

records have,

In the spring of the present year then

to the question

my

occupied

my

moved between

attention Sir

W.

it

not

which

thoughts.

was directed

Hamilton and

Professor De Morgan; and I was induced by the interest which it inspired, to resume the almost-forgotten thread of

former inquiries.

It

appeared

to

me

that,

although Logic

of quantity,* it might be viewed with reference to the idea had also another and a deeper system of relations. If it was lawful to regard it from without, as connecting itself through the

medium

of

Number

was lawful

it

facts of

also

to

regard

it

from within,

based upon

as

another order which have their abode in the consti

of the Mind.

tution

with the intuitions of Space and Time,

which

inquiries

it

The

results

and of the

of this view,

suggested, are embodied, in the following

Treatise. It

the

is

not generally

mode

in

which

permitted to

an Author

his production shall be

to

judged

;

prescribe

but there

two conditions which I may venture to require of those who shall undertake to estimate the merits of this performance. are

The

no preconceived notion of the impossibility of its objects shall be permitted to interfere with that candour and impartiality which the investigation of Truth demands first

is,

that

;

the second shall

not

is,

that their

judgment of the system as a whole

be founded either upon the examination of only * See p. 42.

PREFACE. it, or upon the measure of its conformity with any received system, considered as a standard of reference from

a part of

which appeal

theorems which

It is in the general

denied.

is

occupy the latter chapters of this work, results to which there no existing counterpart, that the claims of the method, as

is

a Calculus of Deductive Reasoning, are most fully set forth.

What may

be the

final estimate

of the value of the system,

have neither the wish nor the right to anticipate. The is not simply determined by its truth

I

estimation of a theory It also

depends upon the importance of

extent of

be

left

its

applications;

human

to the arbitrariness of

of the application

Logic were

its

subject,

to

the science

of the subject permits

danger carried

upon "

:

a principle

which has

Whenever

the nature

the reasoning process to

contrary case

while in the

;

should be so constructed, that there shall be

it

the greatest possible obstacle to a

mere mechanical use of

In one respect, the science of Logic the perfection of

method

its

of the speculative truth of

employment of

be without

on mechanically, the language should be con

structed on as mechanical principles as possible

common

is its

differs

from

principles.

mind an

* Mill s

LINCOLN,

athletic

and

it

the

to the rigour

who knows

and warfare which imparts vigour, and teaches it to contend

to rely

toil

upon

itself in

System of Logic, Ratiocinative

Oct. 29, 1847.

it."*

others;

To supersede

reason, or to subject

the value of that intellectual

difficulties

all

chiefly valuable as an evidence

of technical forms, would be the last desire of one

the

of

solely a question of Notation, I should be content

been stated by an able living writer

with

still

If the utility

Opinion.

of Mathematical forms

to rest the defence of this attempt

to

and the

beyond which something must

emergencies.

and Inductive, Vol.

II.

p. 292.

MATHEMATICAL ANALYSIS OF

LOGIC.

INTRODUCTION. THEY who

are acquainted with the present state of the theory

of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a ques tion on the properties of numbers, under another, that of

a geometrical problem, and under a third, that of a problem This principle is indeed of fundamental of dynamics or optics. it and may with safety be affirmed, that the recent importance ;

advances of pure analysis have been much assisted by the it has exerted in directing the current of

influence which investigation.

But the doctrine

full recognition of the consequences of this important has been, in some measure, retarded by accidental

circumstances.

It

has

happened

in

every

known form

analysis, that the elements to be determined have

of

been con

ceived as measurable by comparison with some fixed standard. The predominant idea has been that of magnitude, or more strictly,

of numerical ratio.

The expression of magnitude, B

or

4

INTRODUCTION.

of operations for for

upon magnitude, has been

-the

express

object

which the symbols of Analysis have been invented, and which their laws have been investigated. Thus the ab

stractions of the

modern Analysis, not

less

than the ostensive

diagrams of the ancient Geometry, have encouraged the notion, that Mathematics are essentially, as well as actually, the Science of Magnitude. The consideration of that view which has already been stated, as embodying the true principle of the Algebra of Symbols,

would, however, lead us

to infer that this conclusion is

by no

means necessary.

If every existing interpretation is shewn to involve the idea of magnitude, it is only by induction that we

can assert that no other interpretation is possible. And it may be doubted whether our experience is sufficient to render such an induction legitimate. The history of pure Analysis is, it may be said, too recent to permit us to set limits to the extent of its

Should we grant

to the inference a high degree and with reason, maintain the still, sufficiency of the definition to which the principle already stated would lead us. We might justly assign it as the definitive

applications.

of probability,

we might

character of a true Calculus, that

it

is

a

method

resting

upon

the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent

That

forms of Analysis a quan titative interpretation is assigned, is the result of the circum stances by which those forms were determined, and is not to

interpretation.

to the existing

be construed into a universal condition of Analysis.

It is

upon

the foundation of this general principle, that I purpose to establish the Calculus of Logic, and that I claim for it a place the acknowledged forms of Mathematical Analysis, re gardless that in its object and in its instruments it must at

among

present stand alone. That which renders Logic possible, is the existence in our minds of general notions, our ability to conceive of a class,

and

to designate its individual

members by

a

common name.

INTRODUCTION.

The theory

of Logic

is

O

thus intimately connected with that of

A

successful attempt to express logical propositions laws of whose combinations should be founded the by symbols, laws of the mental processes which they represent, the upon

Language.

be a step toward a philosophical language. But a view which we need not here follow into detail.*

would, so this

is

far,

Assuming the notion of a

we

class,

are able, from

ceivable collection of objects, to separate

which belong from the rest.

contemplate them apart Such, or a similar act of election, we may con

to the

given

ceive to be repeated. sideration

those

by

any con

a mental act, those

may

be

and

class,

The group

still

to

of individuals left under con

further limited,

among them which belong

to

by mentally selecting some other recognised class,

as well as to the one before contemplated.

And

this process

may be repeated with other elements of distinction, until we arrive at an individual possessing all the distinctive characters which we have taken

into account,

and a member,

at the

same

we have enumerated. It is in fact which we employ whenever, in common

time, of every class which a

method

language,

more

similar to this

we accumulate

descriptive epithets for the sake of

precise definition.

Now

the several mental operations which in the above case

we have supposed

to

be performed, are subject

It is possible to assign relations

to peculiar laws.

among them, whether

as re

spects the repetition of a given operation or the succession of different ones, or It

is,

for

some other

particular,

which are never

violated.

example, true that the result of two successive acts

* This view

is

is well expressed in one of Blanco White s Letters Logic is most part a collection of technical rules founded on classification. The Syllogism is nothing but a result of the classification of things, which the mind naturally and necessarily forms, in forming a language. All abstract terms are or rather the labels of the classes which the mind has settled." classifications Memoirs of the Rev. Joseph Blanco White, vol. n. p. 163. See also, for a "

:

for the

;

very

lucid introduction, Dr.

Latham

s First

Outlines of Logic applied to Language^

Becker s German Grammar, 8$c. Extreme Nominalists make Logic entirely dependent upon language. For the opposite view, see Cudworth s Eternal and Immutable Morality, Book iv. Chap. in. JB2

6

INTRODUCTION.

unaffected by the order in which they are performed ; and there are at least two other laws which will be pointed out in the

proper place. These will perhaps to some appear so obvious as to be ranked among necessary truths, and so little important as to be undeserving of special notice. And probably they are noticed for the first time in this Essay. Yet it may with con fidence be asserted, that

they were other than they are, the entire mechanism of reasoning, nay the very laws and constitu tion of the

human

might indeed

if

intellect,

exist, but

it

would be

vitally

changed.

A

Logic

would no longer be the Logic we

possess.

Such are the elementary laws upon the existence of which, and upon their capability of exact symbolical expression, the method of the following Essay is founded ; and it is presumed that the object which it seeks to attain will be thought to have been very fully accomplished. Every logical proposition, whether categorical or hypothetical, will be found to be capable of exact and rigorous expression, and not only will the laws of

conversion and of syllogism be thence deducible, but the resolu tion of the most complex systems of propositions, the separation of any proposed element, and the expression of its value in terms of the remaining elements, with every subsidiary rela tion involved. Every process will represent deduction, every

mathematical consequence will express a logical inference. The method will even permit us to express arbi generality of the trary operations of the intellect, and thus lead to the demon stration of general theorems in logic analogous, in no slight

No degree, to the general theorems of ordinary mathematics. inconsiderable part of the pleasure which we derive from the the interpretation of external nature, application of analysis to from the conceptions which it enables us to form of the The general formula to universality of the dominion of law. arises

which we

are conducted

seem

to give to that

element a visible

of particular cases presence, and the multitude extent of apply, demonstrate the

its

sway.

Even

to

which they

the

symmetry

INTRODUCTION. of their

expression

analytical

may

no fanciful

in

sense

be

harmony and its consistency. Now I the same sources of do not presume say to what extent The measure of pleasure are opened in the following Essay.

deemed

indicative of

its

to

may be

that extent

left to

the estimate of those

who

shall

think

But I may venture to subject worthy of their study. intellectual of assert that such occasions gratification are not the

The

here wanting.

laws

we have

to

examine are the laws of

one of the most important of our mental faculties. The mathe matics we have to construct are the mathematics of the human

Nor

intellect.

from

of notice. regard to its interpretation, undeserving even a remarkable exemplification, in its general

all

There

are the form and character of the method, apart

is

theorems, of that species of excellence which consists in free dom from exception. -And. this is observed where, in the cor

such a character responding cases of the received mathematics, there is that who think few that The is no means apparent.

by

in analysis

which renders

it

deserving of attention for

its

own

worth while to study it under a form in which be solved and every solution interpreted. can every equation Nor will it lessen the interest of this study to reflect that every will notice in the form of the Calculus peculiarity which they sake,

may

find

it

feature in the constitution of their represents a corresponding

own minds. would be premature to speak of the value which this method may possess as an instrument of scientific investigation. I speak here with reference to the theory of reasoning, and to It

the principle of a true classification of the forms and cases of aim of these investigations Logic considered as a Science.* The

was

in

the

first

instance confined to the expression of the

received logic, and to the forms of the Aristotelian arrangement, * WJiately s Elements of Logic. Indeed Strictly a Science" also "an Art." unless we are willing, with ought we not to reg.ord all Art as applied Science ? Plato, "the multitude/ to consider Art as "guessing and aiming well" "

;

;

Phikbus.

INTRODUCTION. soon became apparent that restrictions were thus intro duced, which were purely arbitrary and had no foundation in

but

it

These were noted as they occurred, and the nature of things. When it became neces will be discussed in the proper place. which sary to consider the subject of hypothetical propositions (in still more, when an for the was demanded general theorems of the interpretation Calculus, it was found to be imperative to dismiss all regard for

comparatively less has been done), and

precedent and authority, and to interrogate the method itself for an expression of the just limits of its application. Still, how

was no special effort to arrive at novel results. among those which at the time of their discovery appeared such, it may be proper to notice the following. ever, there

A logical

But to

be

according to the method of this Essay, the expressible by an equation the form of which determines rules of conversion and of transformation, to which the given proposition

proposition

is

subject.

simple conversion,

is

is,

Thus

the law of what logicians term

determined by the

fact,

that the corre

sponding equations are symmetrical, that they are unaffected by a mutual change of place, in those symbols which correspond to

the convertible classes.

The received laws

of conversion

were thus determined, and afterwards another system, which is thought to be more elementary, and more general. See Chapter,

On the Conversion of Propositions. The premises of a syllogism being expressed by equations, the elimination of a common symbol between them leads to a third equation which expresses the conclusion, this conclusion being

always the most general possible, whether Aristotelian or not. Among the cases in which no inference was possible, it was found, that there were two distinct forms of the final equation. It was a considerable time before the explanation of this fact was discovered, but it was at length seen to depend upon the

presence or absence of a true the premises. is

The

medium

distinction

illustrated in the Chapter,

On

which

of comparison between is

Syllogisms.

thought

to

be new

INTRODUCTION. character of the disjunctive conclusion of a hypothetical syllogism, is very clearly pointed out in the examples of this species of argument.

The nonexclusive

problems illustrated in the chapter, On and the Solution of Elective Equations is conceived to be new

The

class of logical

:

,

is believed that the method of that chapter affords the means of a perfect analysis of any conceivable system of propositions, an end toward which the rules for the conversion of a single it

the first step. categorical proposition are but However, upon the originality of these or

any of these views,

conscious that I possess too slight an acquaintance with the literature of logical science, and especially with its older lite I

am

me to speak with confidence. be not inappropriate, before concluding these obser may a few remarks upon the general question of the offer vations, to rature, to permit It

use of symbolical language in the mathematics. Objections have lately been very strongly urged against this practice, on the ground, that by obviating the necessity of thought, and substituting a reference to general formulae in the room of faculties. personal effort, it tends to weaken the reasoning

Now

the question of the use of symbols may be considered in two distinct points of view. First, it may be considered with reference to the progress of scientific discovery, and secondly, with reference to its bearing upon the discipline of the intellect.

And

with respect to the first view, it may be observed that as it is one fruit of an accomplished labour, that it sets us at arduous toils, so it is a necessary liberty to engage in more result of an advanced state of science, that we are permitted,

and even called upon, to proceed which we before contemplated.

If through the advancing

obvious.

we

find that the

afford

the

no longer a

remedy

tracks,

to

to

is,

to

higher problems, than those

The

proceed

ample

to

seek for difficulties

inference

is

power of scientific methods,

pursuits on which sufficiently

practical

we were once engaged,

field for intellectual effort,

higher inquiries, and, in new And such is, yet unsubdued.

10

INTRODUCTION.

indeed,

the

law of scientific progress. We must be abandon the hope of further conquest, or to

actual

content, either to

employ such

aids of symbolical language, as are proper to the

stage of progress, at to

commit ourselves

which we have arrived. to

so near to the boundaries of possible

the apprehension, that scope will inventive faculties.

In

Nor need we

We have

such a course.

knowledge,

fail

fear

not yet arrived as to suggest

for the exercise of the

second, and scarcely less momentous ques discussing tion of the influence of the use of symbols upon the discipline the

of the intellect, an important distinction ought to be made. It is of most material whether those are consequence, symbols

used with a

understanding of their meaning, with a perfect comprehension of that which renders their use lawful, and an full

expand the abbreviated forms of reasoning which they devolopment or whether they are mere unsuggestive characters, the use of which is suffered

ability to

induce, into their full syllogistic

to rest

upon

;

authority.

The answer which must be given

to the question proposed,

one or the other of these suppositions In the former case an intellectual discipline of a

will differ according as the is

admitted.

high order

is

provided, an exercise not only of reason, but of In the latter case there is no

the faculty of generalization.

mental discipline whatever. It were perhaps the best security against the danger of an unreasoning reliance upon symbols,

on the one hand, and a neglect of their just claims on the other, that each subject of applied mathematics should be treated in the spirit of the

application

methods which were known

was made, but

have assumed.

in the best

The order

would thus bear some

at the

time

when

the

form which those methods

of attainment in the individual

mind

relation to the actual order of scientific

discovery, and the

more

would be offered

to

abstract methods of the higher analysis such minds only, as were prepared to

receive them.

The

relation in

which

this

Essay stands

at

once

to

Logic and

11

INTRODUCTION. to

may

Mathematics,

which has

lately

as to

some notice of the question the relative value of the two

One

of the chief objections which

further justify

been revived,

studies in a liberal education.

have been urged against the study of Mathematics in general, is but another form of that which has been already considered with

And

respect to the use of symbols in particular.

it

need not here

be further dwelt upon, than to notice, that if it avails anything, The it applies with an equal force against the study of Logic. canonical forms of the Aristotelian syllogism are really symbol ; only the symbols are less perfect of their kind than those

ical

of mathematics.

argument, they

If they are as truly

employed

an

supersede the exercise of reason, as does

Whether men

a reference to a formula of analysis.

make

to test the validity of

do, in the

use of the Aristotelian canons, except as present day, a special illustration of the rules of Logic, may be doubted ; yet it

this

cannot be questioned that

when

the authority of Aristotle was

dominant in the schools of Europe, such applications were habit And our argument only requires the admission, ually made. that the case

is

possible.

But the question before us has been argued upon higher grounds. Regarding Logic as a branch of Philosophy, and de fining Philosophy as the

research of tigation

causes,"

of the

"science

and assigning

"

why,

(TO

existence," and "the main business the inves

of a real as its

while Mathematics display Sir W. Hamilton has contended,

Slori,),"

that, (TO oYl)," only the not simply, that the superiority rests with the study of Logic, but that the study of Mathematics is at once dangerous and use "

The trained him

less.*

pursuits of the mathematician

"

have not only not

to that acute scent, to that delicate,

almost instinc

which, in the twilight of probability, the search and discrimination of its finer facts demand; they have gone to cloud

tive, tact

his vision, to indurate his touch, to all

iron chain of demonstration,

and

left

but the blazing light, the

him out of the narrow con

fines of his science, to a passive credulity in *

Edinburgh Review, vol. LXII. p. 409, and Letter

to

any premises, or A.

De Morgan, Esq.

to

INTRODUCTION. an absolute incredulity in In support of these and of other both and argument charges, copious authority are adduced.* I shall not attempt a complete discussion of the topics which all."

are suggested by these remarks. My object is not controversy, and the observations which follow are offered not in the spirit

of antagonism, but in the hope of contributing to the formation of just views upon an important subject. Of Sir W. Hamilton

impossible to speak otherwise than with that respect which

it is is

due

genius and learning. Philosophy is then described as the science of a real existence

and

to

the research

"

word

ton

s

why."

the ancient writers.

that

no doubt may

rest

upon

further said, that philosophy These definitions are common

it is

cause,

mainly investigates the

among

And

of causes.

the meaning of the

Thus Seneca, one of Sir W. Hamil The philosopher seeks LXXXVIIT., "

authorities, Epistle

and knows the causes of natural things, of which the mathe matician searches out and computes the numbers and the mea It may be remarked, in passing, that in whatever the belief has prevailed, that the business of philosophy degree sures."

is

immediately with

science

whose object

causes; is

in

the same degree has every

the investigation of laws, been lightly

esteemed. Thus the Epistle to which we have referred, bestows, by contrast with Philosophy, a separate condemnation on Music

and Grammar, on Mathematics and Astronomy, although Mathematics only that Sir W. Hamilton has quoted.

it

is

that of

Now we

might take our stand upon the conviction of many thoughtful and reflective minds, that in the extent of the mean The business of ing above stated, Philosophy is impossible. true Science, they conclude,

nature of Being, the

mode

is

with laws and phenomena.

The

of the operation of Cause, the why,

* The arguments are in general better than the authorities. Many writers quoted in condemnation of mathematics (Aristo, Seneca, Jerome, Augustine, Cornelius Agrippa, &c.) have borne a no less explicit testimony against other The treatise of the last named sciences, nor least of all, against that of logic. writer

De

Vanitate Scientiantm,

Vide cap. en.

must surely have been

referred to

by mistake.

INTRODUCTION.

13

they hold to be beyond the reach of our intelligence.

But we

do not require the vantage-ground of this position; nor is it doubted that whether the aim of Philosophy is attainable or not, the desire which impels us to the attempt is an instinct of our

Let

higher nature. the

baffled "

"

science of a real that

which

not a hopeless one;

is

existence,"

for

kernel"

be granted that the problem which has

it

of ages,

efforts

and

"

that the

the research of

causes,"

"

is

Philosophy

human

not transcend the limits of the

still

militant,"

I

intellect.

am

do

then

according to this view of the nature of Philosophy, Logic forms no part of it. On the principle of

compelled

to assert, that

we ought no longer to associate Logic and but Metaphysics, Logic and Mathematics. Should any one after what has been said, entertain a doubt a true classification,

upon

must

this point, I

refer

him

to the

afforded in the following Essay.

He

like

truths,

Geometry upon axiomatic

structed

upon

evidence which will be

will there see

and

Logic resting theorems con

its

that general doctrine of symbols,

which

tutes the foundation of the recognised Analysis.

consti

In the Logic

of Aristotle he will be led to view a collection of the formulae of the science, expressed

by another, but, (it is thought) less scheme of symbols. I feel bound to contend for the absolute exactness of this parallel. It is no escape from the con clusion to which it points to assert, that Logic not only constructs perfect

a science, but also inquires into the origin

own "

"

principles,

It is

a distinction which

wholly beyond the domain of

to inquire into the origin

Review, page 415. tion be maintained

clear

and the nature of

denied

to

their

principles."

But upon what ground can such a ?

What

definition of the to allow

distinc

term Science will

such differences

?

application of this conclusion to the question before us

and decisive.

The mental

discipline which

the study of Logic, as an exact science, as that afforded

its

Mathematics.

mathematicians," it is said,

and nature of

be found sufficiently arbitrary

The

is

by the study of Analysis.

is,

is

afforded by in species, the same is

14 Is

INTRODUCTION. then contended that either Logic or Mathematics can

it

supply a perfect discipline to the Intellect

The most

?

careful

and unprejudiced examination of this question leads me to doubt whether such a position can be maintained. The exclusive claims of either must, I believe, be abandoned, nor can any others, par taking of a like exclusive character, be admitted in their room.

an important observation, which has more than once been made, that it is one thing to arrive at correct premises, and another It is

thing to deduce logical conclusions, and that the business of life depends more upon the former than upon the latter. The study of the exact sciences

may

teach us the one, and

some general preparation of knowledge and of attainment of the other, but

it

action, in the field of Practical Logic, the arena of

we

may

give us

practice for the

union of thought with

to the

is

it

Human

Life,

and more perfect accomplishment. I desire here to express my conviction, that with the ad vance of our knowledge of all true science, an ever-increasing

that

are to look for

harmony

will

its

be found

The view winch

fuller

to prevail

among

separate branches.

its

leads to the rejection of one, ought, if con

sistent, to lead to

the rejection of others.

And

indeed

many

of the authorities which have been quoted against the study of Mathematics, are even more explicit in their condemnation of "

Logic.

Natural

science,"

says the Chian Aristo,

Logical science does not concern are founded (as they often are)

us."

When

"

is

above us,

such conclusions

upon a deep conviction of the

preeminent value and importance of the study of Morals, we admit the premises, but must demur to the inference. For it has been well said by an ancient writer, that it is the charac teristic of the liberal sciences, not that they conduct us to Virtue, "

and Melancthon s senti but that they prepare us for Virtue has in abeunt studia mores," ment, passed into a proverb. ;"

"

Moreover, there

is

votaries of truth

a

may

common ground upon which

all

sincere

meet, exchanging with each other the

The works of the language of Flamsteed s appeal to Newton, will be better Providence understood Eternal through your "

labors and

mine."

15

(

)

FIRST PRINCIPLES.

LET

us employ the symbol 1, or unity, to represent the Universe, and let us understand it as comprehending every

conceivable class of objects whether actually existing or not, being premised that the same individual may be found in

it

more than one quality in

class,

inasmuch

common with

as

it

may

possess

other individuals.

more than one

Let us employ the

Z, to represent the individual members of classes, applying to every member of one class, as members of that to every member of another class as particular class, and

letters

X, Y,

X

Y

members

of such class, and so on, according to the received lan

guage of

treatises

Further

let

on Logic.

us conceive a class of symbols x, y,

z,

possessed

of the following character.

The symbol x

operating upon any subject comprehending

individuals or classes, shall be supposed to select from that

Xs which

In like manner the symbol shall be supposed to select from y, operating upon any subject, subject

it

all

all

the

it

contains.

individuals of the class

Y which

are comprised in

it,

and

so on.

When verse) to

no subject is expressed, we shall suppose 1 (the Uni be the subject understood, so that we shall have x = x

(1),

the meaning of either term being the selection from the Universe of all the Xs which it contains, and the result of the operation

16

FIRST PRINCIPLES.

being in common language, the class X, each member is an X.

From

i.

e.

the class of which

these premises

it will follow, that the product xy will represent, in succession, the selection of the class Y, and the selection from the class of such individuals of the class as

Y

are contained in

both

Xs and

represent a

it,

Ys.

X

the result being the class whose And in like manner the

members

product xyz

compound operation

ments are the selection of the such individuals of the class

of which the successive ele

class

Y as

are will

Z, the selection from are contained in

it,

it

of

and the

from the result thus obtained of all the individuals of which it contains, the final result being the class common to X, Y, and Z.

selection

X

the class

From

the nature of the operation which the symbols x, y,

are conceived to represent,

symbols.

An

we

shall designate

them

z,

as elective

expression in which they are involved will be and an equation of which the mem

called an elective function,

bers are elective functions, will be termed an elective equation. It will not be necessary that we should here enter into the analysis of that mental operation

which we have represented by the elective symbol. It is not an aqt of Abstraction according to the common acceptation of that term, because we never lose sight of the concrete, but it may probably be referred .to an ex ercise of the faculties of Comparison and Attention. Our present concern is rather with the laws of combination and of succession,

by which

its

results are governed,

and of these

it

will suffice to

notice the following.

The

result of an act of election is independent of the or of the subject. classification grouping Thus it is indifferent whether from a group of objects con sidered as a whole, we select the class X, or whether we divide 1st.

the group into two parts, select the Xs from them separately, and then connect the results in one aggregate conception. may express this law mathematically by the equation

We

x (u +

v)

= xu + xv,

17

FIRST PRINCIPLES. representing the undivided subject, component parts of it.

u

i

v

2nd.

It is indifferent in

and u and

what order two successive

the

v

acts of

election are performed.

Whether from

the class of animals

we

select sheep,

and from

the sheep those which are horned, or whether from the class of animals we select the horned, and from these such as are sheep,

the result

is

In either case we arrive

unaffected.

at the class

horned sheep.

The

symbolical expression of this law

is

xy = yx.

The result of a given act of election performed twice, number of times in succession, is the result of the same

3rd.

or any

performed once. If from a group of objects we select the Xs, we obtain a class If we repeat the operation of which all the members are Xs. act

no further change will ensue Thus we have

on

this class

we

take the whole.

xx = s*

or

:

in selecting the

Xs

x,

= x

;

and supposing the same operation to be n times performed, we have x n = X)

which

is

the mathematical expression of the law above stated.*

The laws we have

established under the symbolical forms

x (u +

*

The

office of

in the class

X.

= xu + xv xy = yx *n = * v)

the elective symbol

Let the class

whatever the class

Y may be,

X

we

office

least of its

is

(3), individuals comprehended

the

known

"

;

then,

y.

which x performs is now equivalent to the symbol interpretations, and the index law (3) gives

+ which

=

),

(2),

be supposed to embrace the universe have xy

The

x, is to select

(1

= +,

property of that symbol.

-f

,

in

one at

18

F1KST PRINCIPLES.

From

are sufficient for the basis of a Calculus. it

the

first

of these,

appears that elective symbols are distributive, from the second

that they are commutative/ properties

which they possess

in

common with symbols of quantity, and in virtue of which, all the processes of common algebra are applicable to the present The one and sufficient axiom involved in this appli system. cation

is

that equivalent operations

performed upon equivalent

subjects produce equivalent results.*

The

third law (3)

we

shall

denominate the index law.

It is

symbols, and will be found of great impor tance in enabling us to reduce our results to forms meet for peculiar to elective

interpretation.

From

the circumstance that the processes of algebra

applied to the present system, it is

may be

not to be inferred that the

interpretation of an elective equation will be unaffected by such The expression of a truth cannot be negatived by processes. * It is generally asserted by writers on Logic, that all reasoning ultimately What depends on an application of the dictum of Aristotle, de omni et nullo. ever is predicated universally of any class of things, may be predicated in like "

comprehended in that class." But it is agreed that this not immediately applicable in all cases, and that in a majority of What are the instances, a certain previous process of reduction is necessary. elements involved in that process of reduction? Clearly they are as much

manner

of any thing

dictum

is

a part of general reasoning as the dictum itself. Another mode of considering the subject resolves

all reasoning into an appli cation of one or other of the following canons, viz. 1 If two terms agree with one and the same third, they agree with each .

other. 2.

If one term agrees,

and another disagrees, with one and the same

third,

these two disagree with each other. But the application of these canons depends on mental acts equivalent to have to those which are involved in the before-named process of reduction.

We

select individuals

from

classes, to convert propositions, &c., before

we can

avail

of the process of reasoning is insuffi cient, which does not represent, as well the laws of the operation which the mind performs in that process, as the primary truths which it recognises and

ourselves of their guidance.

Any account

applies. It is presumed that the laws in question are adequately represented by the fundamental equations of the present Calculus. The proof of this will be found

in its

and of exhibiting in the results of capability of expressing propositions, that may be arrived at by ordinary reasoning. processes, every result

its

FIRST PRINCIPLES.

The equation are equivalent, member x, and we have

may be Y and Z

a legitimate operation, but it y = z implies that the classes for

member.

Multiply

it

by a

factor

xy =

limited.

xz,

which expresses that the individuals which are common and Z, and vice and are also common to classes

X

Y

19

X

to the versd.

This is a perfectly legitimate inference, but the fact which it declares is a less general one than was asserted in the original proposition.

OF EXPRESSION AND INTERPRETATION.

A

Proposition is a sentence which either affirms or denies, No creature is independent.

as,

All

men

are

mortal,

A Proposition has necessarily two terms,

as men, mortal; the former of which, the one spoken of, is called the subject the latter, or that which is affirmed or denied of the These are connected together subject, the predicate. by the copula is, or is not, or by some other modiEcation of the substantive verb. The substantive verb is the only verb recognised in Logic ; all others are resolvable by means of the verb to be and a participle or adjective, e.g. The Komans conquered"; the word conquered is both copula and predicate, being equivalent to "were (copula) victorious" (predicate). ;

"

A

Proposition must either be affirmative or negative, and must be also either universal or particular. Thus we reckon in all, four kinds of pure categorical Propositions. Universal- affirmative,

1st.

usually represented by A,

Ex. 2nd.

All

Ex. 3rd.

4th.

Xs

are Ys.

Universal-negative, usually represented

by E,

NoXsareYs.

Particular-affirmative, usually represented

by Ex. Some Xs are Ys.

Particular-negative, usually represented

I,

by O,*

Ex. Some Xs are not Ys. 1.

To

express the

class,

not-X, that

is,

the class

including

individuals that are not Xs.

all

The But

class

X and the class not-X together make the Universe. and the class X determined by the

the Universe

is 1,

symbol x 3 therefore the the symbol 1 - x. *

The above

and Whately.

is

taken, with

is

class

not-X

little variation,

will

be determined by

from the Treatises of Aldrich

OF EXPRESSION AND INTERPRETATION.

Hence

the office of the symbol

will

subject

select

to

be,

from

- x attached to a given

1

all

it

not-Xs which

the

it

contains.

And

product xy expresses the entire - x) class whose members are both Xs and Ys, the symbol y (1 will represent the class whose members are Ys but not Xs, in like

and the symbol are neither

Xs

(1

As

the

all

obvious

that

Universe

all

x) (1

-

all

whose members

Xs,

are Ys.

found in the

exist are

is

class

Y,

it

is

Ys, and from at once from the

all

the same as to select

Xs.

x

or

To

Xs

out of the Universe

select

Hence

3.

y) the entire class

the Proposition, All

Xs which to

these to select

-

nor Ys.

To express

2.

as the

manner,

(1

-

xy =

x,

=

0,

y)

(4).

No Xs

express the Proposition,

are Ys.

Xs are Ys, is the same as to assert that Now there are no terms common to the classes X and Y. all individuals common to those classes are represented by xy. Hence the Proposition that No Xs are Ys, is represented by To

assert that

no

the equation

xy =

(5).

Some Xs

4.

To

If

some Xs are Ys, there are some terms common to the and Y. Let those terms constitute a separate class

V,

to

express the Proposition,

are Ys.

X

classes

v,

0,

which there

shall

correspond a separate elective symbol

then v

And

as v includes all

we can

= xy,

terms

indifferently interpret

(6).

common it,

as

to the classes

Some Xs,

or

X

and Y,

Some Ys.

02

OF EXPRESSION AND INTERPRETATION.

Z 5. .

To express

In the

last

the Proposition, Some Xs are not Ys. equation write 1 y for y, and we have

the interpretation of v being indifferently

Some Xs

or

Some

not-Ys.

The above

equations involve the complete theory of cate gorical Propositions, and so far as respects the employment of analysis for the deduction of logical inferences, nothing

But

more

may be satisfactory to notice some par ticular forms deducible from the third and fourth equations, and can be desired.

it

susceptible of similar application.

If

we

multiply the equation (6) by x,

vx = x*y = xy by

Comparing with

(6),

we

we have

(3).

find v = tx,

or

-

v (1

And

x) = 0,

multiplying (6) by y,

(8).

and reducing

in a similar

manner,

we have v

= vy,

v (1 - y} = 0,

or

Comparing

(8)

and

(9),

vx = ty =

And

(9).

v,

(10).

further comparing (8) and (9) with (4),

we have

as the

equivalent of this system of equations the Propositions

All All

The system

Vs Vs

are

Xs\

are

YsJ

(10) might be used to replace (6), or the single

equation

vx = vy,

(11),

vx the interpretation, Some Xs, and to vy the interpretation, Some Ys. But it will be observed that

might be used, assigning

to

23

ON EXPRESSION AND INTERPRETATION.

system does not express quite so much as the single equa from which it is derived. Both, indeed, express the Proposition, Some Xs are Ys, but the system (10) does not this

tion (6),

imply that the and Y. to

class

V

includes all the terms that are

common

X

In like manner, from the equation (7) which expresses the Proposition Some Xs are not Ys, we may deduce the system ra?

= c(l - y) =

(12),

t>,

-

Since in which the interpretation of v (1 y) is Some not-Ys. in this case vy = 0, we must of course be careful not to in terpret vy as If

we

Some Ys.

multiply the

first

equation of the system (12),

ex

by

y,

(I

viz.

y),

we have

.-.

which

-v

-

is

a

form that

vxy =

t-y (1

vxy =

0,

-

y);

(13),

will occasionally present itself.

necessary to revert to the primitive

pret this, for the condition that vx represents

us by virtue of

(5), that its

It is not

equation in order to inter

Some Xs, shews

import will be

Some Xs the subject comprising all the

are not Ys,

Xs

that are found in the class

Universally in these cases, difference of

V.

fosm implies a dif

ference of interpretation with respect to the auxiliary symbol

and each form

is

interpretable by

r,

itself.

Further, these differences do not introduce into the Calculus a needless perplexity.

a precision

be seen that they give conclusions, which could not

It will hereafter

and a definiteness

to

its

otherwise be secured. Finally,

we may remark

that

all

the equations by which

are deducible from

any one general equation, expressing any one general Proposition, from which those particular Propositions arc necessary deductions. particular truths are expressed,

OF EXPRESSION AND INTERPRETATION.

24

This has been partially shewn already, but exemplified in the following scheme.

x =

The general equation implies that the classes

member

;

X

fully

y,

Y

and

much more

it is

are equivalent,

member

for

that every individual belonging to the one, belongs

Multiply the equation by x, and

to the other also.

we have

= xy x = xy,

z* /.

;

which implies, by (4), that all Xs are Ys. equation by y, and we have in like manner

Multiply the same

y = xy* the import of which

is,

that all

Ys

are Xs.

Take

equations, the latter for instance, and writing (i

we may quantity, will be

regard is

it

sought

as

to

- *) = y

it

either of these

under the form

o,

an equation in which y, an be expressed in terms of x.

shewn when we come

unknown

Now

it

to treat of the Solution of Elective

Equations (and the result may here be verified by substitution) that the most general solution of this equation is

y =

*>*,

which implies that All Ys are Xs, and Multiply by x and we have = vx, vy

that

Some Xs

are Ys.

y

which

indifferently implies that

some Ys are Xs and some

Xs

are Ys, being the particular form at which we before arrived. For convenience of reference the above and some other results

have been

classified

in

the annexed Table,

the

first

column of which contains propositions, the second equations, and the third the conditions of final interpretation. It is to be observed, that the auxiliary equations which are given in this

column are not independent

:

they are implied

either

in the equations of the second column, or in the condition for

OF EXPRESSION AND INTERPRETATION. the interpretation of

v.

But

it

has been thought better to write

And it is them separately, for greater ease and convenience. forms different three further to be borne in mind, that although of the particular proposi are given for the expression of each in the first form. included tions, everything is really

TABLE. The

class

X

The

class

not-X

All

Xs

are

Ys i

All

Ys

are

Xs J

All

Xs

are

Ys

No Xs

are

Ys

/vx i-

NoYsareXs

~

vx =

Somenot-XsareYsr C

Some Xs

are

Ys |

v = some

v = xy

or vx = vy

= y) [or vx (1

C

Some Xs

are not Ys ] I

=

v (1 - *) =

,

^

- some Xs

v (1 - x)

=

-

x (1 v y) or vx = v (1 - y) or vxy =

0.

some not-Xs 0.

Xs

or

some Ys

vx = some Xs, vy = some Ys = 0, v (1 - y) = 0. v x) (1

v = some Xs, or some not-Ys vx = some Xs, v (1 - y) = some not-Ys v (1 - x) = 0, vy = 0.

OF THE CONVERSION OF PROPOSITIONS.

A

Proposition is said to be converted when its terms are transposed nothing more is done, this is called simple conversion e.g.

;

when

;

No virtuous man is a tyrant, No tyrant is a virtuous man.

is

converted into

Logicians also recognise conversion per accidens, or by limitation, e.g. All birds are animals, is converted into

Some animals

And

Every poet

He who is

E

are birds.

conversion by contraposition or negation, as is

a

not a

man of genius, converted into man of genius is not a poet.

In one of these three ways every Proposition may be illatively converted, and I simply, A and O by negation, A and E by limitation.

The primary

canonical

viz.

forms already determined for the

expression of Propositions, are

Xs

are Ys,

No Xs

are Ys,

All

On

x

(1

-

=

0,

xy =

0,

y)

Some Xs

are Ys,

v = xy,

Some Xs

are not Ys,

v

we

=x

(1

A. .

.

.

.

E. I.

-

y) ... .0.

E

and I are sym examining metrical with respect to x and y, so that x being changed into y, and y into x, the equations remain Hence E and I these,

perceive that

unchanged.

may be

interpreted into

No Ys

are Xs,

Some Ys

are Xs,

Thus we have the known

respectively. that particular affirmative

admit of simple conversion.

rule of the Logicians,

and universal negative Propositions

OF THE CONVERSION OF PROPOSITIONS.

The equations

Now obtained

be written in the forms

are precisely the forms

these if

A and O may

we had -

in those equations

which we should have changed x into

1

-

y,

in

which would have represented the changing the original Propositions of the Xs into not-Ys, and the

Ys

into not-Xs, the resulting Propositions being

and y into

1

x,

All not-Ys are not-Xs,

Some not-Ys

are not not-Xs (a).

Or we may, by simply inverting the order of the second member of 0, and writing it in the form v = (1 - y)

interpret

it

by

is really

another form of

are Xs,

Hence

(a).

follows the rule,

and particular negative Propositions

that universal affirmative

admit of negative conversion,

by

t

I into

Some not-Ys which

x

factors in the

or, as

it is

also

termed, conversion

contraposition.

The

equations

A

and E, written in the forms (1

-

x=

0,

yz=

0,

y)

give on solution the respective forms

x = vy, x = v the correctness of which

may

(1

-

y),

be shewn by substituting these

values of x in the equations to which they belong, and observing that those equations are satisfied quite independently of the nature

of the symbol

v.

The

first

solution

Some Ys and the second

may be

are Xs,

into

Some not-Ys

are Xs.

interpreted into

OF THE CONVERSION OF PROPOSITIONS.

23

From which

it

appears that universal-affirmative, and universal-

negative Propositions are convertible by limitation, or, as

it

has

been termed, per accidens. The above are the laws of Conversion recognized by Abp. Whately. Writers differ however as to the admissibility of negative conversion. The question depends on whether we will consent to use such terms as not-X, not-Y. Agreeing with those

who

faulty

and defective.

think that such terms ought to be admitted, even although they change the kind of the Proposition, I am con strained to observe that the present classification of them is into All

Ys

No Xs

Thus the conversion of

are not-Xs, though perfectly legitimate,

cognised in the above scheme.

It

may

are Ys,

is

not re

therefore be proper to

examine the subject somewhat more fully. Should we endeavour, from the system of equations we have obtained, to deduce the laws not only of the conversion, but also of the general transformation of propositions, we should be led to recognise the following distinct elements, each connected

with a distinct mathematical process. 1st.

or

The negation into X.

of a term,

i. e.

the changing of

X into not-X,

not-X 2nd.

The

of a

translation

Proposition

we should change Xs are Ys into Some Xs

from one kind

to

another, as if

All

which would be lawful All

Xs

are

Ys

;

are

A

Ys

into I,

or

into

No Xs

are

A

Y.

into

E,

which would be unlawful. 3rd.

The simple conversion

The

conditions in obedience to which these processes

lawfully be performed,

of a Proposition.

may be deduced from

which Propositions are expressed. We have All Xs are Ys x

No Xs

are

Ys

(\

-

may

the equations by

y) =

0.

A,

xy =

0.

E.

OF THE CONVERSION OF PROPOSITIONS. Write

E

in the

form

by

-0 -y)} A into

All

Xs

*{i and

it is

so that

interpretable

o,

are not-Ys,

we may change

No Xs In like manner

Ys

are

A

into All

interpreted by

No Xs so that

=

Xs

E

are not-Ys.

gives

are not-Ys,

we may change All

From

Xs

are

Ys

into

we have

these cases

affirmative Proposition

No Xs

are not-Ys.

the following Rule

:

A universal-

convertible into a universal-negative,

is

and, vice versd, by negation of the predicate.

Again, we have

Some Xs

are

Some Xs

are not

These equations only presence of the term

Ys

v

Ys

= x

....

from those

differ

= xy,

last

(1

y).

considered by the

The same reasoning

v.

-

therefore applies,

and we have the Rule

A particular-affirmative ticular-negative, and

convertible into a par by negation of the predicate. is

proposition

vice versd,

Assuming the universal Propositions All

Xs

No Xs Multiplying by

v,

we

x

are Ysare

-

(\

Ys

y}

=

xy =

find

vx(\ - y) = oxy -

0,

0,

which are interpretable into

Some Xs

are

Some Xs

are not Ys.

Ys

1, .

.

O.

0, 0.

OF THE CONVERSION OF PROPOSITIONS.

30

Hence

a universal-affirmative

convertible into a particular-

is

and a universal-negative into

affirmative,

a particular-negative

without negation of subject or predicate.

Combining the above with the already proved rule of simple conversion,

we

arrive at the following system of independent

laws of transformation. 1st.

An

may be changed

affirmative Proposition

into

its

cor

responding negative (A into E, or I into O), and vice versa,

by negation of the

A

2nd.

predicate. /

universal Proposition

may be changed

sponding particular Proposition, (A 3rd.

into I, or

E

into

its

corre

into O).

In a particular-affirmative, or universal-negative Propo may be mutually converted.

sition, the terms

Wherein negation and

vice versd,

and

of a term

is

the changing of

X into not-X,

not to be understood as affecting the kind

is

of the Proposition. Every lawful transformation

is

reducible to the above rules.

Thus we have All

Xs

No Xs No

are Ys, are not-Ys

not-Ys are

Xs

by

1st rule,

by 3rd

rule,

All not-Ys are not-Xs by 1st rule,

which

is

an example of negative conversion.

No Xs

are Ys,

No Ys

are

All

which

is

Ys

Xs

3rd rule,

are not-Xs

1st rule,

the case already deduced.

Again,

OF SYLLOGISMS.

A

Syllogism consists of three Propositions, the last of which, called the is a logical consequence of the two former, called the premises

conclusion,

;

Prennscs,

Ys

are Xs. z& All Zs are Xs.

(All

\^

Conclusion,

^^

Every syllogism has three and only three terms, whereof that which is the subject of the conclusion is called the minor term, the predicate of the conclusion, the major term, and the remaining term common to both premises, Thus, in ths above formula,

the middle term.

major term,

Y

Z

is

the minor term,

X

the

the middle term.

The figure of a syllogism consists in the situation of the middle term with The varieties of figure are exhibited respect to the terms of the conclusion. the annexed scheme.

m

2nd

3rd Fig.

4th Fig.

YX

XY

YX

XY

ZY ZX

ZY ZX

YZ

YZ ZX

1st Fig.

When we

Fig.

ZX

designate the three propositions of a syllogism by their usual I, O), and in their actual order, we are said to determine

symbols (A, E, the

mood

of the syllogism.

illustration, belongs to the

Thus the syllogism given above, by way

mood

AAA

The moods of all syllogisms commonly received by the vowels in the following mnemonic verses. Fig.

1.

Fig.

2.

Fig.

3.

as valid,

are represented

bArbArA, cElArEnt, dArll, fErlO que pr roris. cEsArE, cAmEstrEs, fEstlnQ bArOkO, secunda?. Tertia dArAptl, dlsAmls, dAtlsI, fElAptOn,

bOkArdO, fErlsO, habet Fig. 4.

of

in the first figure.

:

quarta insuper addit.

brAmAntlp, cAmEnEs, dlmArls, fEsapO, frEsIsOn.

TriE equation by which cerning the

classes

symbols x and

y,

X

we

and Y,

express any Proposition con is

an equation between the

and the equation by which we express any

OF SYLLOGISMS.

Y

and Z, is an equation Proposition concerning the classes and z. from two such equations the If between symbols y

we

eliminate y, the result,

equation between x and

do not vanish, will be an

if it

and

*,

will

constitute

the third

be interpretable

X and Z.

Proposition concerning the classes

And

it

into

a

will then

or Conclusion, of a Syllogism,

member,

of which the two given Propositions are the premises.

The

result of the elimination of

y from the equations

y+*-o, =

0,

ab - a b =

0,

ay is

the equation

+

Now the equations of Propositions being of the first order with reference to each of the variables involved, all the cases of elimination which we shall have to consider, will be re d

ducible to the above case,

the

replaced by functions of x,

and the auxiliary symbol

As

z,

constants

a, b,

,

b

,

being

v.

the choice of equations for the expression of our premises, the only restriction is, that the equations must not both be of the form ay = 0, for in such cases elimination would to

be impossible. necessary to

we

When

both equations are of

solve one of them,

choose for this purpose.

the form xy

=

is

it

this form, it is

indifferent

which we

If that

select

which is

of

0, its solution is

y if

and

(!-*),

(16),

of the form (1 - x) y = 0, the solution will be

y =

vx,

(17),

and these are the only cases which can

The reason

arise.

of this exception will appear in the sequel.

For the sake of uniformity we

shall,

in the

expression of

confine ourselves to the forms particular propositions,

DX = 0y,

vx = v

(1

-

y\

Some Xs

are Ys,

Some Xs

are not Ys,

S3

OF SYLLOGISMS.

a closer analogy with (16) and (17), than the other forms which might be used.

These have

Between the forms about

to

be developed, and the Aristotelian

canons, some points of difference will occasionally be observed, of which it may be proper to forewarn the reader.

To

the right understanding of these it is proper to remark, that the essential structure of a Syllogism is, in some measure, arbitrary. Supposing the order of the premises to be fixed,

and the

major and the minor term to be purely a matter of choice which of

distinction of the

thereby determined,

it

is

the two shall have precedence in the Conclusion.

have settled is

it

this

question in

that this

clear,

is

Logicians favour of the minor term, but

a convention.

Had

that the major term should have the first clusion, a logical

been agreed place in the con it

scheme might have been constructed,

less

convenient in some cases than the existing one, but superior

What

in others.

Convenience but

it is

Now

to

it

lost in barbara, it

would gain

in Iramantip.

perhaps in favour of the adopted arrangement,* be remembered that it is merely an arrangement. is

method we shall exhibit, not having reference one scheme of arrangement more than to another, will

to

the

always give the more general conclusion, regard being paid only to its abstract lawfulness, considered as a result of pure reasoning.

us the

to

And

therefore

we

shall

sometimes have presented

of conclusions, which

spectacle

a logician

would

pronounce informal, but never of such as a reasoning being would account false.

The

Aristotelian canons, however, beside restricting the order

of the terms of a conclusion, limit their this limitation is of

nature also;

and

more consequence than the former.

We

may, by a change of *

figure, replace the particular conclusion

The contrary view was maintained by Hobbes. The question Hallam s Introduction to the Literature of Europe,

fairly discussed in

is

very

vol. in.

In the rhetorical use of Syllogism, the advantage appears to rest with the rejected form.

p. 309.

04

ON SYLLOGISMS.

of Iramantipy

by the general conclusion of barbara; but cannot thus reduce to rule such inferences, as

Some not-Xs Yet

we

are not Ys.

there are cases in which such inferences

may

lawfully

be drawn, and in unrestricted argument they are of frequent Now if an inference of this, or of any other occurrence. lawful in

is

kind,

it

itself,

be exhibited in the results

will

of our method.

We

may by

restricting the

canon of interpretation confine

our expressed results within the limits of the scholastic logic; but this would only be to restrict ourselves to the use of a part of the conclusions to which our analysis entitles us.

The

classification

and we which

name

shall

it

we

shall

adopt will be purely mathematical,

afterwards consider the logical arrangement to It will

corresponds.

be

sufficient, for reference, to

the premises and the Figure in which they are found.

CLASS

1st.

Forms

in

which

v does not enter.

Those which admit of an inference are

AE; EA, Ex. order),

Fig. 2;

A A, A A,

All

Ys

A A, AE,

Fig.

1,

Fig.

4.

Fig.

are Xs,

y

(1

-

z (1 -

Eliminating y by (lo)

x) = 0,

or (1 - x)

y)=

or

1 ;

.-.

mode

0,

zy

y=

0,

=

Q.

- z

we have z (1 - x)

convenient

Fig.

and, by mutation of premises (change of

All Zs are Ys,

A

AA, EA,

4.

=

0,

All Zs are Xs.

of effecting the elimination,

is

to write

the equation of the premises, so that y shall appear only as a factor of one member in the first equation, and only as a factor of the opposite member in the second equation, and

then to multiply the equations, omitting the

we

shall adopt.

y.

This method

OF SYLLOGISMS.

Ex.

AE,

Fig. 2, and,

All

Xs

No

Zs are Ys,

35

E A,

by mutation of premises, x (1 - y) = zy =

are Ys,

x

or

0,

=

Fig,

2.

xy

= zy

0,

;*r=0 .*.

The only

case in

which there

is

No

no inference

AllXsareYs,

*(l-/)=0,

All Zs are Ys,

z (1 - y) = o,

Zs are Xs.

A A,

is

Fig. 2,

x = xy zy = z 2 = #Z .

When

CLASS 2nd.

v

is

.

0=0.

introduced by the solution of an

equation.

The lawful cases directly or indirectly* determinable Aristotelian Rules are AE, A, AE, EA, Fig. 1;

A

EA,

by the Fig. 3;

Fig. 4.

The

lawful cases not so determinable, are

EE,

Fig.

and, by mutation of premises,

EA,

Fig 2; EE, Fig. 3; EE, Fig.

Ex.

AE,

Fig.

1,

All

Ys

No

Zs are Ys,

are

Xs,

y

1

;

EE,

4.

(1

-

zy

= x)

0,

y = vx

=0,

= zy

Fig.

4.

(a)

= vzz :.

Some Xs

are not Zs.

The reason why we cannot are not-Xs,

is

that

by

into Some Zs interpret vzz = the very terms of the first equation (a)

the interpretation of vx is fixed, as Some Xs v is regarded ; as the representative of Some, only with reference to the class *

X.

We

say

directly or indirectly,

mutation or conversion of premises being

some instances required. Thus, AE (fig. 1) is resolvable by Fesapo (fig. 4), or by Ferio (fig. 1). Aristotle and his followers rejected the fourth figure as only a modification of the first, but this being a mere question of form, either scheme may be termed Aristotelian. in

36

OF SYLLOGISMS.

For the reason of our employing a solution of one of the Had primitive equations, see the remarks on (16) and (17).

we

solved the second equation instead of the

we should

first,

have had

(l-a?)y=0, tj(l-*0 =

y,

(a),

v(\-z) (l-*) =

0,

(),

Some not-Zs

.*.

Here

to

it is

be observed, that the second equation

the meaning of v(\ -2), as of the result (b)

Y

are Xs.

Some

not-Zs.

that all the not-Zs

is,

The full meaning which are found in

X, and

are found in the class

is

evident that

the

class

this

could not have been expressed in any other way.

Ex.

AA,

Fig.

All

Ys

are Xs,

y (1

All

Ys

are Zs,

y

2.

(a) fixes

it

3.

(1

-

a?)

2)

=

0,

=

0,

y = vx Q = y(l-z) = vx(\ :.

Some Xs

z)

are Zs.

solved the second equation, we should have had The form of the final equation as our result, Some Zs are Xs.

Had we

particularizes

remark

The is

is

what Xs or what Zs are referred

to,

and

this

general.

EE, Fig. 1, and, by mutation, EE, Fig. 4, a of lawful case not determinable by the Aris an example following,

totelian Rules.

No Ys No Zs

xy = = zy

are Xs, are Ys,

0, 0,

:.

CLASS

3rd.

When

but not introduced by

v

is

met with

solution.

= xy y = v (1 = v (1 -

Some not-Zs

2)

2)

x

are not Xs.

in one of the equations,

OF SYLLOGISMS.

37

The

lawful cases determinable directly or indirectly by the Aristotelian Rules, are AI, El, Fig. 1 ; AO, El, OA, IE,

AI, AO, El, EO, IA, IE, OA, OE, Fig. 3; IA, IE,

Fig. 2; 4.

Fig.

Those not

The I A,

so determinable are

OA, AI, El, AO, Ex.

1.

EO,

;

Fig. 4.

possible, are

is

AO, EO,

AI, EO, IA, OE, Fig. 2; OA, OE,

Fig. 1; 4.

Fig.

AI, Fig. All

1

Fig.

which no inference

cases in

IE,

OE,

Ys

1,

and, by mutation,

are Xs,

Some Zs

y

I A, Fig. 4.

-

(1

are Ys,

x) =

= vy

vz

vz(l -

/.

Ex.

Xs

All

AO,

2.

Fig. 2, and,

are Ys,

Some Zs

*)= Some Zs are Xs,

by mutation, OA, Fig.

x = xy

#(l-y)=o,

are not Ys,

vz =

2.

vy = v(\-z)

v(l-y\

tx = vx(\-z) vzz =

Some Zs

:.

The

interpretation of vz as

observed,

senting the proposition

The

OE,

Some Zs

cases not determinable

Fig.

1,

Some

the equation vz = v

in

and, by mutation,

Some Ys

No

are not

(1

Zs,

-

is

(l

by the

EO,

#)>

but

it

will

be

Aristotelian Rules are

Fig.

4.

vy = v

Xs,

Zs are Ys,

equation of the

-

it

are not Ys.

-

(1

x)

o = Zy

/.

The

implied,

y) considered as repre

= v

c

are not Xs.

(1

-

Some not-Xs

x) z

are not Zs.

premiss here permits us to interpret does not enable us to interpret vz. first-

D2

38

OF SYLLOGISMS.

Of

which no inference

in

cases

is

possible,

we

take

as

examples

AO,

1,

Fig.

and, by mutation,

AllYsareXs, Some Zs are not Ys,

OA,

Fig. 4,

y(\-x)=Q

2/(l-z)=0, vz - v (1 - y)

v(l -z) = vy

(a)

i>(l-*)(l-aO-0

)

= since the auxiliary equation in this case

it is satisfactory to

vz as

Some

do

Zs, but

it

so.

The equation

classes

it is

=

0.

(If),

(a), it is seen, defines

-

z), so that we might and content ourselves with

does not define v

stop at the result of elimination

saying, that

z)

not necessary to perform this reduction, but

it is

Practically

-

v (1

is

(1

not interpretable into a relation between the

X and Z.

Take

as

IA, Fig.

a second

example AT, Fig.

and, by mutation,

2,

2,

AllXsareYs, Some Zs are Ys,

x = xy vy = vz

s(l-y)=0, vz = vy,

vz = vxz

-z)x=Q

0(1

= the auxiliary equation in this case being 0(1

Indeed in every case in

this class,

-

*)=

0,

0.

which no inference

in

the form is possible, the result of elimination is reducible to = 0. Examples therefore need not be multiplied.

CLASS

No

1st.

When

inference

tinction class.

4th.

among The two

When

is

v enters into both equations,

possible in

any

case,

but there exists a

the unlawful cases which

is

dis

peculiar to this

divisions are,

the

result of elimination

to the form auxiliary equations

=

0.

is

The

reducible

by the

cases are II, OI,

39

OF SYLLOGISMS. Fig. 1;

II,

00,

Fig. 2

II,

;

00,

10, 01,

Fig. 3;

Fig.

When

2nd.

the result of elimination

=

auxiliary equations to the form

The

are 1O,

cases

OO,

is

not reducible by the

0.

10, OI, Fig. 2; OI,

Fig. 1;

Let us take

an example of the former case, II, Fig.

as

Some Xs

are Ys,

vx = vy,

Some Zs

are Ys,

vz

=

Now

v

v y,

y = vz

x

Substituting

vx =

=

vz

v,

v

.

we have vv = vv

=

.-.

As an example

,

0.

of the latter case, let us take 10, Fig.

Some Ys are Xs, Some Zs are not Ys,

vy

v (1

-

v (1

y),

reducible.

The above which

it

classification is

We

now

shall

=vy

-z}-

vv

x

is,

that the classes v

indeed evident.

purely founded on mathematical

inquire what

cases of the first class

is the*

logical division

comprehend

two universal premises,

be drawn.

We

the second,

a

all

those in

universal conclusion

see that they include

barbara and celarent in the in

as is

z)

corresponds.

lawful

which, from

may

interpretation

common member,

have nc

distinctions.

Its

-

- x) = 0, v (1 - z) = 0, (1 this form that all similar

Now the auxiliary equations being v It is to the above reduces to vv = 0. cases are

1 ,

vy = vx

= vx,

vz =

vv (I

The

vv z

the auxiliary equations v (1 - x) = 0, v (1 - z) = 0,

give

v

3.

vx = vy

vv

to

00,

4.

Fig.

and

10,

II,

4.

first figure,

the premises of of cesare and camcstrcs

and of bramantip and camcnes in the fourth.

40

OF SYLLOGISMS,

The premises of bramantip

are included, because they admit of an universal conclusion, not in the same although figure.

The

lawful cases of the second class

a particular

conclusion only

are those in

which

deducible from two universal

is

premises.

The

lawful cases of the third class

conclusion

are those in

deducible from two premises, universal and the other particular.

The

which a

one of which

is

is

fourth class has no lawful cases.

Among

the cases in which no inference of any kind

is

pos

we

find six in the fourth class distinguishable from the others by the circumstance, that the result of elimination does sible,

not assume the form

Some Ys

=

0.

The

cases are

are Xs, f Some Ys are not Xs,] f Some Xs are Ys, Zs are not Zs are not \Some Ys, j (Some Zs are not Ys,/ Ys,J \Some f

and

the

"\

"I

three

others

which are

obtained

by mutation of

premises. It might be presumed that some logical peculiarity would be found to answer to the mathematical peculiarity which we

have noticed, and in fact there exists a very remarkable one. If we examine each pair of premises in the above scheme, we shall find that there is virtually

no middle term,

of comparison, in any of them. the individuals spoken of in the

belong

to

the class

Y, but

i.

Thus, in the first

premiss

those spoken

e.

no medium

first

example,

are asserted to

of in

the second

premiss are virtually asserted to belong to the class not-Y: nor can we by any lawful transformation or conversion alter

The comparison will one premiss, and with the

this state of things.

the class

Y

in

still

class

be made with

not-Y in the

other.

Now a

in every case beside the above six, there will be found

middle term, either expressed or implied.

of the most difficult cases.

I

select

two

41

OF SYLLOGISMS. In

AO,

Fig.

viz.

1,

All

Ys

are Xs,

Some Zs

we

are not Ys,

have, by negative conversion of the

first

premiss,

All not-Xs are not-Ys,

Some Zs and the middle term

is

Again, in EO, Fig.

now

seen to be not-Y.

1,

No Ys

.

are not Ys,

are

Some Zs a proved conversion

of the

Xs,

are not Ys, first

premiss (see Conversion of

Propositions), gives

All

Xs

are not-Ys,

Some Zs

are not-Ys,

and the middle term, the true medium of comparison,

is

plainly

although as the not-Ys in the one premiss may be different from those in the other, no conclusion can be drawn. not-Y,

The mathematical

condition in question, therefore, the irre= final the of to the form equation 0, ducibility adequately represents the logical condition of there being no middle term, or

common medium I

am

not aware

of comparison, in the given premises. that

the

distinction

occasioned

by the

presence or absence of a middle term, in the strict sense here understood, has been noticed by logicians before.

The

dis

though real and deserving attention, is indeed by no means an obvious one, and it would have been unnoticed

tinction,

in the present instance but for the peculiarity of

its

mathe

matical expression.

What

appears to be novel in the above case

is

the proof

of the existence of combinations of premises in which there

OF SYLLOGISMS.

When such a medium absolutely no medium of comparison. of comparison, or true middle term, does exist, the condition

is

that

its

ceed

in

quantification

both premises

together

shewn by Professor De Morgan

clearly

ex

shall

and

quantification as a single whole, has been ably

its

to

be necessary to

And lawful inference (Cambridge Memoirs, Vol. vm. Part 3). this is undoubtedly the true principle of the Syllogism, viewed from the standing-point of Arithmetic. I have said that

it

would be

possible to impose conditions

of interpretation which should restrict the results of this cal culus to the Aristotelian forms. Those conditions would be,

That we should agree not

1st.

to interpret the

forms v(l -

x),

v(l-z). That we should agree

2ndly.

to reject

which the order of the terms should

every interpretation in

violate the Aristotelian rule.

Or, instead of the second condition,

it

might be agreed

that,

the conclusion being determined, the order of the premises should,

if

necessary, be changed, so as to

make

the syllogism

formal.

From

the general character of the system

it is

indeed plain,

may be made to represent any conceivable scheme of logic, by imposing the conditions proper to the case con that

it

templated.

We have

found it, in a certain class of cases, to be necessary the two equations expressive of universal Propo to replace sitions,

that

it

done * It

by their solutions; and it may be proper to remark, would have been allowable in all instances to have

this,*

may we

Barbara,

so

that every case of the Syllogism, without ex-

be satisfactory to illustrate this statement by an example.

should have All

Ys

are Xs,

y

All Zs are Ys,

z

z .

.

= vx =vy = vv x

All Zs are Xs.

In

43

OF SYLLOGISMS. ception,

might have been treated by equations comprised in

the general forms

y = vx, y = v (1 vy = vx, _ = vy v (i Or,

we may

y vx = or y + vx - v = = vy vx

or x),

- = vy + vx v

x),

multiply the resulting equation by

1

.

.

.

.

A,

.

.

. .

E,

.

.

.

.

.

.

.

.

- x, which

I,

O.

gives

.(l-.)-O, conclusion, All Zs are Xs. additional examples of the application of the system of equations in the text to the demonstration of general theorems, may not be inappropriate.

whence the same

Some

Let y be the term to be eliminated, and let x stand indifferently for either of the other symbols, then each of the equations of the premises of any given syllogism may be put in the form

+

ay if

the premiss

is

affirmative,

(a)

0,

and in the form

ay if it is

=

bx

+ 6(1-*) =

0,

(/3)

and b being either constant, or of the form v. To prove us examine each kind of proposition, making y successively

negative, a

this in detail, let

subject and predicate.

A, All Ys are Xs, All Xs are Ys, E,

I,

No Ys No Xs

y x

are Xs, are Ys,

Some Xs Some Ys

y

The

affirmative equations (y), equations (), (tj) and (0), to (/3).

vy

(<$)

is

vy

-

vx

-

_

(i

a?)

= =

0,

(y),

0,

(*),

= =

0,

vy = - x) =

_

It

0,

(),

0,

() (),

0,

vx

=v

3.)

=0

y),

(1

(0).

,

and the negative is seen that the two last negative equa of interpretation. In the former

and (), belong

a difference v (1

utility of the

v (1

v (1

vy

...

in the latter,

-

are Ys, are Xs,

tions are alike, but there

vx

xy

O, Some Ys are not Xs, Some Xs are not Ys,

The

-

to.

(a),

= Some not-Xs, - a?) = 0. (1

ar)

v

two general forms of

reference,

() and

from the following application. 1st. A conclusion drawn from two affirmative propositions By (a) we have for the given propositions, ay

ay

+

bx

-\-

bz

=

0, 0,

(/3),

is itself

will appear

affirmative.

44

OF SYLLOGISMS.

Perhaps the system we have actually employed is better, which v only may be employed,

as distinguishing the cases in

and eliminating

which

is

of the form (a)

A

2nd.

>

ab z

Hence,

.

_

a bx

if

there

drawn from an

conclusion

= is

,

a conclusion,

affirmative

it is

affirmative.

and a negative proposition

is

negative.

By

and

(a)

(/3),

we have

for the given propositions

ay

ay + .

which

is

of the form

A

3rd.

(/3)

.

-

a bx

b

conclusion draicn

+ -

ab (1

Hence the

.

(1

bx

0,

*)

=

0,

z)

=

0,

conclusion, if there

is

one,

is

negative.

from two

negative premises will involve a negation, predicate, and will therefore be inadmissible in

(no-X, not-Z) in both subject and the Aristotelian system, though just in itself.

For the premises being

+b ay + b ay

the conclusion will be ab (1

which

-

2)

(1 (1

-

-

a?)

z)

a b (1

= = -

0, 0,

ar)

=

0,

only interpretable into a proposition that has a negation in each term. 4th. Taking into account those syllogisms only, in ichich the conclusion is the most general, that can be deduced from the premises, if, in an Aristotelian is

minor premises be changed in quality (from affirmative to negative negative to affirmative), whether it be changed in quantity or not, no con clusion will be deducible in the same figure. syllogism, the

or

from

An

Aristotelian proposition does not admit a term of the form not-Z in the Now on changing the quantity of the minor proposition of a syllogism, it from the general form

subject, we transfer

ay to the general

form

+

bz

=

0,

+ & (1 - *) = 0, And therefore, in the

ay

see (a) and (/3), or vice versd. there will be a change from z to

equation of the conclusion,

But this is equivalent to the change of Z into not-Z, or not-Z into Z. Now the subject of the original conclusion must have involved a Z and not a not-Z, therefore the subject of the new conclusion will involve a not-Z, and the conclusion will not be admissible in the Aristotelian forms, except a change of Figure.

1

*,,

or vice versd.

by conversion, which would render necessary

Now the conclusions of this calculus are always the most general that can be drawn, and therefore the above demonstration must not be supposed to extend to a syllogism, in which a particular conclusion is deduced, when a universal one is possible. This is the case with bramantip only, among the Aristotelian forms, and therefore the transformation of bramantip into camenes, and vice versd, is

the case of restriction contemplated in the preliminary statement of the

theorem.

45

OF SYLLOGISMS. from those in which

must.

it

But

demonstration of

for the

above system

the certain general properties of the Syllogism,

from

is,

its

and from the mutual analogy of

simplicity,

forms, very convenient.

We

shall

apply

its

to the following

it

theorem.*

is

Given the three propositions of a Syllogism, prove that there but one order in which they can be legitimately arranged,

and determine that order. All the forms above given for the expression of propositions, are particular cases of the general form,

a + bx + cy =

0.

con

an Aristotelian syllogism, we substitute If for the minor premiss of in the same figure. deducible is tradictory, no conclusion case of bramantip, all the others It is here only necessary to examine the last the proposition. being determined by have AO, On changing the minor of bramantip to its contradictory, we its

5th.

Fig.

4,

and

this admits of

no legitimate inference.

Hence the theorem is true without exception. may in like manner be proved.

Many other

general theorems

* This communicated by the Rev. Charles Graves, elegant theorem was to whom the Fellow and Professor of Mathematics in Trinity College, Dublin, for a very his acknowledgments record grateful Author desires further to for some new of the former portion of this work, and examination judicious method. The example of Reduction ad impossible applications of the among the number

is

following

:

Reducend Mood, All Xs Baroko

1

are Ys,

Some Zs

axe not

Ys

Some Zs

are not

Xs

Reduct Mood, All Xs are Ys All Zs are Xs Barbara All Zs are

Ys

-

y

=

t>

w=v vz 1

* (\

* (1

-

-

y

x)

y)

- .r) - y) - x) (1

(1 (1

= vv = V (1 = * = 0.

*)

be the contradictory of the The conclusion of the reduct mood is &c. It may just be remarked that the Whence, minor premiss. suppressed one mathematical test of contradictory propositions is, that on eliminating seen to

the other elective symbol vanishes. symbol between their equations, involves no difficulty. Bokardo and Baroko of reduction The ostensive elective

= vy for the Professor Graves suggests the employment of the equation a: that on and are Xs remarks, All Ys, primary expression of the Proposition - y) = 0, the equation from 1 - y, we obtain .r (1 members both by multiplying which we set out in the text, and of which the previous one is a solution.

46

OF SYLLOGISMS.

Assume then

for the premises of the given syllogism, the

equations

then, eliminating

a + bx + cy =

0,

(18),

a + bz+ cy =

0,

(19),

we

shall

have for the conclusion

y>

ad - a c +

bc x - b cz = 0,

(20).

Now

taking this as one of our premises, and either of the original equations, suppose (18), as the other, if by elimination of a common term x, between them, we can obtain a result

equivalent to the remaining premiss (19), it will appear that there are more than one order in which the Propositions may

be lawfully written is

;

but

if

otherwise, one arrangement only

lawful.

Effecting then the elimination, 1

be (a + b z

+

c

we have

y}=

0,

(21),

Now on equivalent to (19) multiplied by a factor be. the of this value in factor the examining equations A, E, I, O, we find it in each case to be v or - v. But it is evident, which

is

given Proposition be mul derived from another equa factor,

that if an equation expressing a tiplied by an extraneous tion,

its

will

interpretation

Thus there

impossible.

either

will either

be

limited

be no result

or

rendered

at all, or the

result will be a limitation of the remaining Proposition. If,

however, one of the original equations were

x =

the factor be would be ation

of the

or

y, 1,

other premiss.

x - y =

0,

and would not

Hence

if

limit the interpret

the

first

member

of

syllogism should be understood to represent the double proposition All Xs are Ys, and All Ys are Xs, it would be a

indifferent written.

in

what order the remaining Propositions were

47

OF SYLLOGISMS.

more general form of the above investigation would be, the equations express the premises by

A to

a + bx + cy + dxy = a +b z + cy + d zy =

0,

(22),

0,

(23).

we should

After the double elimination of y and x (be

and

it

-

ad} (a +

bz +

would be seen that the

cy + d

zy)

factor be

=

find

;

- ad must in every

of meaning. case either vanish or express a limitation

The determination ficiently obvious.

of the order of the Propositions

is

suf

OF HYPOTHETICALS.

A

hypothetical Proposition is defined to be two or more categorical* united by a copula (or conjunction), and the different kinds of hypothetical Propositions are named from their respective conjunctions, viz. conditional (if)

disjunctive

(either, or), &c.

is

In conditionals, that categorical Proposition from which the other called the antecedent, that which results from it the consequent.

Of the 1st.

conditional syllogism there are two, and only

The

constructive,

two

results

formula?.

A is B, then C is D, A is B, therefore C is D.

If

But 2nd.

The

Destructive, If

But C

is

A is B, then

C

not D, therefore

is

D,

A is not B.

A

dilemma is a complex conditional syllogism, with several antecedents in the major, and a disjunctive minor.

IF we examine either of the forms of conditional syllogism above given, we shall see that the validity of the argument does not depend upon any considerations which have reference the terms A, B, C, D, considered as the representatives of individuals or 6f classes. may, in fact, represent the is B, C is D, Propositions and by the arbitrary symbols to

We

A

respectively,

following

X

Y

and express our syllogisms in such forms

as the

:

X But X If

Y is true, is true, therefore Y is true. is

Thus, what we have

true, then

to consider is not objects

and

classes

of objects, but the truths of Propositions, namely, of those

49

OF HYPOTHKTICALS,

which are embodied

in the terms of

elementary Propositions our hypothetical premises.

we the symbols X, Y, Z, representative of Propositions, elective symbols x, y, z, in the following appropriate the

To may

sense. shall comprehend hypothetical Universe, 1, circumstances. of able cases and conjunctures

The

The

elective

symbol x attached

such cases shall select is

true,

If

and similarly

we

conceiv

to

any subject expressive those cases in which the Proposition

for

Y

of

X

and Z.

contemplation of a given pro and hold in abeyance every other consideration,

confine ourselves to the

X,

position

all

then two cases only are conceivable, viz. and secondly that it Proposition is true,

first is

that the given

false*

As

these

of the Proposition, and cases together make up the Universe elective symbol x, the latter as the former is determined by the is

determined by the symbol

1

-

x.

But if other considerations are admitted, each of these cases less extensive, the will be resolvable into others, individually that a Proposition is either true or false, * It was upon the obvious principle future events, endeavoured that the Stoics, applying it to assertions respecting to their argument, that been has It Fate. of replied doctrine the to establish which is id quod the precise meaning of an abuse of the word true, involves true nor false." Copleston res est. An assertion respecting the future is neither however, pre on Necessity and Predestination, p. 36. Were the Stoic axiom, a given event will take place, sented under the form, It is either certain that fail to meet the difficulty. would above the not will it reply that ; or certain settle the The proper answer would be, that no merely verbal definition can of Nature. When we is the actual course and constitution what question, take place, or certain that affirm that it is either certain that an event will that the order of events is necessary, it will not take place, we tacitly assume that the state of things that the Future is but an evolution of the Present so But this (at least as re be. shall which that which is, completely determines Exhibited is the very question at issue. moral of conduct the agents) spects abuse of terms, an involve not does Stoic the reasoning under its proper form, "

;

but a petitio principii. of the doctrine of Necessity It should be added, that enlightened advocates and through the as end the appointed only in in the present day, viewing which are the reproa means, justly repudiate those practical 01 consequences of Fatalism.

50

ON HYPOTHETICALS.

number

of which will depend

upon the number of foreign con Thus if we associate the Propositions X number of conceivable cases will be found as

siderations admitted.

and Y, the

total

exhibited in the following scheme. Cases.

1st

2nd 3rd 4th If

Elective expressions.

X true, Y true X true, Y false X false, Y true X false, Y false

we add

x (I

xy (1

y)

- z) y

-*)(!-

(1

the elective expressions for the two

above cases the sum

is

x,

which

is

y)

(24)-

first

of the

the elective symbol appro being true independently

X

more general case of of any consideration of and if we add the elective expres ; sions in the two last cases together, the result is 1 - x which y priate to the

Y

is

of

the elective expression appropriate to the

X being

more general

case

false.

Thus the extent of the hypothetical Universe does not at all depend upon the number of circumstances which are taken

And

it is

to

those circumstances

may

be, the

into account.

be noted that however few or

sum

many

of the elective expressions

representing every conceivable case will be unity. Thus let us consider the three Propositions, X, It rains, Y, It hails,

Z, It freezes.

The

possible cases are the following

Cases.

:

Elective expressions.

and freezes, xyz and hails, but does not freeze xy (1 - z) rains and freezes, but does not hail xz (1 - y) freezes and hails, but does not rain yz (1 - x)

1st

It rains, hails,

2nd

It rains

3rd

It

4th

It

-

-

5th

It rains, but neither hails nor freezes

6th

but neither rains nor freezes y (1 - x) (1 - z) It freezes, but neither hails nor rains z (I - x)(l y) - x)(l It neither rains, hails, nor freezes (1 y) (1

7th 8th

x

(1

y) (1

z)

It hails,

1

= sum

z)

51

OF HYPOTHET1CALS. Expression of Hypothetical Propositions.

X

To

is true. express that a given Proposition 1 - x selects those cases in which the Proposi But if the Proposition is true, there are no is false.

The symbol tion

X

such cases in

its

hypothetical Universe, therefore 1

- x =

0,

x =

1,

or

(25).

X

is false. To express that a given Proposition The elective symbol x selects all those cases in

Proposition

is

and therefore

true,

x =

And

in every case,

0,

if

the Proposition

which the

is false,

(26).

having determined the elective expression

assert the truth of that appropriate to a given Proposition, we elective the expression to unity, and Proposition by equating its

by equating the same expression to 0. and Y, are simulta To express that two Propositions, falsehood

X

neously true.

The

elective

symbol appropriate

the equation sought

To

express that

to this case is xy, therefore

is

xy =

(27).

1,

two Propositions,

X and Y, are simultaneously

false.

The

condition will obviously be

(!-*)(! -y)= x + y

or

To

is

Y

is

*>

0,

(28).

is to

X

true, or the

two Propositions not true, that they are both false.

either one or the other of

assert that

it is

the elective expression appropriate to their both being - x} (1 the equation required is y), therefore

false is (1

(1

or

is

true.

assert that

true,

Now

xy =

express that either the Proposition

Proposition

To

-

-*)(!- sO=0, x + y - xy

=

1,

(29).

OF HYPOTHETICALS.

And, by

indirect considerations of this kind,

may every

dis

junctive Proposition, however numerous its members, be ex But the following general Rule will usually be pressed. preferable.

RULE. Consider what are

those distinct and mutually exclusive it is which in the statement of the given Propo of implied sition, that some one of them is true, and equate the sum of their

cases

elective expressions to unity.

This will give the equation of the

given Proposition.

For the sum of the

elective expressions for all distinct con

ceivable cases will be unity.

and

some all

Now all these cases being mutually

being asserted in the given Proposition that one case out of a given set of them is true, it follows that

exclusive,

it

which are not included in that

set are false,

elective expressions are severally equal to 0.

and

that their

Hence

the

sum

of the elective expressions for the remaining cases, viz. those included in the given set, will be unity. Some one of those

and as they are mutually more than one should be true.

cases will therefore be true, it

is

impossible that

exclusive,

Whence

the Rule in question.

And

in the application of this

Rule

it is

to

be observed, that

if the cases contemplated in the given disjunctive Proposition

are not mutually exclusive, they

must be resolved

into,

an equi

valent series of cases which are mutually exclusive. if

we

Either

X

Thus, viz.

take the Proposition of the preceding example, is true, and assume that the two true, or

Y

is

members of this Proposition are not exclusive, insomuch that in the enumeration of possible cases, we must reckon that of the Propositions

X

and

exclusive cases which

Y

fill

being both true, then the mutually up the Universe of the Proposition,

with their elective expressions, are 1st,

2nd, 3rd,

X true and Y false, Y true and X false, X true and Y true,

x (I - y), - x\ y(\ xy,

OF HYPOTHETIC A LS.

and the sum of these

elective expressions equated to unity gives

x + y -xy =

we suppose

1.

(30),

members of

the disjunctive to be con cases the only Proposition to be exclusive, then sidered are

But

as before*

1st,

2nd,

if

the

X true, Y false, Y true, X false,

x y

(1 (1

-

and the sum of these elective expressions equated

xThe

To

Ixy + y =

1,

y),

- x\ to 0, gives

(31).

subjoined examples will further illustrate this method.

express the Proposition, Either

X

is

Y

not true, or

is

not

true, the members being exclusive.

The mutually 1st,

2nd,

exclusive cases are

X not true, Y true, Y not true, X true,

and the sum of these equated

to

is

(1

x

(1

#), y),

unity gives

x - 2xy + y = which

-

y

(32),

1,

the same as (31), and in fact the Propositions which

they represent are equivalent.

To

X

express the Proposition, Either members not being exclusive.

is

not true, or

Y

is

not

true, the

To

we must add

the cases contemplated in the last Example,

the following, viz.

X not true, Y not true, The sum

(i

-

y)

+ y or

To is

-

x) (1

-

y).

of the elective expressions gives

#

Y

(1

- *) + 0, xy

- *)

(!

-

y) =

^

(33).

X

express the disjunctive Proposition, Either Z is true, the members being exclusive.

is

true, or

true, or

E 2

OF HYPOTHETIC A LS.

Here

the mutually exclusive cases are

1st,

X true, Y false,

Z

2nd,

Y

3rd,

Z

X false, Y false,

Z

true,

false,

X false,

true,

and the sum of the

x(\ - y)

false,

y

(1

-

(1

z) (1

-

2),

x),

* (1 - a) (1 - y),

elective expressions equated to 1, gives,

upon reduction, x + y +

- 2 (xy + yz + zz) 4 Say* =

z

The

of

the same Proposition, expression are in no sense exclusive, will be

-

(1

And

it

-

x) (1

-

y) (1

=

z)

0,

1,

(34).

when

the

members

(35).

easy to see that our method will apply to the

is

expression of any similar Proposition, subject to any specified

To

express the

whose members amount and character of exclusion.

conditional Proposition,

X

If

is

are

true,

Y

is true.

Here

it

cases of

by the

is

Y

implied that

the cases of

The former

being true.

elective

all

symbol

and the

x,

X

being true, are

cases being determined

latter

by

y,

we

have, in

virtue of (4),

*(l-y)=0,

(36).

To

express the conditional Proposition, If not true.

The

equation

is

be true,

Y

is

obviously

*y-0, this is equivalent to (33),

X

X

and in

(37); fact the disjunctive Proposition,

Y not true, and the conditional not true, are equivalent. true, Y Proposition, If X To express that If X not true, Y not true. Either

is

not true, or is

is

is

is

In (36) write

1

is

- x for x, and (i

I

-

- *) y

y

for y,

o.

we have

55

OF HYPOTHETICALS.

The resuhs which we have obtained admit in

many

ticular

Let

different ways.

of verification

it suffice to take for

examination the equation

x-2xy

+

y=l,

(38),

which expresses the conditional Proposition, Either or

Y

is

more par

true, the

members being

X

First, let the Proposition stituting,

X

is

true,

in this case exclusive.

z=\, and

be true, then

sub

we have -

1

= 2y + y

which implies that let

Secondly,

Y

X

y =

0,

be not true, then x =

y =

Y

y =

or

0,

not true.

is

gives

which implies that

-

/.

1,

is

i

(39),

9

In like manner

true.

with the assumptions that

and the equation

0,

Y is true,

or that

x = Again, in virtue of the property the equation in the form x 1 - Ixy +

y*

=

Y

x, y*

=

we may proceed is false.

y,

we may

write

1,

and extracting the square root, we have x - y = 1, (40), and or

this

as when represents the actual case; for, is respectively false or true, we have

X

is

true

false, Y

x =

/.

There

will

be no

y = x - y

1

or 0, or

=

1

1,

or -

1.

of other cases. difficulty in the analysis

Examples of Hypothetical Syllogism. of every form of hypothetical Syllogism will consist in forming the equations of the premises, and eliminating the symbol or symbols which are found in more than one of them. The result will express the conclusion.

The treatment

56

OF HYPOTHETICALS. 1st.

Disjunctive Syllogism.

Either

But

X

X

is

is

true,

Y is not

Therefore

X

Either

But

X

is

Y is

true, or

true,

Y

true, or

x + y - 2 xy = x- 1

true (exclusive),

x

true (not exclusive),

is

y=

/.

.

not true, Therefore is true,

+

is

Y

1

.*.

y xy = = x y= 1

1

2nd. Constructive Conditional Syllogism.

X X

If

is

But

true, is

Y is true,

true,

Y

Therefore

is

..

true,

1

x (1 - y) = x = I or y = y =

1.

3rd. Destructive Conditional Syllogism.

If

X

But

is

true,

Y

is

x

true,

-

(I

y)

Y is not true,

Therefore

X

is

not true,

=

y = x=

.-.

Simple Constructive Dilemma, the minor premiss ex

4th. clusive.

X

Y If Z is true, Y If

is

But Either

From x and

is

true,

X

is

is

x

true,

Z

is

true,

In whatever way we

y) = 0,

we have

the equations (41), (42), (43),

z.

-

z (1 - y) = x-\-z -2xz =

true,

true, or

(1

(41),

0,

(42),

1,

(43).

to eliminate

effect this, the result is

y-ii whence 5th.

it

appears that the Proposition

Y

Complex Constructive Dilemma,

is

true.

the minor premiss not

exclusive.

X

If If

"W

is is

Either

From

true,

Y

true,

Z

X

is

is is

x

true,

W

is

x

true,

these equations, eliminating x,

y +

-

z

-

yz =

1,

y)

=

w (1 z) = + w - xw = -

true,

true, or

(l

we have

0,

0, 1.

57

OF HYPOTHETICALS. which expresses the Conclusion, Either the

Y

is

true, or

Z

is

true,

members being non-exclusive. Complex Destructive Dilemma, the minor premiss ex

6th. clusive.

X

If

is

W

If

is

From

true,

Y

Either

result

true,

is

these

Y

is

true,

Z

is

true,

x(\ - y) = w (1 - 2) =

Z

not true, or

is

we must

equations

is

xw =

eliminate

arid

y

1

.

The

z.

Qj

X

which expresses the Conclusion, Either not true, the members not being exclusive.

Complex Destructive Dilemma,

7th.

y + z 2yz =

not true,

is

not true, or

Y

is

the minor premiss not

exclusive.

X

If

is

W

If

is

Either

On

true, true,

Y

is

true,

Z

is

true,

x(\ - y} = 10 ( 1 - z) =

Y is not true, or Z

elimination of y and

is

0.

we have

z,

xw -

0,

which indicates the same Conclusion It

yz =

not true,

as the previous

example.

.

appears from these and similar cases, that whether the of the minor premiss of a Dilemma are exclusive

members

or not, the

members of

The above which

the (disjunctive) Conclusion are never

This fact has perhaps escaped the notice of logicians.

exclusive.

are the principal forms of hypothetical Syllogism

logicians have recognised.

It would be easy, however, the list, especially by blending of the disjunctive and the conditional character in the same Proposition, of which

to

extend the

the following If

X

But

is

Y

is

an example.

true, then either

Y

is

true, or

Z

is

true,

x(\-y-z + yz)=Q is

y=

not true,

Therefore If

X

is

true,

Z

is

true,

/.

x(\ -

o z)

=

0.

58

OF HYPOTHETICALS.

That which logicians term a Causal Proposition is properly a conditional Syllogism, the major premiss of which is sup pressed.

The

X

assertion that the Proposition

Proposition Y

is

is

true,

is

true, because the

equivalent to the assertion,

Y

The

is true, Proposition Therefore the Proposition

X

is

true;

and these are the minor premiss and conclusion of the con ditional Syllogism,

If

Y is true, X is true, Y is true,

But

Therefore

X

is

true.

And

thus causal Propositions are seen to be included in the applications of our general method.

Note, that there

a family of disjunctive and conditional

is

Propositions, which do

not, of right, belong to the class

con

sidered in this Chapter. Such are those in which the force of the disjunctive or conditional particle is expended upon the predicate of the Proposition, as if, speaking of the inhabitants of a particular island, we should say, that they are all either Europeans or Asiatics; meaning, that it is true of each indi vidual, that he

the

appropriate

is

Europeans, and z

to Asiatics,

then the equation of the above

is

Proposition

x = xy + to

European or an Asiatic. If we symbol x to the inhabitants, y to

either a

elective

or

xz,

z(l-y-z)=0,

which we might add the condition yz =

(a);

0, since

no Europeans

The nature

of the symbols x, y, z, indicates that the Proposition belongs to those which we have before de are Asiatics.

Very different from the above is the signated as Categorical. all the inhabitants are Europeans, or they Either Proposition, are

all Asiatics.

positions.

The

sent Chapter;

Here the case

is

disjunctive particle separates

Pro

that contemplated in (31) of the pre

and the symbols by

which

it

is

expressed,

59

OF HYPOTHETICALS. although subject to the same laws as those of

(a),

have a

totally

different interpretation.*

The distinction is real and important. Every Proposition which language can express may be represented by elective symbols, and the laws of combination of those symbols are in cases the

same

but in one

class of instances the

symbols have reference to collections of objects, in the other, to the all

;

truths of constituent Propositions. * Some writers, among whom is Dr. Latham (First Outlines), regard it as the exclusive office of a conjunction to connect Propositions, not words. In this view I am not able to agree. The Proposition, Every animal is either rational or irrational, cannot be resolved into, Either every animal is rational, or every animal is irrational. The former belongs to pure categoricals, the latter to

such conversions would seem to be hypotheticals. In singular Propositions, allowable. This animal is either rational irrational, is equivalent to, Either t>r

this

animal

is

rational, or it is irrational.

would almost

This peculiarity of singular Pro

justify our ranking them, a separate class, as Ramus and his followers did.

positions

though truly universals, in

PROPERTIES OF ELECTIVE FUNCTIONS.

symbols combine according to the laws of quantity, we may, by Maclaurin s theorem, expand a given

SINCE

elective

(x), in

function

ascending powers of x,

excepted.

0(*)=
Now

known

cases of failure

Thus we have

=

z,

a?

=

(0)* +

+

+

2

+

&c,

(44).

whence

x, &c.,

W = 0(0)

^*

^{0

(0)

+

^r~-)

+ &c.},

(45).

i>

Now

if in

(44)

we make x =

0(l) = 0(0) +

1,

we have

(0)+^

)

+

&c.,

whence

Substitute this value for the coefficient of x in the second

member

of (45), and (x)

=

we have*

(0)

+ (0 (1) -

(0)} x,

(46),

* Although this and the following theorems have only been proved for those forms of functions which are expansible by Maclaurin s theorem, they may be regarded as true for all forms whatever this will appear from the applications. The reason seems to be that, as it is only through the one form of expansion ;

that elective functions

become

interpretable,

no conflicting interpretation

is

possible.

The development of (x) may mula for expansion in factorials, <#>

also

be determined thus.

By

the

known

for

PROPERTIES OF ELECTIVE FUNCTIONS. which we

shall also (*)

employ under the form

=(!)*+

Every function of x, are alone involved,

The

order.

(U

0(0) (1-*),

(47).

which integer powers of that symbol by this theorem reducible to the first

in

is

$

quantities

of the function

(0),

we

(1),

moduli

shall call the

are of great importance in the theory of elective functions, as will appear from the succeeding

They

(x).

Propositions.

PROP.

Any two

1.

functions

(x),

$

are

(x),

equivalent,

whose corresponding moduli are equal. This

is

a plain consequence of the last Proposition. For since

it

is

(x)

evident that

=

if



(0) +

(0)

<

=

{

^

-

(1)



(0),

(0)} x,

(1)

=

^

(1),

the two

expansions will be equivalent, and therefore the functions which

they represent will be equivalent also. The converse of this Proposition is

equally true, viz.

If two functions are equivalent, their corresponding moduli are equal.

Among we may

the most important applications of the above theorem,

notice the following. it

Suppose function

<

required to determine for

(x), the

following equation {4-

-

an elective symbol, x (x - 1) second, vanish. Also A0 (0) = (1)



.<*>{*

O).

<

(*)}"

Now x being

= 0(0)} +

what forms of the

is satisfied, viz.

= 0,

so that all the terms after the

(0),

whence

{(!)-

(0)}ar.

The mathematician may be only case in which the

compound

interested in the remark, that this an expansion stops at the second term. The

operative functions

is

not the

expansions of



(

+

and

x~ l }

respectively,

See Cambridge Mathematical Journal, Vol.

iv. p. 219.

/a?

+

[

iV*!,

^^

PROPERTIES OF ELECTIVE FUNCTIONS.

68

Here we

at

once obtain for the expression of the conditions

in question,

= {<#>

(0)}"

Again, suppose

(0).



{(!

The general theorem (0)

+ (0) =

This result

^ 0*0

X

(#)>

(48).

under required to determine the conditions

it

which the following equation



)}"-*(!),

X

is satisfied, viz.

once gives

at (0)-

^(

(!)

1

)

=

X( 1 )

>

(#), be proved by substituting for coefficients the and equating expanded forms,

also

may

tne ^ r

<j>

of the resulting equation properly reduced.

All the above theorems

may be extended

to functions of

For, as different elective

more

symbols combine

than one symbol. with each other according to the same laws as symbols of quan a given function with reference to any we can first tity,

particular

expand symbol which

with reference

to

and then expand the result so on in succession, the and any other symbol, it

contains,

order of the expansions being quite indifferent. Thus the given function being (xy) we have



= (xy)

and expanding the

(ay)

=

$

(00) 4

coefficients

{<

+ {

to



(xO) +

(10) (1 1)





{

(si)

-

(*0)} y,

<

with reference to x, and reducing (00)} (10)

x + {



(01)

(01) +

4>

-

(00)}y

<

(00)} xy,

(50),

which we may give the elegant symmetrical form

(51),

y,

wherein we

accordance with the

shall, in

employed, designate moduli of the function

(00),

<



<

(01),

language already

(10),

^

(11),

as

the

(xy).

it will appear that By inspection of the above general form, functions of two variables are equivalent, whose correspond

any

ing moduli are

all

equal.

PROPERTIES OF ELECTIVE FUNCTIONS. tbr satisfaction of

Thus the conditions upon which depends the equation, are seen to be {<*>

And

(oo)}-

=

n

(oo),

<

{<*>

=

(oi)}

(oi),

<

(52)>

upon which depends the

the conditions

satisfaction of

the equation are

(00)

(10)

^ (00) = x (00), i|r

(10) -


*

X 00),

01 ) =

00 ^ O

1)

=

X( X OI). l

)>

(53).

from (47) and (51), the very easy to assign by induction It is evident function. elective an of form expanded general It is

that if the

number

moduli will be 2

m ,

of elective symbols

and that

m, the number of the

is

their separate values will

be obtained

in the in every possible way the values 1 and several The function. the of given places of the elective symbols of which the moduli serve as coefficients, terms of the

by interchanging

will

expansion then be formed by writing for each

that recurs

1

under the

functional sign, the elective symbol x, &c., which it represents, and for each the corresponding 1 - x, &c., and regarding these

the modulus from product of which, multiplied by are obtained, constitutes a term of the expansion.

as factors, the

which they Thus, if we represent the moduli of any <

(xy

.

.

.)

by a l9

0,,

.

.

in which

tf^. .tr

itself,

when expanded

to the moduli, will

assume the form

a r , the function

and arranged with reference

are functions of x, y.

1 - x, 1 - y, of the forms x y y,. individually the index relations .

and the further

.

V=

*,"-*i

.

.

2

>

.,

&c

..

*=

&c

resolved into factors

These functions

&c.

-

relations,

*=0

elective function

-

satisfy

64

PROPERTIES OF ELECTIVE FUNCTIONS.

the product of any two of them This will at once vanishing. be inferred from inspection of the particular forms (47) and (51). Thus in the latter we have for the values of tl9 t^ &c., the forms

and

it is

evident that these satisfy the index relation, and that

We

their products all vanish.

stituent functions of

shall designate tJ2

and we

(xy),

<j>

.

.

as the

con

shall define the peculiarity

of the vanishing of the binary products, by saying that those And indeed the classes which they

functions are exclusive.

represent are mutually exclusive. The sum of all the constituents of an expanded function is An elegant proof of this Proposition will be obtained unity.

by expanding 1 as a function of any proposed = 1, we have Thus if in (51) we assume (xy)

elective symbols.

<

$(10)=1, 1

= xy + x

It is obvious

(01)=l,

(1

-

y) +

(1

<(00)=1,

- x) + y

(1

-



(1 1)

=

1,

and (51) gives

x) (1

-

y),

(57)..

indeed, that however numerous the symbols whence the sum

the moduli of unity are unity, of the constituents is unity. all

involved,

We

now prepared

are

to enter upon the question of the For this purpose general interpretation of elective equations. we shall find the following Propositions of the greatest service.

PROP.

2.

=

If the

first

member

of

the

general

equation

be expanded in a series of terms, each of which (xy...) of the form at, a being a modulus of the given function, then is for every numerical modulus a which does not vanish, we shall <

0,

have the equation

at

= Q

and the combined interpretations of these several equations express the full significance of the original equation. For, representing the equation under the form

a t\ -Multiplying by

t

lt

*

a

we

A

+ a Jr =0,

have, by (56), a.t.

=

0,

(59),

(58).

will

PROPERTIES OF ELECTIVE FUNCTIONS.

whence

if

o

(55

a numerical constant which does not vanish,

is 1

f,-0, and similarly for all the moduli which do not vanish. And inasmuch as from these constituent equations we can form the given equation, their interpretations will together express

its

entire significance.

Thus

if

the given equation were

x - y =

we should have

0,

Xs and Ys

<(11)=

>

<(

are identical,

10 )=

l >

<#>

* (i -y) - y U - *) whence, by the above theorem, x (1 - y) = 0,

y results It

(1

x) =

=

^

1,

(00) = 0,

would assume the form

so that the expansion (51)

-

(01)

(60),

-

0,

o,

All

Xs

are Ys,

All

Ys

are

which are together equivalent

Xs,

to (60).

may happen that the simultaneous satisfaction may require that one or more of

thus deduced,

of equations the elective

symbols should vanish. This would only imply the nonexistence it may even happen that it may lead to a final of a class :

result of the

form

1

= 0,

which would indicate the nonexistence of the

Such

logical Universe.

when we attempt to unite contra single equation. The manner in which

cases will only arise

dictory Propositions in a

the difficulty seems to be evaded in the result is characteristic. It appears from this Proposition, that the differences in the interpretation

of elective

functions

depend

solely

upon the

number and position of the vanishing moduli. No change in the value of a modulus, but one which causes it to vanish, the equation in produces any change in the interpretation of

which

it

is

found.

values which

we

If

among

the infinite

number

are thus permitted to give to the

of different

moduli which

do not vanish in a proposed equation, any one value should be

66

PROPERTIES OF ELECTIVE FUNCTIONS.

preferred, either

unity, for when the moduli of a function are the function itself satisfies the condition

it is

or

1

,

=

{*(y and

all


)}"

once introduces symmetry into our Calculus, and provides- us with fixed standards for reference. this at

PROP.

3.

If

w=

(xy



and

if

.

w, x, y,

.),

.

being elective symbols,

.

member be completely expanded and arranged terms of the form at, we shall be permitted

the second

in a series of

every term in which the modulus a does not satisfy the condition

to equate separately to

an =

and

to leave for the

value of

.

a,

the

sum

of the remaining terms.

As

the nature of the demonstration of this Proposition is unaffected quite by the number of the terms in the second member, we will for simplicity confine ourselves to the sup position of there being four,

two

first

and suppose that the moduli of the

only, satisfy the index law.

We have then w

=

a^ + af

+ a/3 + af^ = az

z

with the relations

a lt

a"

a"

(61),

,

in addition to the two sets of relations connecting in accordance with (55)

Squaring (61),

and

=

ah + a^

and subtracting (61) from

+

K -4K=

4,

2

being an hypothesis, that the

;

coefficients of these terms

2,

whence (61) becomes 2 utility

3,

+ a\t^ 4 a\tit

do not vanish, we have, by Prop.

The

2,

this,

-03K it

lf

we have

w

and

t

(56).

.

a^

+ a&.

of this Proposition will hereafter appear.

PROP. shall

4.

PROPERTIES OF ELECTIVE FUNCTIONS.

67

The

we

functions

being mutually exclusive,

,,. ,tr

always have

^ (A + A

+

0r*r)

+ t 00 *o ^ fa) ^ + ^ (,) a or the form of values of a^ =

*

8

whatever may be the Let the function af^ + then the moduli a^a2

.

#

2 2

$(11..), Also ty (a^ + a z

=

^

(11

{

t.

z

.

.

+

=

a,tr ) .

->|r

+ ^r

= It

^r

t (J

-f

.

( Ol ) ary.

^ +

^r(

^ (,)

would not be

a: 8)

(1

+

*,

difficult

to

{

{<#>

-,

=

-\Jr.

-

(00.

(...)<

(10..);

xy.

.)}

.

r

+ a r tr be represented by

.

(:ry.

.

.

),

(ajy. .)} a:

(10)}

^r

y)

.)

{

...

f (0

-

(1

(00)}

+

^

tr ,

we have noticed are sufficient The following Proposition may

(Of ) (1

-*)(l-y)... - *)

(1

-

y).,.

(64).

extend the

for

...

y)

(1

of interesting But those which

list

which the above are examples.

properties, of



a r will be given by the expressions

.

.

.

.

(^j,

-

-

our present requirements.

serve as an illustration of their

utility.

PROP.

5.

Whatever process of reasoning we apply

to a single

given Proposition, the result will either be the same Proposition or a limitation of

its

it.

Let us represent the equation of the given Proposition under most general form,

= 0, tf^-f a z tz .. + a r tr resolvable into as

many

(65),

equations of the form

t

=

as there are

moduli which do not vanish.

Now

the most general transformation of this equation ^r (ajt, +

provided that allowing

it

we

even

proposed relation

A

.

.

-f

attribute to to involve

atr ) = i|r

^ (0),

is

(66),

a perfectly arbitrary character,

new

elective symbols, having

to the original ones.

P

any

68

PROPERTIES OF ELECTIVE FUNCTIONS.

The development

^ To reduce

*,

(<)

of (66) gives,

^ (O

+

t

z

this to the general

.

.

by the +Vr (a

last

=

t

f)

Proposition,

r

1r (0).

form of reference,

it is

only neces

sary to observe that since

-

^+^,.4 we may

1,

r

^ (0),

write for

whence, on substitution and transposition,

{* (a,) - * (0)}

{^ (Oj) - ^ (0)}

H,

*z

.

{t (a,) -

+

.

* (0)}

*,

-

0.

appears, that if a be any modulus of the of the transformed original equation, the corresponding modulus

From which

it

equation will be <\fr

If a = 0, then

(a)

yfr

-

^

(a)

(0)

-

=

ty (0). i/r

(0)

-

-f (0)

=

whence

0,

there are no new terms in the transformed equation, and there fore there are no new Propositions given by equating its con stituent

members

to 0.

(a) ty (0) may vanish without a vanishing, Again, since in the transformed equation which existed terms may be wanting

^

in the primitive.

Thus some

may

original Proposition

of the constituent truths of the

entirely disappear from the interpre

tation of the final result.

Lastly, if

^

(a)

-

^r (0)

a numerical constant, or

it

do not vanish,

it

must involve new

In the former case, the term in which

it is

must

either be

elective symbols.

found will give

*-0, which

is

one of the constituents of the original equation we shall have

:

in the

latter case

(^ in

which

t

(a

-

+ (0)}

has a limiting factor.

equation, therefore,

is

t

=

0,

The

interpretation of this

a limitation of the interpretation of (65).

PROPERTIES OF ELECTIVE FUNCTIONS.

69

The purport

of the last investigation will be more apparent to the mathematician than to the As from any mathe logician. matical equation an infinite it

seemed

to

be necessary

number of others may be deduced, to shew that when the original

equation expresses a logical Proposition, every member of the derived series, even when obtained by expansion under a func tional sign, admits of exact

and consistent interpretation.

F2

OF THE SOLUTION OF ELECTIVE EQUATIONS.

IN whatever way an elective symbol, considered as unknown, may be involved in a proposed equation, it is possible to assign its complete value in terms of the remaining elective symbols considered as known. It is to be observed of such equations,

that from the very nature of elective symbols, they are neces sarily linear,

and that

their solutions

have a very close analogy

with those of linear differential equations, arbitrary elective

symbols in the one, occupying the place of arbitrary constants

The method

in the other. illustrate

by

of solution

we

shall in the first place

particular examples, and, afterwards, apply to the

investigation of general theorems.

Given

(1

terms of

As y

-

x) y = 0,

(All

Ys

are Xs),

to

determine y in

x.

is

a function of x,

we may assume y = vx +

v (1 -

x\

(such being the expression of an arbitrary function of x), the moduli v and v remaining to be determined. have then

We

(1

or,

-x) [vx +

v (l

-#)} =

0,

on actual multiplication, v (1 - x) = 0:

may be generally true, without imposing any restriction we must assume v = 0, and there being no condition to upon X, limit v y we have that this

y = This that

y

is

vx,

(67).

the complete solution of the equation. The condition an elective symbol requires that v should be an elective

is

71

OF THE SOLUTION OF ELECTIVE EQUATIONS. symbol

also (since

index law),

satisfy the

must

it

its

interpre

tation in other respects being arbitrary.

= 0, Similarly the solution of the equation, xy

= v

y

Given mine

(1

0, (All

is

y = v

are Zs are Xs), to deter

Ys which

a function of x and

_

(i

_

x) (i

we

substituting,

+

*)

t,

whence

v

=

y = v t) t>",

",

-

(1

(1

v"x

-

*) + v

"zx.

get

x) (1

-

z)

x)z =

0,

solution

+

-

"#

is

therefore

*)

+

(69),

t? "a*,

and the rigorous being arbitrary elective symbols,

Every Y is and Z. and not-Z, or an

interpretation of this result

and not-Z, or an It is

- X) z +

(l

The complete

0.

we may assume

2,

v (1 -

t/,

(68).

*),

y.

As y

And

- x) = zy

-

(1

is

X

is,

that

either a

not-X

X

in con deserving of note that the above equation may, two the solved be of its linear form, by adding

sequence

with reference to x and z ; and replacing particular solutions the arbitrary constants which each involves by an arbitrary function of the other symbol, the result

is

y -**(*) + (!-*)*(*)

(70).

equivalent to the other, it is for the arbitrary functions $ (z), only necessary to substitute

To shew

$

(x), their

that this solution

equivalents

wz +

we

get

is

w

(1

-

y = wxz + (w +

z)

w")

and

w"x

x(\ -

+

z)

w"

+

w"

(1

(1

-

#),

- z)

(1

-

z).

In consequence of the perfectly arbitrary character of w",

we may

replace their

y = wxz

which agrees with

-i-

wx

(69).

sum by (1

-

z)

a single

+

w"

(1

w

and

symbol w, whence

-

- z), x) (1

OF THE SOLUTION OF ELECTIVE EQUATIONS.

7

The

=

(1

- w)



(1

-

y]z =

expressed by

0,

is

arbitrary functions,

z

wx

solution of the equation

(xy) + (1 x)

$

(toy)

+ yx (wx\

(71).

These instances may serve to shew the analogy which exists between the solutions of elective equations and those of the corresponding order of linear differential equations. Thus the expression of the integral of a partial differential equation, either

by

arbitrary functions or

ficients, is in strict

last

To pursue

examples.

ter to

by a

with arbitrary coef

series

analogy with the case presented in the two this

comparison further would minis

curiosity rather than to utility.

We

shall prefer to con template the problem of the solution of elective equations under its most general aspect, which is the object of the succeeding

investigations.

To

solve the general equation

If we expand we have



(xy) = 0, with reference to y.

the given equation with reference to

x and

y = the coefficients

Now

(72),

(00) &c. being numerical constants.

the general expression of y, as a function of x,

y = v

0,

y,

ttzr-f

v (1 -

and v being unknown symbols

this value in (72),

we

to

is

x\

be determined. Substituting may be written in

obtain a result which

the following form, [<

(10) +

{

(11)

-

4 (10)}

v]x+U>

(00) +

{

(00)

(1

and

in order that this equation

way

restricting the generality of x,

<

(00)

-i{<

(01)

be

satisfied may we must have

(f>

= 0, (00)} v

$

(00)} v ]

-*)=0;

without any

73

OF THE SOLUTION OF ELECTIVE EQUATIONS. from which we deduce

0(10)

(00)

,

"

wherefore

the original equation with respect to y

Had we expanded only,

we should have had = 0(zO) + (0(*l)-0(*0)}y 0;

but

it

might have

startled those

who

are unaccustomed to the

had processes of Symbolical Algebra, deduced

(x

we from

this

equation

0)

because of the apparently meaningless character of the second

Such a

member.

result

would however have been perfectly

lawful, and the expansion of the second member would have given us the solution above obtained. I shall in the following example employ this method, and shall only remark that those to

whom

it

may appear

doubtful,

may

verify

its

conclusions

by

the previous method.

= 0, or in other words (xyz) to determine the value of z as a function of x and y.

To

solve the general equation

Expanding the given equation with reference to (xyO) + {0 (xy\} (xyO)} z = .

z,

we have

;

...(74),

and expanding the second member as a function of x and y by aid of the general theorem, we have
*(qio)

QQQ)

CTI

0(110) -0(111)

n

x(l

0(100) -0(101)

_,w,


OO

)

_

(1 v

(000)- 0(001)

.

(75)

OF THE SOLUTION OF ELECTIVE EQUATIONS.

74

and this is the complete solution required. By the same method we may resolve an equation involving any proposed number of elective symbols. In the interpretation of any general solution of

this nature, the following cases may present themselves. The values of the moduli 0(00), (01), &c. being constant, one or more of the coefficients of the solution may assume (f>

the form

In the former case, the indefinite symbol g must be replaced by an arbitrary elective symbol v. In the g

or

J.

case, the term, which is multiplied by a factor J (or by numerical constant except 1), must be separately any equated

latter

to 0,

and

This

is

will indicate the existence of a subsidiary Proposition.

evident from (62).

Ex. Given x

-

(1

y)=

All

0,

Xs

are Ys, to determine

y

as

a function of x.

Let

(xy)

(00) =

;

=

x(l-

y),

whence, by

-*

then 0(10) =

1,



(11)-

(01) = 0,

0,

(73),

+

(!-*),

(76),

The interpretation of this of the entire class with an

v being an arbitrary elective symbol. result is that the class indefinite

Y consists

remainder of not-Xs.

the highest sense,

t. e.

it

may

X

This remainder

vary from

up

is

indefinite in

to the entire class

of not-Xs.

Ex. Given x

(\

-

z)

+ z =y,

(the

class

Y

consists of the

entire class Z, with such not-Zs as are Xs), to find Z.

= x (1 - z) - y + z, whence (xyz) lowing set of values for the moduli,

we have

Here

0(110)= 0, 0(010)=-!,

0(111)=

0,

0(011) =

0,

0(100)= 1, 0(000)=0,

and substituting these in the general formula

the fol

0(101)=

1,

0(001) =

1,

(75),

we have

OF THE SOLUTION OF ELECTIVE EQUATIONS.

75

the infinite coefficient of the second term indicates the equation

x

(1

-

and the indeterminate

by

v,

=

y)

2

=

two

are not

Of

;

term being replaced

we have

- x] y + vxy,

(1

the interpretation of which

Ys which

Ys

are

coefficient of the first

an arbitrary elective symbol,

are Xs.

Xs

All

0,

is,

Xs, and an

Z

that the class indefinite

consists of all the

remainder of Ys which

course this indefinite remainder

vanish.

may

The

we have

obtained are logical inferences (not very obvious ones) from the original Propositions, and they give us results

all

the information which

its

constituent elements.

it

contains respecting the class Z, and

X consists of

all

Ex. Given x - y (1 - z) + z(\ - y). The class Ys which are not-Zs, and all Zs which are not-Ys

:

required

the class Z.

We have Oy*) - s - y <(110)=

0,

(i

-

)

- *

(i

(lll)=l,

0(010) = -

1,

0(011)=

0,

-

y),

whence, by substituting in (7 5), z = x(\-y) + y(\-x}, the interpretation of which which are not Ys, and of all

is,

0(101) = 0, 0(001) = -!;

0(100)= 1, 0(000)=0,

the class

Ys which

(78),

Z

consists of all

are not

Xs

;

Xs

an inference

strictly logical.

Ex. Given y (l - z Proceeding

(1

-

as before to

#)}

=

0,

All

Ys

are Zs

and not-Xs,

form the moduli, we have, on sub

stitution in the general formulae,

z

=

or z = y (1

\

- y), *) + g (1 *) (1 y) + y (1 xy + \x (1 x) (1 --y) x] + vx (1 y} + v (1 = y(l-*) + (l-y)0(*), (79),

with the relation

xy =

from these

No Ys

it

appears that

:

are

Xs, and that the

class

Z

OF THE SOLUTION OF ELECTIVE EQUATIONS.

76

consists of all

Ys which

are not

Xs, and of an

indefinite re

mainder of not-Ys. This method, in combination with Lagrange s method of indeterminate multipliers, may be very elegantly applied to the treatment of simultaneous equations. Our limits only permit us to offer a single example, but the subject is well deserving of

further investigation.

Given the equations x (1 - z) = 0, z (1 - y) = 0, All Xs are Zs, All Zs are Ys, to determine the complete value of z with any subsidiary relations connecting x and y.

Adding

the second equation multiplied by an indeterminate

constant A, to the

first,

x

(1

we have -

z)

+ \z

(1

-

whence determining the moduli, and

y)

=

0,

substituting in (75),

*(i-) + 80-*)y*

(so),

from which we derive z

= xy

-f

v (1 - x) y,

with the subsidiary relation

*(1 -y)=0: the former of these expresses that the class

Z

consists of all

that are Ys, with an indefinite remainder of not-Xs that are

the latter, that All

Xs

Xs Ys

;

are Ys, being in fact the conclusion

of the syllogism of which the two given Propositions are the premises. assigning an appropriate meaning to our symbols, all the in equations we have discussed would admit of interpretation

By

hypothetical, but

it

may

suffice to

have considered them

as

examples of categoricals.

That peculiarity of

elective symbols, in virtue of

which every = 0,

elective equation is reducible to a system of equations

= 3

0, &c., so constituted, that all the binary products

tf,

/2 tj# ,

represents a general doctrine in Logic with re ference to the ultimate analysis of Propositions, of which it &c.,

vanish,

may be

desirable to offer

some

illustration.

OF THE SOLUTION OF ELECTIVE EQUATIONS. of these constituents

Any

of the forms x, y,...l - w, fore

\

t

l9

-

77

&c. consists only of factors In categoricals it there

*

3,

z, Sec.

a compound class, i. e. a class defined by the of certain presence Dualities, and by the absence of certain other qualities.

represents

Each

constituent equation

existence of

some

^=

&c. expresses a denial of the

0,

and the

class so defined,

different classes are

mutually exclusive.

Thus

all

categorical Propositions are resolvable into

the existence

of compound a member of another.

class being

The x

(1

class

-

certain

classes,

a denial of

no member of one such

Xs are Ys, expressed by the equation resolved into a denial of the existence of a

Proposition, All y} =

0, is

whose members are

Xs and

not-Ys.

The

= xy, Proposition Some Xs are Ys, expressed by resolvable as follows. On expansion, v - xy = vx (1 y) + vy (1 x) + v (1 - x) (1 - y) - xy (1 t>

is

t>);

The

three

first

imply that there

no

class whose members unknown Some, and are 1st, Xs and not Ys; 2nd, Ys and not Xs; 3rd, not-Xs and not-Ys. The fourth implies that there is no class whose members are Xs and Ys without belonging to this unknown Some. From the same analysis it appears that all is

to a certain

belong

hypothetical Propo be resolved into denials of the coexistence of the truth or falsity of certain assertions.

sitions

may

Thus

the Proposition, If

equation x and the falsity of

by

its

(1

-

y)

=

X

is

true,

Y

is

true,

resolvable

is

0, into a denial that the truth

of

Y coexist.

And

the Proposition Either is

exclusive,

both true

But

it

;

X

is

true, or

resolvable into a denial,

secondly, that

may be

asked,

is

first,

Y

is

true,

that

X and Y are both false.

X

X

members

and

Y

are

not something more than a system of

negations

necessary to the constitution of

position?

is

an affirmative Pro

not a positive element required?

Undoubtedly

OF THE SOLUTION OF ELECTIVE EQUATIONS.

78 there

is

need of one; and

this positive

element

is

supplied

by the assumption (which may be regarded as ^ prerequisite of reasoning in such cases) that there is a Uni verse of conceptions, and that each individual it contains either in categoricals

belongs to a proposed class or does not belong to it ; in hypotheticals, by the assumption (equally prerequisite) that there a Universe of conceivable cases, and that any given Pro Indeed the question of the position is either true or false.

is

existence of conceptions (el e<m) is preliminary to any statement of their qualities or relations (ri ecrri). Aristotle, Anal. Post. lib. ii.

It

cap. 2.

would appear from the above,

regarded as resting foundation.

Nor

at

is

once upon

that Propositions

a positive and

upon

may be

a negative

such a view either foreign to the

spirit

of Deductive Reasoning or inappropriate to its Method; the latter ever proceeding by limitations, while the former contem plates the particular as derived

from the general.

Demonstration of the Method of Indeterminate Multipliers, as applied to Simultaneous Elective Equations.

To

avoid needless complexity, it will be sufficient to consider the case of three equations involving three elective symbols, It will those equations being the most general of the kind.

marked by every feature affecting the character of the demonstration, which would present itself in the discussion of the more general problem in which the be seen that the case

number

is

of equations and the

number of

variables are

both

unlimited.

Let the given equations be = 0, $ (xyz) = (xyz)

0,

x (XVZ) =

>

C 1 )-

Multiplying the second and third of these by the arbitrary constants h and k, and adding to the first, we have (xyz) + h

$ (xyz} +

79

METHOD OF INDETERMINATE MULTIPLIERS. and we are to

to

shew, that in solving this equation with reference by the general theorem (75), we shall obtain

any variable z

not only the general value of z independent of h and k, but

any subsidiary relations which may exist between x and y

also

independently of

z.

form represent the general equation (2) under the form F(xyz) = 0, its solution may by (75) be written in the If

we

x(\ -y)

y(l -a?) _

F(l\0)

.F(OOO)

F(010)

JF(IOO)

and we have seen, that any one of these four terms is to be equated to 0, whose modulus, which we may represent by M, does not satisfy the condition M"=M, or, which is here the or 1 same thing, whose modulus has any other value than Consider the modulus (suppose 3/,) of the first term,

and giving

to the

symbol

F

its

full

.

viz.

meaning,

F(llQ)

we have 0(110) It is evident that the condition

M* =

M

l

cannot be satisfied

and unless the right-hand member be independent of h and k in order that this may be the case, we must have the function ;

dent of h and

ind

L

^(110)+ Assume then *(110).+ Jty(110) + *x(110) c

being independent of h and k ;

tions

and equating

coefficients,

$(lll)c0(110), whence, eliminating

c,

0(110)

we

have, on clearing of frac

METHOD OF INDETERMINATE MULTIPLIERS.

80

being equivalent

to the triple

system

)^(111) = (M (3);

0(110) -

and it appears that if any one of these equations the modulus will not satisfy the condition

M

l

the

first

we

shall

is

not satisfied,

M* = M

lt

term of the value of z must be equated have xy v9

to

whence 0,

and

a relation between x and y independent of z. Now if we expand in terms of z each pair of the primitive

equations (1),

we

have + GryO) (0 (ayl) -f tf shall

(syO)

jtf

Cryl)

and successively eliminating tions, we have (xyl)

z

= (xyQ)} z

0,

^(*yO)} z =

0,

between each pair of these equa

$ (ayO) -

(ayO)

i

(ay 1)

(syl)

which express

by the

=

0,

=

0,

the relations between x and y that are formed the elimination of z. Expanding these, and writing in full first

term,

all

we have

xO 11)}

xy + &c. =

0,

and it appears from Prop. 2, that if the coefficient of xy in any of these equations does not vanish, we shall have the equation

xy=

0;

but the coefficients in question are the same as the

first

members

of the system (3), and the two sets of conditions exactly agree. Thus, as respects the first term of the expansion, the method of

indeterminate coefficients leads to

same result as ordinary obvious that from their similarity of form, ; the same reasoning will apply to all the other terms. elimination

and

it is


METHOD OF INDETERMINATE MULTIPLIERS.

81

Suppose, in the second place, that the conditions (3) are satis is independent of h and k. It will then indif

fied so that

M

l

ferently assume the equivalent forms

M

1

1

i-lliii) .

^(

i

0(110)

m

1

)

xO")"

i

x(no)

^(110)

These are the exact forms of the

panded values of

z,

first modulus in the ex deduced from the solution of the three

If this

primitive equations singly. or

=

common

the term will be retained in z

value of

M

is

1

l

v, ; any other constant = value (except 0), we have a relation xy 0, not given by elimi nation, but deducible from the primitive equations singly, and

Thus

similarly for all the other terms.

pression of the subsidiary relations of the process of solution. It is evident,

upon

its

in every case the ex

a necessary accompaniment

consideration, that a similar proof will

to the discussion of a

apply both of

is

if

symbols and of

system indefinite as to the number

its

equations.

POSTSCRIPT. SOME

explanations and references which have during the printing of this work are subjoined. The remarks on the connexion between Logic and Language, Both the one and the p. 5, are scarcely sufficiently explicit. other I hold to depend very materially upon our ability to form additional

occurred to

me

general notions by the faculty of abstraction. Language is an instrument of Logic, but not an indispensable instrument. To the remarks on Cause, p. 1 2, I desire to add the following Considering Cause as an invariable antecedent in Nature, (which is Brown s view), whether associated or not with the idea of :

by Sir John Herschel, the knowledge of its knowledge which is properly expressed by the word that (TO orl), not by why (TO Biorl). It is very remarkable that the two greatest authorities in Logic, modern and ancient, agree ing in the latter interpretation, differ most widely in its applica tion to Mathematics. Sir W. Hamilton says that Mathematics Power,

as suggested

existence

is

a

POSTSCRIPT. exhibit only the that (TO orl)

:

Aristotle says,

The why belongs

they have the demonstrations of Causes. Anal. Post. lib. i., cap. xiv. It must be added that Aristotle s view is consistent with the sense (albeit an erroneous one) which in various parts of his writings he virtually to the to mathematicians, for

assigns

word Cause,

an antecedent in Logic, a sense according to which the premises might be said to be the cause of the conclu sion. This view appears to me to give] even to his viz.

physical of their peculiar character. Upon reconsideration, I think that the view on p. 41, as to the presence or absence of a medium of comparison, would readily follow from Professor De Morgan s doctrine, and I therefore The mode in which it relinquish all claim to a discovery. inquiries

much

appears in this treatise is, however, remarkable. I have seen reason to change the opinion expressed in pp. 42, 43. The system of equations there given for the expres sion of Propositions in Syllogism is always preferable to the one before employed

first,

in generality

secondly, in facility of

interpretation.

In virtue of the principle, that a Proposition is either true or every elective symbol employed in the expression of and 1, which are the hypotheticals admits only of the values forms of an elective only quantitative symbol. It is in fact possible, setting out from the theory of Probabilities (which is

false,

purely quantitative), to arrive at a system of methods and pro cesses for the treatment of hypotheticals exactly similar to those

which have been given. The two systems of elective symbols and of quantity osculate, if I may use the expression, in the, and 1. It seems to me to be implied by this, that points unconditional truth (categoricals) and probable truth meet to gether in the constitution of contingent truth;

(hypotheticals).

The

general doctrine of elective symbols and all the more cha racteristic applications are quite independent of any quantitative origin.

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