Collected Papers of Charles Sanders Peirce, Volumes III and IV: Exact Logic (Published Papers) and The Simplest Mathematics [Vol 3 and 4] 9780674138018

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evatpiceenete pst aeperi

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ALBRIGED

COELEGE

LIBRARY

READING, PENNSYLVANIA

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COLLECTED PAPERS OF CHARLES SANDERS PEIRCE

VOLUME

III

EXACT LOGIC (Published Papers) AND

VOLUME IV THE SIMPLEST

MATHEMATICS

EDITED BY CHARLES

HARTSHORNE AND

PAUL

TWO

WEISS

VOLUMES

IN

ONE

THE BELKNAP PRESS OF HARVARD UNIVERSITY PRESS CAMBRIDGE,

MASSACHUSETTS 1960

© CopyrRiGHT

AND FELLOWS

1933 BY THE

OF HARVARD

SECOND

PRESIDENT

COLLEGE

PRINTING

Volumes I-VI of the Collected Papers of Charles Sanders Peirce are now published by the Belknap Press of Harvard University Press, but responsibility for the content of these volumes and for the care of the entire corpus of Peirce Papers remains with the Department of Philosophy of Harvard University. The present volumes are reproduced by offset lithography from the first edition. Lists of errata are included.

DISTRIBUTED OXFORD

IN

GREAT

UNIVERSITY

BRITAIN

PRESS,

BY

LONDON

PRINTED IN THE UNITED STATES OF AMERICA LIBRARY

OF CONGRESS

CATALOG

CARD

NUMBER

60-9172

es) JY 3 ay mJ

VOLUME

III

EXACT LOGIC (Published Papers)

Errata

3.19n1 for vol. 9 read 8.1-6 3.88, line 11 for poo read p® 3.88, line 12 for cox read oo” 3.97n (p. 60) prefix f and ¢ respectively to the last two lines of the page 3.97 (p. 61, lines 4-6) for II read II throughout 3.110, line 13 for x2 read x? 3.110 (p. 67, line 19) for JA’x — 1 read /4* — 1 3.110 (p. 67, line 20) for /Ax read /4* 3.110 (p. 67, line 22) for log /Ax read log /4* 3.112 (p. 69, line 16) for (1—“) read '-™) 3.112 (p. 69, line 18) for (1—”) read (1—u) 3.114, line 26 for {/~1} read {/—1} 3.136n4 for contains read ‘contains’ 3.139 (p. 92, line 12) for Ox read 0* Sal 50) line

18 for 111119 read

111, 119

3.181n1, line 2 for vol. 4 read vol. 10 3.184n1 (p. 117) for (1851), p. 104]. read (1851)], p. 104. 3.195, line 1 for contradiction read contraposition 3.200n* (p. 128) for vol. 3 read vol. 3 [of the American Journal of Mathematics| *3,202-ine ly tor/be— G@ = Oiread bo =eon ce = 0 +3202, line 2 fora = ad = co read ¢ = co d= 10 3.213, line 24 for zxw—2x:x.§

Then unity denotes the class of which any class is a part; that is, what ts or ens.

8. It is plain that if for the moment we allow a:b to denote the maximum value of a;b, then

(16)

£=1—xr=0-7.4|

* As [0=(«+x)]=[0+~=«], 0 added to a class is the class. O represents the minimum which results from subtracting a class from itself.

+ If x is a minimum, a;b=¢,b=a; if a maximum, a;b, =a, 6+ ab. t The class a; is equal to the class of that which is both a and d plus the indeterminate class of what is neither a nor 6 on the condition that there are no a’s which are not b. § As a:b represents the maximum or upper limit of a;5 (see 8) unity represents the maximum which results from dividing any class by itself. As (l=x;x) = (x,1=) the result of multiplying a class by 1 is the class.

q As a:b=a+ 6, O:x==(0+ c= x).

6

BOOLE’S

CALCULUS

OF LOGIC

[3.9

So that (17)

x,(1—x)=0

x+O0:x=1.

9. The rules for the transformation of expressions involving logical subtraction and division would be very complicated. The following method is, therefore, resorted to.* It is plain that any operations consisting solely of logical addition and multiplication, being performed upon interpretable symbols, can result in nothing uninterpretable. Hence, if g+ X« signifies such an operation performed upon symbols of which x is one, we have et Xx=a,4+5,(1 =8) 7

where a and 2 are interpretable. It is plain, also, that all four operations being performed in any way upon any symbols, will, in general, give a result of which one term is interpretable and another not; although either of these terms may disappear. We have then

gx=t,at+j,(1—zx).! We have seen that if either of these coefficients 7 and 7 is

uninterpretable, the other factor of the same term is equal to nothing, or else the whole expression is uninterpretable. But

g(1)=7 and 9(0) 7. Hence

(18)

gx= g(1),x+ 9(0),(1—x)

g(x and y)= (1 and 1),x,y+ (1 and 0),x,9+ (0 and 1),#,y + 9(0 and 0),%,9. (18’) yx=(e(1)+-£),(e(0) +)? g(x and y)=(¢(1 and 1)-+%+-9),(e(1 and 0)+%+,y), (p(0 and 1)-+x-+9),(~(0 and Q)+x+y). * See Lewis’ Survey of Symbolic Logic, pp. 58-67, 82 and 132-174 for a very clear presentation and a development of these ‘‘transformations.”

t f(x) =*,0+2,b. 1 The proof offered for this is fallacious inasmuch as 7 and 7 have not been proved to be independent of x. — 1870. [Peirce follows this remark with a proof which is too long and of insufficient interest to be reproduced]. 2 Identity (18’) is reducible to (18) by development by second member by

(18). — 1870. 7

3.10]

EXACT

LOGIC

Developing by (18) x+y, we have,

a+y=(1+1),x,y+(1+0),2,5+ 0+ 1),¥,9+ (0+ 0),,5. So that, by (11), (19) (i-l)=2. 10.

1-0=1.

0+1+(0+1).

0--0=0.

Developing x;y in the same way, we have’ xyy=1;lx,y+150,2,5+031,%,y+0;0,2,9.

So that, by (14), (20)

1l=1

1;0=(1;0)

0;1=0

Boole gives (20),* but not (19). In solving identities we must remember that

(21)

(a+b)—b=a

(22)

(ab) +b=a.

From a6 the value of 5 cannot be obtained. (23)

(a,b) +b=a

(24)

a;b,b=a.

From a;b the value of 5 cannot be determined. 11.

Given theidentity

o¢x=0.

Required to eliminate x.

o(1)=2,¢(1)+(1—2),¢(1) o(0)==x,9(0) +(1—x), ¢(0). Logically multiplying these identities, we get

9(1),¢(0)=x,¢(1),¢(0) + (1-2), e(1),¢(0). For two terms disappear because of (17). But we have, by (18),

e(1),x+ 9(0),(L—x)= ex=0. Multiplying logically by x we get

e(1),x=0 and by (1—-2) we get

(0),(1—x)=0. 1 @;b, c must always be taken as (@;8), c, not as a;(8, c). * Laws of Thought, vol. 2, p. S7ff.

8

0;0=».

BOOLE’S

CALCULUS

OF LOGIC

(3.12

Substituting these values above, we have (25)

12.

¢(1),¢(0)=0 when gx=0.

Given

eae i,

Required to eliminate x. Let

¢y’x=1—

~x=0

¢'(1),¢'(0)=(1— ¢(1)),— ¢(0)) =0 1—(1—¢(1)),(1—¢(0)) +1. Now, developing as in (18), only in reference to ¢(1) and ¢(0) instead of to x and y,

1—(1—¢(1)),(1— 9(0)) = ¢(1),¢(0) + o(1),(1 — ¢(0)) + (0),(1—¢(1)). But by (18) we have also,

¢(1)+ 9(0)= ¢(1), (0) + ¢(1),(1— ¢(0)) + ¢(0),(1— ¢(1)). So that

(26)

g(1)+ ¢(0)+=1 when gx=1.

Boole gives (25),* but not (26). 13. We pass now from the consideration of identities to that of equations.t Let every expression for a class have a second meaning, which is its meaning in an equation. Namely, let it denote the proportion of individuals of that class to be found among all the individuals examined in the long run. Then we have (27)

(28)

If a=b

a=b

a+b=(a+6)+(a,b).

14. Let b, denote the frequency of b’s among the a’s. Then considered as a class, if a and b are events, b, denotes the fact that if a happens b happens.

(29)

ab, =a,b.t

* Tiid., p. 10. + Le., from the logical relations of class identity in extension to the “arithmetical” relations of numerical equality. Cf. 44. t ‘Arithmetical’ multiplication is represented by juxtaposing the terms.

9

3.15]

EXACT

LOGIC

It will be convenient to set down some obvious and funda-

mental properties of the function ),. (30)

ab, = bay,

(31) (32)

y(b, and ¢,) =(¢(b and c))a (1—b),=1-8, b

1

(33)

b=

(34)

ei

(35)

ae ba-o(1-4)

by1-a)

(ga)a=(¢(1))a-

The application of the system to probabilities may best be exhibited in a few simple examples, some of which I shall select from Boole’s work, in order that the solutions here given may be compared with his. 15. Example 1. Given the proportion of days upon which it hails, and the proportion of days upon which it thunders. Required the proportion of days upon which it does both. Let 1=days, p=days when it hails, g=-days when it thunders, r=days when it hails and thunders. Vist

Then by (29),

1= 2,0 = PIp = Ia

Answer. The required proportion is an unknown fraction of the least of the two proportions given. By p might have been denoted the probability of the major, and by q that of the minor premiss of a hypothetical syllogism of the following form: If a noise 1s heard, an explosion always takes place; If a match is applied to a barrel of gunpowder, a noise is heard; If a match is applied to a barrel of gunpowder, an explosion always takes place. In this case, the value given for 7 would have represented the probability of the conclusion. Now Boole (page 284) solves this problem by his unmodified method, and obtains the following answer:

r=pqta(1—g) 10

BOOLE’S

CALCULUS

OF LOGIC

(3.15

where a is an arbitrary constant. Here, if g=1 and p=0, r=(Q. That is, his answer implies that if the major premiss be false and the minor be true, the conclusion must be false. That

this is not really so is shown by the above example. Boole (page 286) is forced to the conclusion that ‘propositions which, when true, are equivalent, are not necessarily equivalent when regarded only as probable.” This is absurd, because probability belongs to the events denoted, and not to forms of expression. The probability of an event is not altered by translation from one language to another. Boole, in fact, puts the problem into equations wrongly (an error which it is the chief purpose of a calculus of logic to prevent), and proceeds as if the problem were as follows: It being known what would be the probability of Y, if X were to happen, and what would be the probability of Z, if Y were to happen; what would be the probability of Z, if X were to happen? But even this problem has been wrongly solved by him. For, according to his solution, where p=lYx

q=Zy

1=Le,

r must be at least as large as the product of p and g. But if X be the event that a certain man is a negro, Y the event that he is born in Massachusetts, and Z the event that he is a white man, then neither p nor q is zero, and yet 7 vanishes. This problem may be rightly solved as follows: Let p’=Y,>=X,Y Q=ZqFX,Z

WS

OILS

Then, 171,950

P09.

Developing these expressions by (18) we have =p,’

=P

+1'pa' (0,7) +9'5.a(P',7)

t1's.a( Pg) +1 5.a(P'g’)-

The comparison of these two identities shows that r= p,q +9'5.4@(P',7’') :

il

EXACT LOGIC

3.16]

Now

b'

=P'- 2 7+

BG And

-V 0"

7-27, =P —V 0"

B=?

— Pa FT-Vo0’

B=

-Vp P=— Pa

Then let y,2 £,V, 2+4,9,24+-4,9,2+2%,9,

1—x,y

— 2 2TEI TUITE, P=yj2 =

X,Y

52

)—a+n(q -D*») 16. Example 2. (See Boole, page 276.) to find p.

Given r and q;

p=r;q=1+,(1—q) because p is interpretable. Answer. The required proportion lies somewhere between the proportion of days upon which it both hails and thunders, and that added to one minus the proportion of days when it thunders.

17. Example 3. (See Boole, page 279.) Given, out of the number of questions put to two witnesses, and answered by yes or no, the proportion that each answers truly, and the proportion of those their answers to which disagree. Required, 12

BOOLE’S

CALCULUS

OF LOGIC

[3.18

out of those wherein they agree, the proportion they answer truly and the proportion they answer falsely.* Let 1=the questions put to both witnesses, p=those which the first answers truly, g=those which the second answers truly, r==those wherein they disagree, w=those which both answer truly, w’=those which both answer falsely.

w= Pg

w'=P,g

r= phq—w+ptq-w’.

,

Now by (28)

p+q=pt+q—w

Pri=PaPrl—qnwr

Substituting and transposing, 2w=pt+q-r

2w’=2—p—gq-r.

Now w.,= wan

but w(1—r)=w.

W i+. = w',(1-9) l—r

but w’(1—r) =w’

—r

ape

BE

Zea bisLd

ED Ey)

BIAS pA oh

eee

me

18. The differences of Boole’s system, as given by himself, from the modification of it given here, are three. First. Boole does not make use of the operations here termed logical addition and subtraction. The advantages obtained by the introduction of them are three, viz., they give unity to the system; they greatly abbreviate the labor of working with it; and they enable us to express particular propositions. This last point requires illustration. Let 7 be a class only determined to be such that only some one individual of the class a comes under it. Then a--i, a is the expression for some a. Boole cannot properly express some a. Second. Boole uses the ordinary sign of multiplication for logical multiplication. This debars him from converting every logical identity into an equality of probabilities. Before the transformation can be made the equation has to be brought * Cf. 2.674. 13

3.19]

EXACT

LOGIC

into a particular form, and much labor is wasted in bringing it to that form. Third.

Boole has no such function as ap. This involves him

in two difficulties. When the probability of such a function is required, he can only obtain it by a departure from the strictness of his system. And on account of the absence of that symbol, he is led to declare that, without adopting the principle that simple, unconditioned events whose probabilities are given are independent, a calculus of logic applicable to probabilities would be impossible. 19. The question as to the adoption of this principle is certainly not one of words merely. The manner in which it is answered, however, partly determines the sense in which the term ‘“‘probability”’ is taken. In the propriety of language, the probability of a fact either is, or solely depends upon, the strength of the argument in its favor, supposing all relevant relations of all known facts to constitute that argument. Now, the strength of an argument is only the frequency with which such an argument will yield a true conclusion when its premisses are true. Hence probability depends solely upon the relative frequency of a specific event (namely, that a certain kind of argument yields a true conclusion from true premisses) to a generic event (namely, that that kind of argument occurs with true premisses). Thus,

when an ordinary man says that it is highly probable that it will rain, he has reference to certain indications of rain — that is, to a certain kind of argument that it will rain — and means to say that there is an argument that it will rain, which is of a kind of which but a small proportion fail. ‘‘ Probability,” in the untechnical sense, is therefore a vague word, inasmuch as it does not indicate what one, of the numerous subordinated and codrdinated genera to which every argument belongs, is the one the relative frequency of the truth of which is ex-

pressed. It is usually the case, that there is a tacit understanding upon this point, based perhaps on the notion of an infima species of argument. But an infima species is a mere fiction in logic. And very often the reference is to a very wide genus. The sense in which the term should be made a technical one is that which will best subserve the purposes of the calculus in 14

BOOLE’S

CALCULUS

OF LOGIC

[3.19

question. Now, the only possible use of a calculation of a probability is security in the long run. But there can be no question that an insurance company, for example, which assumed that events were independent without any reason to think that they really were so, would be subjected to great hazard. Suppose, says Mr. Venn,* that an insurance company knew that nine tenths of the Englishmen who go to Madeira die, and that nine tenths of the consumptives who go there get well. How should they treat a consumptive Englishman? Mr. Venn has made an error in answering the question, but the illustration puts in a clear light the advantage of ceasing to speak of probability, and of speaking only of the relative frequency of this event to that.! * Logic of Chance, ch. 9, section 24. 1 See a notice, Venn’s Logic of Chance, in the North American Review for July

1867 [vol. 9].

15

II

UPON

THE

LOGIC

OF MATHEMATICS*

Part It

§1. THE

BOOLIAN

CALCULUS*

20. The object of the present paper is to show that there are certain general propositions from which the truths of mathematics follow syllogistically, and that these propositions may be taken as definitions of the objects under the consideration of the mathematician without involving any assumption in reference to experience or intuition. That there actually are such objects in experience or pure intuition is not in itself a part of pure mathematics. 21. Let us first turn our attention to the logical calculus of Boole. I have shown in a previous communication to the Academy,t that this calculus involves eight operations, v7z., Logical Addition, Arithmetical Addition, Logical Multiplication, Arithmetical Multiplication, and the processes inverse to these. DEFINITIONS

1. 2.

3. 4.

Identity. a=b expresses the two facts that any a is b and any 0 is a. Logical Addition. a+b denotes a member of the class which contains under it all the a’s and all the b’s, and nothing else. Logical Multiplication. a,b denotes only whatever is both a and b. Zero denotes nothing, or the class without extent, by which we mean that if a is any member of any class, a+0 is a.

* Proceedings of the American Academy of Arts and Sciences, vol. 7, pp. 402412, September_ 1867.

t No other part seems to have been written. t See Paper No. I.

16

LOGIC 5. 6.

7.

OF MATHEMATICS

[3.22

Unity denotes being, or the class without content, by which we mean that, if a is a member of any class, a isa, 1. Arithmetical Addition. a+b,ifa,b=0, is the same as a+), but, if a and 6 are classes which have any extent in common, it is not a class. Arithmetical Multiplication. ab represents an event when a and 0 are events only if these events are independent of each other, in which case ab=a,b. By the events being

independent is meant that it is possible to take two series of terms, A,, Ao, A;, etc., and B;, Bo, B3, etc., such that the

following conditions will be satisfied.

(Here x denotes any

individual or class, not nothing; Am, An, Bm, Bn, any members of the two series of terms, and 2A, 2B, >(A,B) logical sums of some of the A,’s, the B,’s, and the (A,,B,)’s

respectively.)

22.

From

Condition 1. Condition-2.~

No A, is Ap. No 6.1s-B,.

Condition Condition Condition Condition

x= (A,B) a=2ZA.

these

3. 4. 5. 6.

b=ZB. Some A, is B,.*

definitions

a series

of theorems

follow

syllogistically, the proofs of most of which are omitted on account of their ease and want of interest.

THEOREMS

I 23.

li a=, then b=c.

II 24.

If a=b, and b=c, then a=c.

III 25.

If a+b=c, then b+a=c.

* a and b are independent

if they are summations of terms (4 and 5) each

of whose members is distinct (1 and 2), so that there is a class of the terms of

a and b together (3) and a term in a has a corresponding member in b (6). There are as many members of a,b as there are combinations of a member of a with one of b. Cf. 33.

17

3.26]

EXACT

LOGIC

IV 26.

If atb=m

and b+c=n

and atn=x,

then m+ce=~x.

Corollary. These last two theorems hold good also for arithmetical addition. V 27. If atb=c and a’+b=c, then a=a’, or else there is nothing not 0. This theorem does not hold with logical addition. But from definition 6 it follows that

No a is 6 (supposing there is any a) No

a’ is b (supposing there is any a’)

neither of which propositions would be implied in the corresponding formule of logical addition. Now from definitions 2 and 6, Any aisc “. Any a isc not b But again from definitions 2 and 6 we have Any ¢ not 6 is a’ (if there is any not d)

.. Any a is a’ (if there is any not b) And in a similar way it could be shown that any a’ is a (under the same supposition). Hence by definition 1, a=a’ if there is anything not b. Scholium. In arithmetic this proposition is limited by the supposition that 6 is finite.* The supposition here though similar to that is not quite the same.

VI 28.

If a,b==c, then b,a=c.

29.

VII If a,b=m and b,c=n and a,n=x, then m,c=x. Vill

30.

If mn=6b

and a+m=u

and a+n=v

and a+b=x,

then u,v=~. * —because in transfinite arithmetic finite quantities can be added to infinites without affecting the total— § +x= N +y= N where x and y are finite and § is the smallest transfinite cardinal.

18

LOGIC

OF MATHEMATICS

[3:31

IX 31.

If m+n=b

and aym=u

and a,n=v and a,b=~x, then

Ut-V=KX.

The proof of this theorem may be given as an example of the proofs of the rest. It is required then (by definition 3) to prove three propositions, vz. First. That any w is x. Second. That any 2 is x. Third. That any x not wu is v. First

PROPOSITION

Since u=a,m, by definition 3 Any u is m, and since m+n=b, by definition 2

whence

Any mis b, Any wu is 0,

But since u=a,m, by definition 3 Any u is a, whence Any w is both a and 6, But since a,b==x, by definition 3 Whatever is both a and d is x

whence

Any u is x. SECOND

PROPOSITION

This is proved like the first. THIRD

PROPOSITION

Since a,m==u, by definition 3, Whatever is both a and m is u.

or Whatever is not wu is not both a and m. or Whatever is not wu is either not @ or not m. or Whatever is not w and is a is not m. But since a,b=~, by definition 3 Any x is a, whence Any x not wis not u and is a,

whence

Any x not w is not m. 19

3.32]

EXACT

But since a,b=~x, by Any whence Any Any

LOGIC

definition 3 ~« is }, x not u is 8, «x not u is 6, not m.

But since m+-n=), by definition 2

whence

Any 6 not mis n, Any « not u is n,

and therefore

Any x not u is both a and n.*

But since a,n==v, by definition 3 Whatever is both a and 77 is 2,

whence

Any x not w is v.

32. Corollary 1. This proposition readily extends itself to arithmetical addition. Corollary 2, The converse propositions produced by transposing the last two identities of theorems vii and 1x are also true.

Corollary 3.

Theorems v1, v1, and 1x hold also with arith-

metical multiplication. This is sufficiently evident in the case of theorem v1, because by definition 7 we have an additional premiss, namely, that a and 0 are independent, and an additional conclusion which is the same as that premiss. 33. In order to show the extension of the other theorems, I shall begin with the following lemma. If a and 0 are independent, then corresponding to every pair of individuals, one of which is both a and 3, there is just one pair of individuals one of which is a and the other 6; and conversely, if the pairs of individuals so correspond, a and b are independent. For, suppose a and b independent, then, by definition 7, condition 3, every class (A,,,B,) is an individual. If then A, denotes any A, which is a, and B, any By, which is b, by condition 6 (Az, By) and (A,,,B,) both exist, and by conditions 4 and 5 the former is any individual a, and the latter any individual 6. But given this pair of individuals, both of the pair (A,,B,) and (Am, B,) exist by condition 6. But one individual of this pair is both aand 6. Hence the pairs correspond, as stated above. Next, * Originally m. t Originally u.

20

LOGIC

OF MATHEMATICS

(3.34

suppose a and 6 to be any two classes. Let the series of A,,’s be a and not-a; and let the series of B,’s be all individuals separately. Then the first five conditions can always be satisfied. Let us suppose, then, that the sixth alone cannot be

satisfied. Then A, and B, may be taken such that (A,,B,) is nothing. Since A, and B, are supposed both to exist, there must be two individuals (A,, B,) and (Am, B,) which exist. But there is no corresponding pair (Am,B,) and (A,,B,). Hence, no case in which the sixth condition cannot be satisfied simultaneously with the first five is a case in which the pairs rightly correspond; or, in other words, every case in which the pairs correspond rightly is a case in which the sixth condition can be satisfied, provided the first five can be satisfied. But the first five can always be satisfied. Hence, if the pairs correspond as stated, the classes are independent. 34. In order to show that theorem vil may be extended to arithmetical multiplication, we have to prove that if a and 3, b and c, and a and (6,c), are independent, then (a,b) and ¢ are independent. Let s denote any individual. Corresponding to every s with (a,b,c), there is an a and (0,c). Hence, corresponding to every s with s and with (a,b,c) (which is a particular case of that pair), there is an s with a and with (0,c). But for every s with (0,c) there is a 6 with c; hence, corresponding to every a with s and with (0,c), there is an a with 6 and with c. Hence, for every s with s and with (a,b,c) there is an a with b and with c. For every a with 6 there is an s with (a,6); hence, for every a with } and with c, there is an s with (a,0) and c. Hence, for every s with s and with (a,b,c) there is an s with (a,b) and with c. Hence, for every s with (a,b,c) there is an (a,b) withc. The converse could be proved in the same way. Hence; etc.

35. Theorem Ix holds with arithmetical addition of whichever sort the multiplication is. For we have the additional premiss that “‘No m is m”’; whence since ‘“‘any u is m”’ and “any vis,” ‘no wis v,” which is the additional conclusion. Corollary 2, so far as it relates to theorem Ix, holds with arithmetical addition and multiplication. For, since no m is n, every pair, one of which is a and either m or 1, is either a pair, one of which is a and m, or a pair, one of which is a and n, and is not both. Hence, since for every pair one of which 21

3.36]

EXACT

LOGIC

is a and m, there is a pair one of which is a and the other m, and since for every pair one of which is a,n there is a pair one of which is a and the other 1; for every pair one of which is a and either m or n, there is either a pair one of which is a and the other m, or a pair one of which is a and the other 7, and not both; or, in other words, there is a pair one of which is a and the other either m or n. (It would perhaps have been better to give this complicated proof in its full syllogistic form. But as my principal object is merely to show that the various theorems could be so proved, and as there can be little doubt that if this is true of those which relate to arithmetical addition it is true also of those which relate to arithmetical multiplication, I have thought the above proof (which is quite apodeictic) to be sufficient. The reader should be careful not to confound a proof which needs itself to be experienced with one which requires experi-

ence of the object of proof.) x 36.

If ab=c and a’b=c, then a =a’, or no 6 exists.

This does not hold with logical, but does with arithmetical multiplication. For if a is not identical with a’, it may be divided thus a=a,a’+a,a’ if @’ denotes not a’. Then a,b=

(a,a’)

b+

(a,a’),b

and by the definition of independence the last term does not vanish unless (a,d’)=0, or all a is a’; but since a,b=a' b= (a,a’),b+(d,a’),b, this term does vanish, and, therefore, only ais a’, and in a similar way it could be shown that only a’ is a.

XI Ole

ekcae= bs

This is not true of arithmetical addition, for since by definition 7, ly, l=aal by theorem Ix

x,(1+¢a)=a2(1+a)=xl+xa=x+4+