Paper 2: Upon the Logic of Mathematics Proceedings of the American Academy of Arts and Sciences, vol. 7, pp. 402-412, September 1867. ^†1

PART I No other part seems to have been written. ^†2

§1. The Boolean CalculusE

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, See Paper No. I. ^†3 that this calculus involves eight operations, viz., Logical Addition, Arithmetical Addition, Logical Multiplication, Arithmetical Multiplication, and the processes inverse to these.

Definitions

1. Identity. a b expresses the two facts that any a is b and any b is a.

2. 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.

3. Logical Multiplication. a,b denotes only whatever is both a and b.

4. 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.

5. Unity denotes being, or the class without content, by which we mean that, if a is a member of any class, a is a, 1.

6. Arithmetical Addition. a+b, if a,b 0, is the same as a b, but, if a and b are classes which have any extent in common, it is not a class.

7. Arithmetical Multiplication. ab represents an event when a and b 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[1], A[2], A[3], etc., and B[1], B[2], B[3], etc., such that the following conditions will be satisfied. (Here x denotes any individual or class, not nothing; A[m], A[n], B[m], B[n], any members of the two series of terms, and ΣA, ΣB, Σ(A,B) logical sums of some of the A[n]'s, the B[n]'s, and the (A[n],B[n])'s respectively.)

Condition 1. No A[m] is A[n].

Condition 2. No B[m] is B[n].

Condition 3. x Σ(A,B)

Condition 4. a ΣA.

Condition 5. b ΣB.

Condition 6. Some A[m] is B[n]. 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. ^†1

22. From these 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. If a b, then b a.

II

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

III

25. If a b c, then b a c.

IV

26. If a b m and b c n and a n x, then m c x.

Corollary. These last two theorems hold good also for arithmetical addition.

V

27. If a + b c and a'+b c, then a a', or else there is nothing not b.

This theorem does not hold with logical addition. But from definition 6 it follows that

No a is b (supposing there is any a)

No a' is b (supposing there is any a')

neither of which propositions would be implied in the corresponding formulæ of logical addition. Now from definitions 2 and 6,

Any a is c

∴ Any a is c not b

But again from definitions 2 and 6 we have

Any c not b is a' (if there is any not b)

∴ 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 b is finite. —because in transfinite arithmetic finite quantities can be added to infinites without affecting the total — ℵ + x = ℵ + y = ℵ where x and y are finite and ℵ is the smallest transfinite cardinal. ^†1 The supposition here though similar to that is not quite the same.

VI

28. If a,b c, then b,a c.

VII

29. If a,b m and b,c n and a,n x, then m,c x.

VIII

30. If m,n b and a m u and a n v and a b x, then u,v x.

IX

31. If m n b and a,m u and a,n v and a,b x, then u v x.

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, viz.

First. That any u is x.

Second. That any v is x.

Third. That any x not u is v.

FIRST PROPOSITION

Since u a,m, by definition 3

Any u is m,

and since m n b, by definition 2

Any m is b,

whence

Any u is b,

But since u a,m, by definition 3

Any u is a,

whence

Any u is both a and b,

But since a,b x, by definition 3

Whatever is both a and b is x

whence

Any u is x.

SECOND PROPOSITION

This is proved like the first.

THIRD PROPOSITION

Since a,m u, by definition 3,

But since a,b x, by definition 3

But since a,b x, by definition 3

But since m n b, by definition 2

But since a,n v, by definition 3

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 VIII and IX are also true.

Corollary 3. Theorems VI, VII, and IX hold also with arithmetical multiplication. This is sufficiently evident in the case of theorem VI, because by definition 7 we have an additional premiss, namely, that a and b 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 b are independent, then corresponding to every pair of individuals, one of which is both a and b, there is just one pair of individuals one of which is a and the other b; 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 (Am,Bn) is an individual. If then Aa denotes any Am which is a, and Bb any Bm which is b, by condition 6 (Aa,Bn) and (Am,Bb) both exist, and by conditions 4 and 5 the former is any individual a, and the latter any individual b. But given this pair of individuals, both of the pair (Aa,Bb) and (Am,Bn) exist by condition 6. But one individual of this pair is both a and b. Hence the pairs correspond, as stated above. Next, suppose a and b to be any two classes. Let the series of Am's be a and not-a; and let the series of Bm'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 Ap and Bq may be taken such that (Ap,Bq) is nothing. Since Ap and Bq are supposed both to exist, there must be two individuals (Ap,Bn) and (Am,Bq) which exist. But there is no corresponding pair (Am,Bn) and (Ap,Bq). 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 VII may be extended to arithmetical multiplication, we have to prove that if a and b, b and c, and a and (b,c), are independent, then (a,b) and c are independent. Let s denote any individual. Corresponding to every s with (a,b,c), there is an a and (b,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 (b,c). But for every s with (b,c) there is a b with c; hence, corresponding to every a with s and with (b,c), there is an a with b 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 b there is an s with (a,b); hence, for every a with b and with c, there is an s with (a,b) 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) with c. 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 n"; whence since "any u is m" and "any v is n," "no u is 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 n, 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 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 n; 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 n, 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 experience of the object of proof.)

X

36. If ab c and a'b c, then a a', or no b 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,ā'

if ā' denotes not a'. Then

a,b (a,a'),b + (a,ā'),b

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

XI

37. 1 a 1.

This is not true of arithmetical addition, for since by definition 7,

1x,1 x1

by theorem IX

x,(1+a) x(1+a) x1 + x a x + x a.

Whence x a 0, while neither x nor a is zero, which, as will appear directly, is impossible.

XII

38. 0,a 0.

Proof. For call 0, a x. Then by definition 3

x belongs to the class zero.

∴ by definition 4

x 0.

Corollary 1. The same reasoning applies to arithmetical multiplication.

Corollary 2. From theorem x and the last corollary it follows that if ab 0, either a 0 or b 0.

XIII

39. a,a a. See 4n. ^†1

XIV

40. a a a. See 4n. ^†1

These do not hold with arithmetical operations.

41. General Scholium. This concludes the theorems relating to the direct operations. As the inverse operations have no peculiar logical interest, they are passed over here.
In order to prevent misapprehension, I will remark that I do not undertake to demonstrate the principles of logic themselves. Indeed, as I have shown in a previous paper, these principles considered as speculative truths are absolutely empty and indistinguishable. See 2.467. Cf. Paul Weiss, Erkenntnes Bd. 2, H. 4, S. 242-8 where it is shown that all the logical propositions are variations of some such form as PQ+P̅Q+PQ̅+~(PQ) ^†2 But what has been proved is the maxims of logical procedure, a certain system of signs being given.

The definitions given above for the processes which I have termed arithmetical plainly leave the functions of these operations in many cases uninterpreted. Thus if we write

a+b b+a

a+(b+c) (a+b)+c

bc cb

(ab)c a(bc)

a(m+n) a m+a n

we have a series of identities whose truth or falsity is entirely undeterminable. In order, therefore, fully to define those operations, we will say that all propositions, equations, and identities which are in the general case left by the former definitions undetermined as to truth, shall be true, provided they are so in all interpretable cases.

§2. On Arithmetic. The ideas here presented for the derivation of arithmetic from logic are somewhat similar to those employed in the Principia Mathematica, Whitehead and Russell, vol 2, section A (1912). ^†1

42. Equality is a relation of which identity is a species.

If we were to leave equality without further defining it, then by the last scholium all the formal rules of arithmetic would follow from it. And this completes the central design of this paper, as far as arithmetic is concerned.

43. Still it may be well to consider the matter a little further. Imagine, then, a particular case under Boole's calculus, in which the letters are no longer terms of first intention, but terms of second intention, and that of a special kind. Genus, species, difference, property, and accident, are the well-known terms of second intention. These relate particularly to the comprehension See vol. 2, bk. II, ch. 5. for a discussion of these terms. ^†2 of first intentions; that is, they refer to different sorts of predication. Genus and species, however, have at least a secondary reference to the extension See vol. 2, bk. II, ch. 5. for a discussion of these terms. ^†2 of first intentions. Now let the letters, in the particular application of Boole's calculus now supposed, be terms of second intention which relate exclusively to the extension of first intentions. This leads to a somewhat similar definition of a cardinal number as that given in the Principia Mathematica. But see 4.333 where this paper is characterized as being the worst Peirce ever published.^†3 Let the differences of the characters of things and events be disregarded, and let the letters signify only the differences of classes as wider or narrower. In other words, the only logical comprehension which the letters considered as terms will have is the greater or less divisibility of the classes. Thus, n in another case of Boole's calculus might, for example, denote "New England States"; but in the case now supposed, all the characters which make these States what they are being neglected, it would signify only what essentially belongs to a class which has the same relations to higher and lower classes which the class of New England States has, — that is, a collection of six.

44. In this case, the sign of identity will receive a special meaning. For, if m denotes what essentially belongs to a class of the rank of "sides of a cube," then m n will imply, not that every New England State is a side of a cube, and conversely, but that whatever essentially belongs to a class of the numerical rank of "New England States" essentially belongs to a class of the rank of "sides of a cube," and conversely. Identity of this particular sort may be termed equality, and be denoted by the sign =. Thus, in one point of view, identity is a species of equality, and, in another, the reverse is the case. This is because the Being of the copula may be considered on the one hand (with De Morgan [Formal Logic, p. 59]) as a special description of "inconvertible, transitive relation," while, on the other hand, all relation may be considered as a special determination of Being. If a Hegelian should be disposed to see a contradiction here, an accurate analysis of the matter will show him that it is only a verbal one. ^†P1 Moreover, since the numerical rank of a logical sum depends on the identity or diversity (in first intention) of the integrant parts, and since the numerical rank of a logical product depends on the identity or diversity (in first intention) of parts of the factors, logical addition and multiplication can have no place in this system. Arithmetical addition and multiplication, however, will not be destroyed. ab = c will imply that whatever essentially belongs at once to a class of the rank of a, and to another independent class of the rank of b belongs essentially to a class of the rank of c, and conversely. Cf. Principia Mathematica, vol. 2, section A.^†1 a + b = c implies that whatever belongs essentially to a class which is the logical sum of two mutually exclusive classes of the ranks of a and b belongs essentially to a class of the rank of c, and conversely. Cf. Principia Mathematica, vol. 2, section A.^†1 It is plain that from these definitions the same theorems follow as from those given above. Zero and unity will, as before, denote the classes which have respectively no extension and no comprehension; only the comprehension here spoken of is, of course, that comprehension which alone belongs to letters in the system now considered, that is, this or that degree of divisibility; and therefore unity will be what belongs essentially to a class of any rank independent of its divisibility. These two classes alone are common to the two systems, because the first intentions of these alone determine, and are determined by, their second intentions. Finally, the laws of the Boolian calculus, in its ordinary form, are identical with those of this other so far as the latter apply to zero and unity, because every class, in its first intention, is either without any extension (that is, is nothing), or belongs essentially to that rank to which every class belongs, whether divisible or not.

These considerations, together with those advanced [in 1.556], will, I hope, put the relations of logic and arithmetic in a somewhat clearer light than heretofore.

Paper 3: Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Coneptions of Boole's Calculus of Logic Memoirs of the American Academy, vol. 9, pp. 317-78 (1870). Reprinted separately by Welch, Bigelow and Company, Cambridge, Mass., 1870, pp. 1-62. "In 1870 I made a contribution to this subject [logic] which nobody who masters the subject can deny was the most important excepting Boole's original work that ever has been made." — From the "Lowell Lectures," 1903. ^†1

§1. De Morgan's NotationE

45. Relative terms usually receive some slight treatment in works upon logic, but the only considerable investigation into the formal laws which govern them is contained in a valuable paper by Mr. De Morgan in the tenth volume of the Cambridge Philosophical Transactions. "On the Syllogism No. IV, and on the Logic of Relations," pp. 331-58, dated 1859.^†2 He there uses a convenient algebraic notation, which is formed by adding to the well-known spiculæ of that writer the signs used in the following examples.

X . . LY signifies that X is some one of the objects of thought which stand to Y in the relation L, or is one of the L's of Y.

X . LMY signifies that X is not an L of an M of Y.

X . . (L,M)Y signifies that X is either an L or an M of Y.

LM' an L of every M. L[,]M an L of none but M's.

L[-1]Y something to which Y is L. l (small L) non-L.

This system still leaves something to be desired. Moreover, Boole's logical algebra has such singular beauty, so far as it goes, that it is interesting to inquire whether it cannot be extended over the whole realm of formal logic, instead of being restricted to that simplest and least useful part of the subject, the logic of absolute terms, which, when he wrote, was the only formal logic known. The object of this paper is to show that an affirmative answer can be given to this question. I think there can be no doubt that a calculus, or art of drawing inferences, based upon the notation I am to describe, would be perfectly possible and even practically useful in some difficult cases, and particularly in the investigation of logic. I regret that I am not in a situation to be able to perform this labor, but the account here given of the notation itself will afford the ground of a judgment concerning its probable utility.

46. In extending the use of old symbols to new subjects, we must of course be guided by certain principles of analogy, which, when formulated, become new and wider definitions of these symbols. As we are to employ the usual algebraic signs as far as possible, it is proper to begin by laying down definitions of the various algebraic relations and operations. The following will, perhaps, not be objected to.

§2. General Definitions of the Algebraic Signs

47. Inclusion in or being as small as is a transitive relation. The consequence holds that I use the sign ⤙ in place of ≦. My reasons for not liking the latter sign are that it cannot be written rapidly enough, and that it seems to represent the relation it expresses as being compounded of two others which in reality are complications of this. It is universally admitted that a higher conception is logically more simple than a lower one under it. Whence it follows from the relations of extension and comprehension, that in any state of information a broader concept is more simple than a narrower one included under it. Now all equality is inclusion in, but the converse is not true; hence inclusion in is a wider concept than equality, and therefore logically a simpler one. On the same principle, inclusion is also simpler than being less than. The sign ≦ seems to involve a definition by enumeration; and such a definition offends against the laws of definition.^†P1

48. Equality is the conjunction of being as small as and its converse. To say that x = y is to say that x ⤙ y and y ⤙ x.

49. Being less than is being as small as with the exclusion of its converse. To say that x < y is to say that x ⤙ y, and that it is not true that y ⤙ x.

50. Being greater than is the converse of being less than. To say that x > y is to say that y < x.

51. Addition is an associative operation. That is to say, I write a comma below the sign of addition, except when (as is the case in ordinary algebra) the corresponding inverse operation (subtraction) is determinative, [i.e., except when the addition is arithmetical.] ^†P1

(x y) z = x (y z).

Addition is a commutative operation. That is,

x y = y x.

52. Invertible I.e., arithmetical. ^†1 addition is addition the corresponding inverse of which is determinative. The last two formulæ hold good for it, and also the consequence that

53. Multiplication is an operation which is doubly distributive with reference to addition. That is,

x(y z) = xy xz,

(x y)z = xz yz.

Multiplication is almost invariably an associative operation. Cf. 55, 69f. ^†3

(xy)z = x(yz).

Multiplication is not generally commutative. If we write commutative I.e., "non-relative," or what was before called "logical." ^†4 multiplication with a comma, Cf. 73, 74n. ^†5 we have

x,y = y,x.

54. Invertible I.e., arithmetical. See 75. ^†1 multiplication is multiplication whose corresponding inverse operation (division) is determinative. We may indicate this by a dot; The symbolism of the earlier papers is here slightly modified: the simple conjunction of terms now represents relative, instead of arithmetical, multiplication, and the dot is introduced to represent arithmetical multiplication. The comma, however, is still retained for logical multiplication. ^†2 and then the consequence holds that

55. Functional multiplication Cf. 71 and 72.^†4 is the application of an operation to a function. It may be written like ordinary multiplication; but then there will generally be certain points where the associative principle does not hold. Thus, if we write (sin abc) def, there is one such point. If we write (log (base abc) def) ghi, there are two such points. The number of such points depends on the nature of the symbol of operation, and is necessarily finite. If there were many such points, in any case, it would be necessary to adopt a different mode of writing such functions from that now usually employed. We might, for example, give to "log" such a meaning that what followed it up to a certain point indicated by a † should denote the base of the system, what followed that to the point indicated by a ‡ should be the function operated on, and what followed that should be beyond the influence of the sign "log." Thus log abc † def ‡ ghi would be (log abc) ghi, the base being def. In this paper I shall adopt a notation very similar to this, which will be more conveniently described further on.

56. The operation of involution obeys the formula In the notation of quaternions Hamilton has assumed (xy)z=x(zy)instead of(xy)z=x(yz), although it appears to make but little difference which he takes. Perhaps we should assume two involutions, so that (xy)z=x(yz),z(yx)=(zy)x. But in this paper only the former of these is required. [See 113ff. for the latter.] ^†P1

(xy)z = x(yz).

Involution, also, follows the indexical principle.

xy z = xy,xz.

Involution, also, satisfies the binomial theorem. See 77. ^†1

(x y)z = xz Σpxz-p,yp yz,

where Σp denotes that p is to have every value less than z, and is to be taken out of z in all possible ways, and that the sum of all the terms so obtained of the form xz-p,yp is to be taken.

57. Subtraction is the operation inverse to addition. We may write indeterminative I.e., logical; see 10. ^†2 subtraction with a comma below the usual sign. Then we shall have that

(x y) y = x,

(x - y) + y = x,

(x + y) - y = x.

58. Division is the operation inverse to multiplication. Since multiplication is not generally commutative it is necessary to have two signs for division. I shall take

(x:y)y = x,

x y/x = y.

59. Division inverse to that multiplication which is indicated by a comma may be indicated by a semicolon. So that

(x;y),y = x. See 10 (24). ^†3

60. Evolution and taking the logarithm are the operations inverse to involution.

(x√y)x = y,

xlogxy = y.

61. These conditions are to be regarded as imperative. But in addition to them there are certain other characters which it is highly desirable that relations and operations should possess, if the ordinary signs of algebra are to be applied to them. These I will here endeavour to enumerate.

1. It is an additional motive for using a mathematical sign to signify a certain operation or relation that the general conception of this operation or relation should resemble that of the operation or relation usually signified by the same sign. In particular, it will be well that the relation expressed by ⤙ should involve the conception of one member being in the other; addition, that of taking together; multiplication, that of one factor's being taken relatively to the other (as we write 3 X 2 for a triplet of pairs, and Dφ for the derivative of φ); and involution, that of the base being taken for every unit of the exponent.

2. In the second place, it is desirable that, in certain general circumstances, determinate numbers should be capable of being substituted for the letters operated upon, and that when so substituted the equations should hold good when interpreted in accordance with the ordinary definitions of the signs, so that arithmetical algebra should be included under the notation employed as a special case of it. For this end, there ought to be a number known or unknown, which is appropriately substituted in certain cases, for each one of, at least, some class of letters.

3. In the third place, it is almost essential to the applicability of the signs for addition and multiplication, that a zero and a unity should be possible. By a zero I mean a term such that

x 0 = x,

whatever the signification of x; and by a unity a term for which the corresponding general formula

x1 = x

holds good. On the other hand, there ought to be no term a such that ax=x, independently of the value of x.

4. It will also be a strong motive for the adoption of an algebraic notation, if other formulæ which hold good in arithmetic, such as

xz,yz = (x,y)z,

1x = x,

x1 = x,

x0 = 0,

continue to hold good; if, for instance, the conception of a differential is possible, and Taylor's Theorem holds, and The symbol represents the base of Naperian logarithms, the symbol represents π and represents the square root of the negative in Benjamin Peirce's Linear Associative Algebras, §15ff (1870). ^†1 or (1+i)1/i plays an important part in the system, if there should be a term having the properties of The symbol represents the base of Naperian logarithms, the symbol represents π and represents the square root of the negative in Benjamin Peirce's Linear Associative Algebras, §15ff (1870). ^†1 (3.14159), or properties similar to those of space should otherwise be brought out by the notation, or if there should be an absurd expression having the properties and uses of The symbol represents the base of Naperian logarithms, the symbol represents π and represents the square root of the negative in Benjamin Peirce's Linear Associative Algebras, §15ff (1870). ^†1 or the square root of the negative.

§3. Application of the Algebraic Signs to Logic

62. While holding ourselves free to use the signs of algebra in any sense conformable to the above absolute conditions, we shall find it convenient to restrict ourselves to one particular interpretation except where another is indicated. I proceed to describe the special notation which is adopted in this paper.

Use of the Letters

63. The letters of the alphabet will denote logical signs. Now logical terms are of three grand classes. The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as "a —." These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (quale); for example, as horse, tree, or man. These are absolute terms. The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms. The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of — to —, or buyer of — for — from —. These may be termed conjugative terms. The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object. Cf. the discussion on the categories in vol. 1, bk. III, and on signs in vol. 2, bk. II. ^†1 No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Cf. 421 and 1.347. ^†2 Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives. I shall denote absolute terms by the Roman alphabet, a, b, c, d, etc.; relative terms by italics, a, b, c, d, etc.; and conjugative terms by a kind of type called Kennerly, 𝐚, 𝐛, 𝐜, 𝐝, etc.

I shall commonly denote individuals by capitals, and generals Cf. 69n. ^†3 by small letters. General symbols for numbers will be printed in black-letter, thus, 𝖆, 𝖇, 𝖈, 𝖉, etc. The Greek letters will denote operations.

64. To avoid repetitions, I give here a catalogue of the letters I shall use in examples in this paper, with the significations I attach to them.

Numbers Corresponding to Letters

65. I propose to use the term "universe" to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's, "On the Structure of the Syllogism," Section 1, Cambridge Philosophical Transactions, vol. 8 (1846); Formal Logic, p. 37 (1847). ^†1 is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains. Cf. 2.518ff. ^†2

I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (men), the number of "tooth of" would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus [t].

The Signs of Inclusion, Equality, etc.

66. I shall follow Boole Laws of Thought, p. 27.^†3 in taking the sign of equality to signify identity. Thus, if v denotes the Vice-President of the United States, and p the President of the Senate of the United States,

v = p

means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President. The sign "less than" is to be so taken that

f < m

means every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense. According to De Morgan, Formal Logic, p. 334. De Morgan refers to the first edition of Drobisch's Logic. The third edition contains nothing of the sort. ^†P1 It will follow from these significations of = and < that the sign ⤙ (or ≦, "as small as") will mean "is." Thus,

f ⤙ m

means "every Frenchman is a man," without saying whether there are any other men or not. So,

m ⤙ l

will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of = and < plainly conform to the indispensable conditions. Upon the transitive character of these relations the syllogism depends, for by virtue of it, from

that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal. But not only do the significations of = and < here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. Cf. 42-44. A class is the extension of a collection; see e.g., 537n. ^†1 So, to write 5 < 7 is to say that 5 is part of 7, just as to write f < m is to say that Frenchmen are part of men. Indeed, if f < m, then the number of Frenchmen is less than the number of men, and if v = p, then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.

The Signs for Addition

67. The sign of addition is taken by Boole, Op. cit., p. 33. ^†2 so that

x + y

denotes everything denoted by x, and, besides, everything denoted by y. Thus

m + w

denotes all men, and, besides, all women. This signification for this sign is needed for connecting the notation of logic with that of the theory of probabilities. But if there is anything which is denoted by both the terms of the sum, the latter no longer stands for any logical term on account of its implying that the objects denoted by one term are to be taken besides the objects denoted by the other. For example,

f + u

means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are besides themselves. For this reason alone, in a paper which is published in the Proceedings of the Academy for March 17, 1867, In 3. ^†1 I preferred to take as the regular addition of logic a noninvertible process, such that

m b

stands for all men and black things, without any implication that the black things are to be taken besides the men; and the study of the logic of relatives has supplied me with other weighty reasons for the same determination. Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in a tract called Pure Logic, or the Logic of Quality, [1864] Ch. 6, §63; ch. 15, §177ff.^†2 had anticipated me in substituting the same operation for Boole's addition, although he rejects Boole's operation entirely and writes the new one with a + sign while withholding from it the name of addition. In another book [Substitution of Similars (1869) and subsequent works] he uses the sign ·|· instead of +. ^†P1 It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of taking together involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves—provided all the terms summed are mutually exclusive. Addition being taken in this sense, nothing is to be denoted by zero, for then

x 0 = x,

whatever is denoted by x; and this is the definition of zero. Cf. 82. ^†1 This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of zero and that of nothing, and because we shall thus have

[0] = 0

The Signs for Multiplication

68. I shall adopt for the conception of multiplication the application of a relation, in such a way that, for example, lw shall denote whatever is lover of a woman. This notation is the same as that used by Mr. De Morgan, although he appears not to have had multiplication in his mind. s(m w) will, then, denote whatever is servant of anything of the class composed of men and women taken together. So that

s(m w) = sm sw.

(l s)w will denote whatever is lover or servant to a woman, and

(l s)w = lw sw.

(sl)w will denote whatever stands to a woman in the relation of servant of a lover, and

(sl)w = s(lw).

Thus all the absolute conditions of multiplication are satisfied.

The term "identical with—" is a unity for this multiplication. That is to say, if we denote "identical with—" by 1 we have

x1 = x,

whatever relative term x may be. For what is a lover of something identical with anything, is the same as a lover of that thing.

69. A conjugative term like giver naturally requires two correlates, one denoting the thing given, the other the recipient of the gift. We must be able to distinguish, in our notation, the giver of A to B from the giver to A of B, and, therefore, I suppose the signification of the letter equivalent to such a relative to distinguish the correlates as first, second, third, etc., so that "giver of — to —" and "giver to — of —" will be expressed by different letters. Let g denote the latter of these conjugative terms. Then, the correlates or multiplicands of this multiplier cannot all stand directly after it, as is usual in multiplication, but may be ranged after it in regular order, so that

gxy

will denote a giver to x of y. But according to the notation, x here multiplies y, so that if we put for x owner (o), and for y horse (h),

goh

appears to denote the giver of a horse to an owner of a horse. But let the individual horses be H, H', H'', etc. Then

h h is a variable designating an unspecified H. See 84, 94, 111. ^†1 = H H' H'' etc.

goh = go(H H' H'' etc.) = goH goH' goH'' etc.

Now this last member must be interpreted as a giver of a horse to the owner of that horse, and this, therefore, must be the interpretation of goh. This is always very important. A term multiplied by two relatives shows that the same individual is in the two relations. If we attempt to express the giver of a horse to a lover of a woman, and for that purpose write

glwh,

we have written giver of a woman to a lover of her, and if we add brackets, thus,

g(lw)h,

we abandon the associative principle of multiplication. A little reflection will show that the associative principle must in some form or other be abandoned at this point. But while this principle is sometimes falsified, it oftener holds, and a notation must be adopted which will show of itself when it holds. We already see that we cannot express multiplication by writing the multiplicand directly after the multiplier; let us then affix subjacent numbers after letters to show where their correlates are to be found. The first number shall denote how many factors must be counted from left to right to reach the first correlate, the second how many more must be counted to reach the second, and so on. Then, the giver of a horse to a lover of a woman may be written

g12l1wh = g11l2hw = g2-1hl1w.

70. Of course a negative number indicates that the former correlate follows the latter by the corresponding positive number. A subjacent zero makes the term itself the correlate. Thus,

l0

denotes the lover of that lover or the lover of himself, just as goh denotes that the horse is given to the owner of itself, for to make a term doubly a correlate is, by the distributive principle, to make each individual doubly a correlate, so that

l0 = L0 L0' L0'' etc.

A subjacent sign of infinity may indicate that the correlate is indeterminate, so that

l∞

will denote a lover of something. We shall have some confirmation of this presently. See 73. ^†1

If the last subjacent number is a one it may be omitted. Thus we shall have

l1 = l,

g11 = g1 = g.

This enables us to retain our former expressions lw, goh, etc.

71. The associative principle does not hold in this counting of factors. Because it does not hold, these subjacent numbers are frequently inconvenient in practice, and I therefore use also another mode of showing where the correlate of a term is to be found. This is by means of the marks of reference, † ‡ || § ¶, which are placed subjacent to the relative term and before and above the correlate. Thus, giver of a horse to a lover of a woman may be written

g†‡†l||||w‡h.

The asterisk I use exclusively to refer to the last correlate of the last relative of the algebraic term.

72. Now, considering the order of multiplication to be: — a term, a correlate of it, a correlate of that correlate, etc., — there is no violation of the associative principle. The only violations of it in this mode of notation are that in thus passing from relative to correlate, we skip about among the factors in an irregular manner, and that we cannot substitute in such an expression as goh a single letter for oh. I would suggest that such a notation may be found useful in treating other cases of non-associative multiplication. By comparing this with what was said above In 55. ^†1 concerning functional multiplication, it appears that multiplication by a conjugative term is functional, and that the letter denoting such a term is a symbol of operation. I am therefore using two alphabets, the Greek and Kennerly, where only one was necessary. But it is convenient to use both.

73. Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives. Now the absolute term "man" is really exactly equivalent to the relative term "man that is —," and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term. Then man that is black will be written

m,b.

But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one. Then

l,sw l, s is the logical product of l and s; ls, on the other hand, is the relative product of l and s. See 54n, 68. ^†2

will denote a lover of a woman that is a servant of that woman. The comma here after l should not be considered as altering at all the meaning of l, but as only a subjacent sign, serving to alter the arrangement of the correlates. In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates. So

m,,b,r,

means a man that is a rich individual and is a black that is that rich individual. But this has no other meaning than

m,b,r ,

or a man that is a black that is rich. Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number. If, therefore, l,,sw is not the same as l,sw (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates. And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should), we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series "that is—and is—and is—etc." Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates? Any term may be regarded as having an infinite number of factors, those at the end being ones, thus,

l,sw = l,sw,1,1,1,1,1,1,1, etc.

A subjacent number may therefore be as great as we please. But all these ones denote the same identical individual denoted by w; what then can be the subjacent numbers to be applied to s, for instance, on account of its infinite "that is" 's? What numbers can separate it from being identical with w? There are only two. The first is zero, which plainly neutralizes a comma completely, since

s,0w = sw, A servant of herself who is also a woman is the same as a servant of a woman? ^†1

and the other is infinity; for as 1∞ is indeterminate in ordinary algebra, so it will be shown hereafter to be here, so that to remove the correlate by the product of an infinite series of ones is to leave it indeterminate. Accordingly,

m,∞

should be regarded as expressing some man. Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros.

"Something" may then be expressed by

1∞. "Something" is whatever is identical with an undetermined thing. ^†1

I shall for brevity frequently express this by an antique figure one (𝟏).

"Anything" by

10. "Anything" is whatever is identical with itself.^†2

I shall often also write a straight 1 for anything.

74. It is obvious that multiplication into a multiplicand indicated by a comma is commutative, It will often be convenient to speak of the whole operation of affixing a comma and then multiplying, as a commutative multiplication, the sign for which is the comma. But though this is allowable, we shall fall into confusion at once if we ever forget that in point of fact it is not a different multiplication, only it is multiplication by a relative whose meaning — or rather whose syntax — has been slightly altered; and that the comma is really the sign of this modification of the foregoing term. ^†P1 that is,

s,l = l,s.

This multiplication is effectively the same as that of Boole in his logical calculus. Boole's unity is my 1, that is, it denotes whatever is.

75. The sum x + x generally denotes no logical term. But x,∞ + x,∞ may be considered as denoting some two x's. It is natural to write

where the dot shows that this multiplication is invertible. We may also use the antique figures so that

Then 𝟐 alone will denote some two things. But this multiplication is not in general commutative, and only becomes so when it affects a relative which imparts a relation such that a thing only bears it to one thing, and one thing alone bears it to a thing. For instance, the lovers of two women are not the same as two lovers of women, that is,

l2.w and 2.lw

are unequal; but the husbands of two women are the same as two husbands of women, that is,

76. The conception of multiplication we have adopted is that of the application of one relation to another. So, a quaternion being the relation of one vector to another, the multiplication of quaternions is the application of one such relation to a second. Even ordinary numerical multiplication involves the same idea, for 2 X 3 is a pair of triplets, and 3 X 2 is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives.

If we have an equation of the form

xy = z,

and there are just as many x's per y as there are per things, things of the universe, then we have also the arithmetical equation,

[x][y] = [z].

For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then

[t][f] = [t f]

holds arithmetically. So if men are just as apt to be black as things in general,

[m,][b] = [m,b],

where the difference between [m] and [m,] must not be overlooked. It is to be observed that

[1] = 𝟏.

Boole was the first to show this connection between logic and probabilities. Op. cit., p. 243ff. ^†1 He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation.

Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.

The Sign of Involution

77. I shall take involution in such a sense that xy will denote everything which is an x for every individual of y. Thus lw will be a lover of every woman. Then (sl)w will denote whatever stands to every woman in the relation of servant of every lover of hers; and s(lw) will denote whatever is a servant of everything that is lover of a woman. So that

(sl)w = s(lw).

A servant of every man and woman will be denoted by sm w This is more accurately read as: a servant of all those who are either men or women. ^†1, and sm, sw will denote a servant of every man that is a servant of every woman. So that

sm w = sm,sw.

That which is emperor or conqueror of every Frenchman will be denoted by (e c)f, and ef Σpef-p, cp cf will denote whatever is emperor of every Frenchman or emperor of some Frenchmen and conqueror of all the rest, or conqueror of every Frenchman. Consequently,

(e c)f = ef Σpef-p,cp cf.

Indeed, we may write the binomial theorem so as to preserve all its usual coefficients; for we have

(e c)f = ef[f].ef-†1,c𝟏†(([f].([f]-𝟏))/𝟐).ef-‡𝟐,c𝟐‡ etc. Cf. J. N. Lambert. "Sechs Versuche einer Zeichenkunst in der Vernunftlehre"; in Logische u. Philosophische Abhandlungen, vol. 1, ed. J. Bernouilli, Berlin, (1782) for the use of this "Newtonian Formula" in an intensional logic of absolute terms. ^†2

That is to say, those things each of which is emperor or conqueror of every Frenchman consist, first, of all those individuals each of which is a conqueror [emperor!] of every Frenchman; second, of a number of classes equal to the number of Frenchmen, each class consisting of everything which is an emperor of every Frenchman but some one and is a conqueror of that one; third, of a number of classes equal to half the product of the number of Frenchmen by one less than that number, each of these classes consisting of every individual which is an emperor of every Frenchman except a certain two, and is conqueror of those two, etc. This theorem holds, also, equally well with invertible addition, and either term of the binomial may be negative provided we assume

(—x)y = (—)[y].xy.

In addition to the above equations which are required to hold good by the definition of involution, the following also holds,

(s,l)w = sw,lw, I.e., a servant and lover of every woman is a servant of every woman and a lover of every woman.^†1

just as it does in arithmetic.

78. The application of involution to conjugative terms presents little difficulty after the explanations which have been given under the head of multiplication. It is obvious that betrayer to every enemy should be written

𝐛a,

just as lover of every woman is written

lw,

but 𝐛 = 𝐛11 and therefore, in counting forward as the subjacent numbers direct, we should count the exponents, as well as the factors, of the letter to which the subjacent numbers are attached. Then we shall have, in the case of a relative of two correlates, six different ways of affixing the correlates to it, thus:

If both correlates are absolute terms, the cases are

These interpretations are by no means obvious, but I shall show that they are correct further on. See 145. ^†1

79. It will be perceived that the rule still holds here that

(𝐛a)m = 𝐛(am)

that is to say, that those individuals each of which stand to every man in the relation of betrayer to every enemy of his are identical with those individuals each of which is a betrayer to every enemy of a man of that man.

80. If the proportion of lovers of each woman among lovers of other women is equal to the average number of lovers which single individuals of the whole universe have, then

[lw] = [lW'] [lW''] [lW'''] etc.=[l][w].

Thus arithmetical involution appears as a special case of logical involution.

§4. General Formulæ

81. The formulæ which we have thus far obtained, exclusive of mere explanations of signs and of formulæ relating to the numbers of classes, are:

(1)	If x ⤙ y and y ⤙ z, then x ⤙ z.
(2)	(x y) z = x (y z).	(Jevons)
(3)	x y = y x.	(Jevons)
(4)	(x y)z = xz yz.
(5)	x(y z) = xy xz.
(6)	(xy)z = x(yz).
(7)	x,(y z) = x,y x,z.	(Jevons)
(8)	(x,y),z = x,(y,z).	(Boole)
(9)	x,y, = y,x.	(Boole)
(10)	(xy)z = x(yz).
(11)	xy z = xy,xz.
(12)	(x y)z = xz Σp(xx-p,yp) yp
	= xz[z].xz-†𝟏,y†𝟏 (([z].[z-𝟏])/𝟐).xz-‡𝟐,y‡𝟐
	(([z].[z-𝟏].[z-𝟐])/(𝟐.𝟑)).xz-||𝟑,y||𝟑 etc.
(13)	(x,y)z = xz,yz.
(14)	x + 0 = x.	(Boole)
(15)	x1 = x.
(16)	(x + y) + z = x + (y + z).	(Boole)
(17)	x + y = y + x.	(Boole)
(18)	x + y - y = x.	(Boole)
(19)	x,(y + z) = x,y + x,z.	(Boole)
(20)	(x + y)z = x + [z].xz-†1,y†𝟏 + etc.

We have also the following, which are involved implicitly in the explanations which have been given.

(21)

x ⤙ x y. This proposition is the source of the famous so-called paradoxes of material implication. †1

This, I suppose, is the principle of identity, for it follows from this that x = x. This sentence seems to have been misplaced and should have appeared after (22) or (23). ^†2

(22)	x x = x.	(Jevons)
(23)	x,x = x.	(Boole)
(24)	x y = x + y - x,y.

The principle of contradiction is

where n stands for "not." The principle of excluded middle is

(26)

x nx = 1.

It is an identical proposition, that, if φ be determinative, we have

The six following are derivable from the formulæ already given:

(28)	(x y),(x z) = x y,z.
(29)	(x - y) (z - w) = (x z)-(y w) + y,z,(1-w) + x,(1 - y),w.

In the following, φ is a function involving only the commutative operations and the operations inverse to them.

(30)	φx = (φ1),x + (φ0),(1 - x).	(Boole)
(31)	φx = (φ1(1-x)),(φ0x).
(32)	If φx = 0 (φ1),(φ0) = 0.	(Boole)
(33)	If φx = 1 φ1 φ0 = 1.

The reader may wish information concerning the proofs of formulæ (30) to (33). When involution is not involved in a function nor any multiplication except that for which x,x=x, it is plain that φx is of the first degree, and therefore, since all the rules of ordinary algebra hold, we have as in that

φx = φ0 + (φ1 - φ0),x.

We shall find, hereafter, that when y has a still more general character, we have,

φx = φ0 + (φ1 - φ0)x.

The former of these equations by a simple transformation gives (30).

If we regard (φ1), (φ0) as a function of x and develop it by (30), we have

(φ1),(φ0) = x,(φ1),(φ0) + (φ1),(φ0),(1-x).

Comparing these terms separately with the terms of the second member of (30), we see that

(φ1),(φ0) ⤙ x.

This gives at once (32), and it gives (31) after performing the multiplication indicated in the second member of that equation and equating φx to its value as given in (30). If (φ1 φ0) is developed as a function of x by (31), and the factors of the second member are compared with those of the second member of (31), we get

φx ⤙ φ1 φ0,

from which (33) follows immediately.

Properties of Zero and Unity

82. The symbolical definition of zero is

x+0 = x,

Hence, from the invertible character of this addition, and the generality of (14), we have

x,0 = 0.

By (24) we have in general,

	x 0 = x + 0 - x,0 =	x,
or	x 0 =	x.
By (4) we have	a x = (a 0)x =	a x 0x.
But if a is an absurd relation,	a x =	0,
so that	0x =	0,
which must hold invariably.

From (12) we have	ax = (a 0)x = ax 0x etc.,
whence by (21)	0x ⤙ ax. I.e., by (21) 0x ⤙ ax 0x and as by (12) ax + 0x = ax, 0x ⤙ ax. †1

But if a is an absurd relation, and x is not zero,

ax = 0.

83. Any relative x may be conceived as a sum of relatives X, X', X'', etc., such that there is but one individual to which anything is X, but one to which anything is X', etc. Thus, if x denote "cause of," X,X',X'' would denote different kinds of causes, the causes being divided according to the differences of the things they are causes of. Then we have

X y = X(y 0) = X y X0,

whatever y may be. Hence, since y may be taken so that

and in a similar way,

We have, then,

x0 = (X X' X'' X''' etc.)0
= X0 X'0 X''0 X'''0 etc. = 0.

84. If the relative x be divided in this way into X,X',X'', X''', etc., so that x is that which is either X or X' or X'' or X''', etc., then non-x is that which is at once non-X and non-X' and non-X'', etc.; that is to say,

non-x = non-X, non-X', non-X'', non-X''', etc.;

where non-X is such that there is something (Z) such that everything is non-X to Z; and so with non-X', non-X'', etc. Now, non-x may be any relative whatever. Substitute for it, then, y; and for non-X, non-X', etc., Y,Y', etc. Then we have

where Z',Z'',Z''' are individual terms which depend for what they denote on Y',Y'',Y'''.

Then we have

1 = Y'Z' = Y'Z' = Y'(Z' 0) = Y'Z',Y'0 = Y'Z',Y'0,
or Y'0 = 1,	Y''0 = 1,	Y'''0= 1,	etc.
Then	y0 = (Y', Y'', Y''', etc.)0 = Y'0, Y''0, Y'''0, etc. = 1.

Now a may express any relation whatever, but things the same way related to everything are the same. Hence,

where X,X' etc, denote individuals (and by the very meaning of a general term this can always be done, whatever x may be)

We have by (24)	x 1 = x + 1 - x,1 = x + 1 - x = 1,
or	x 1 = 1.

85. We may divide all relatives into limited and unlimited. Limited relatives express such relations as nothing has to everything. For example, nothing is knower of everything. Unlimited relatives express relations such as something has to everything. For example, something is as good as anything. For limited relatives, then, we may write

p1 = 0.

The converse of an unlimited relative expresses a relation which everything has to something. Thus, everything is as bad as something. Denoting such a relative by q,

q1 = 1.

These formulæ remind one a little of the logical algebra of Boole; because one of them holds good in arithmetic only for zero, and the other only for unity.

We have by (10)	1x = (q0)x = q(0x) = q0 = 1,
or	1x = 1.
We have by (4)	1x = (a 1)x = a x 1x,
or by (21)	a x ⤙ 1x.

But everything is somehow related to x unless x is 0; hence unless x is 0,

1x = 1.

If a denotes "what possesses," and y "character of what is denoted by x,"

Since 1 means "identical with," l,1w denotes whatever is both a lover of and identical with a woman, or a woman who is a lover of herself. And thus, in general,

x,1 = x[0],.

86. Nothing is identical with every one of a class; and therefore 1x is zero, unless x denotes only an individual when 1x becomes equal to x. But equations founded on interpretation may not hold in cases in which the symbols have no rational interpretation.

Collecting together all the formulæ relating to zero and
unity, we have

(34)	x 0 = x.	(Jevons)
(35)	x 1 = 1.	(Jevons)
(36)	x0 = 0.
(37)	0x = 0.
(38)	x,0 = 0.	(Boole)
(39)	x0 = 1.
(40)	0x = 0, provided x > 0. On his own copy, Peirce substitutes the condition "x is an unlimited relative," for "x > 0." †1
(41)	1x = x.
(42)	x,1 = x0,.
(43)	x1 = x.
(44)	1x = 0, unless x is individual, when 1x = x.
(45)	q1 = 1, where q is the converse of an unlimited relative.
(46)	1x = 1, provided x > 0. See 145. †1
(47)	x,1 = x.	(Boole)
(48)	p1 = 0, where p is a limited relative.
(49) 1	x = 1.

These, again, give us the following:

(50)	0 1 = 1	(64)	01=0
(51)	0 1 = 1	(65)	1 1=1
(52)	00 = 0	(66)	1,1=1
(53)	0,0 = 0	(67)	11=1
(54)	00 = 1	(68)	11 = 1
(55)	10 = 0	(69)	1,1 = 1
(56)	01 = 0	(70)	11 = 1
(57)	0,1 = 0	(71)	11=1
(58)	01 = 0	(72)	11 = 1
(59)	10 = 1	(73)	1,1 = 1
(60)	01 = 0	(74)	11 = 1
(61)	10 = 0	(75)	11 = 0
(62)	0,1 = 0	(76)	1, = 1
(63)	10 = 1

From (64) we may infer that 0 is a limited relative, and from (60) that it is not the converse of an unlimited relative. From (70) we may infer that 1 is not a limited relative, and from (68) that it is the converse of an unlimited relative.

Formulæ Relating to the Numbers of Terms

87. We have already seen that

(77)	If x ⤙ y, then [x] ⤙ [y].
(78)	When x,y = 0, then [x y] = [x] [y],
(79)	When [xy]:[nxy] = [x]:[nx], then [xy] = [x][y].
(80)	When [x𝖓y] = [x][𝖓y][1], then [x y] = [x][y].

It will be observed that the conditions which the terms must conform to, in order that the arithmetical equations shall hold, increase in complexity as we pass from the more simple relations and processes to the more complex.

88. We have seen that

Most commonly the universe is unlimited, and then

and the general properties of 1 correspond with those of infinity. Thus,

x 1 = 1	corresponds to	x + ∞ = ∞,
q1 = 1	corresponds to	q ∞ = ∞,
1x = 1	corresponds to	∞ x= ∞,
p1 = 0	corresponds to	p ∞ = 0,
1x = 1	corresponds to	∞ x = ∞.

The formulæ involving commutative multiplication are derived from the equation 1, = 1. But if 1 be regarded as infinite, it is not an absolute infinite; for 10 = 0. On the other hand, 11 = 0.

It is evident, from the definition of the number of a term, that

We have, therefore, if the probability of an individual being x to any y is independent of what other y's it is x to, and if x is independent of y,

§5. General Method of Working with This Notation

89. Boole's logical algebra contains no operations except our invertible addition and commutative multiplication, together With the corresponding subtraction and division. He has, therefore, only to expand expressions involving division, by means of (30), so as to free himself from all non-determinative operations, in order to be able to use the ordinary methods of algebra, which are, moreover, greatly simplified by the fact that

x,x = x.

90. Mr. Jevons's modification In his Pure Logic. ^†1 of Boole's algebra involves only non-invertible addition and commutative multiplication, without the corresponding inverse operations. He is enabled to replace subtraction by multiplication, owing to the principle of contradiction, and to replace division by addition, owing to the principle of excluded middle. For example, if x be unknown, and we have

x m = a,

or what is denoted by x together with men make up animals, we can only conclude, with reference to x, that it denotes (among other things, perhaps) all animals not men; that is, that the x's not men are the same as the animals not men. Let m̅ denote non-men; then by multiplication we have

xm̅, m,m̅ = x,m̅ = a,m̅,

because, by the principle of contradiction,

m,m̅ = 0.

Or, suppose, x being again unknown, we have given

a,x = m.

Then all that we can conclude is that the x's consist of all the m's and perhaps some or all of the non-a's, or that the x's and non-a's together make up the m's and non-a's together. If, then, ā denote non-a, add ā to both sides and we have

	a,x ā = m ā.
Then by (28)	(a ā),(x ā) = m ā.

But by the principle of excluded middle,

	a ā = 1
and therefore	x ā = m ā.

I am not aware that Mr. Jevons actually uses this latter process, but it is open to him to do so. In this way, Mr. Jevons's algebra becomes decidedly simpler even than Boole's.

It is obvious that any algebra for the logic of relatives must be far more complicated. In that which I propose, we labor under the disadvantages that the multiplication is not generally commutative, that the inverse operations are usually indeterminative, and that transcendental equations, and even equations like

abx = cdex + fx + x,

where the exponents are three or four deep, are exceedingly common. It is obvious, therefore, that this algebra is much less manageable than ordinary arithmetical algebra.

91. We may make considerable use of the general formulæ already given, especially of (1), (21), and (27), and also of the following, which are derived from them:

(86)	If a ⤙ b	then there is such a term x that a x = b.
(87)	If a ⤙ b	then there is such a term x that b,x = a.
(88)	If b,x = a	then a ⤙ b.
(89)	If a ⤙ b	c a ⤙ c b.
(90)	If a ⤙ b	ca ⤙ cb.
(91)	If a ⤙ b	ac ⤙ bc.
(92)	If a ⤙ b	cb ⤙ ca Where c = not, this becomes the formula for contraposition. See 142 and 2.550. In 186, however, a different formula for contraposition is given. †1
(93)	If a ⤙ b	ac ⤙ bc.
(94)	a,b ⤙ a

There are, however, very many cases in which the formulæ thus far given are of little avail.

92. Demonstration of the sort called mathematical is founded on suppositions of particular cases. The geometrician draws a figure; the algebraist assumes a letter to signify a single quantity fulfilling the required conditions. But while the mathematician supposes an individual case, his hypothesis is yet perfectly general, because he considers no characters of the individual case but those which must belong to every such case. The advantage of his procedure lies in the fact that the logical laws of individual terms are simpler than those which relate to general terms, because individuals are either identical or mutually exclusive, and cannot intersect or be subordinated to one another as classes can. Mathematical demonstration is not, therefore, more restricted to matters of intuition than any other kind of reasoning. Indeed, logical algebra conclusively proves that mathematics extends over the whole realm of formal logic; and any theory of cognition which cannot be adjusted to this fact must be abandoned. We may reap all the advantages which the mathematician is supposed to derive from intuition by simply making general suppositions of individual cases.

93. In reference to the doctrine of individuals, Cf. the definition of individuals in 611-13.^†1 two distinctions should be borne in mind. The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied. For, let A be such a term. Then, if it is neither true that all A is X nor that no A is X, it must be true that some A is X and some A is not X; and therefore A may be divided into A that is X and A that is not X, which is contrary to its nature as a logical atom. Such a term can be realized neither in thought nor in sense. Not in sense, because our organs of sense are special — the eye, for example, not immediately informing us of taste, so that an image on the retina is indeterminate in respect to sweetness and non-sweetness. When I see a thing, I do not see that it is not sweet, nor do I see that it is sweet; and therefore what I see is capable of logical division into the sweet and the not sweet. It is customary to assume that visual images are absolutely determinate in respect to color, but even this may be doubted. I know no facts which prove that there is never the least vagueness in the immediate sensation. In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates. A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate. Such a term as "the second Philip of Macedon" is still capable of logical division—into Philip drunk and Philip sober, for example; but we call it individual because that which is denoted by it is in only one place at one time. It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them. Such differences we habitually disregard in the logical division of substances. In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others. There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse, and if I be a term which in consequence of such neglect becomes indivisible in that discourse, we have in that discourse,

[I] = 1.

This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual {to atomon} and singular {to kath' hekaston}; but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense. The absolute individual can not only not be realized in sense or thought, but cannot exist, properly speaking. For whatever lasts for any time, however short, is capable of logical division, because in that time it will undergo some change in its relations. But what does not exist for any time, however short, does not exist at all. All, therefore, that we perceive or think, or that exists, is general. So far there is truth in the doctrine of scholastic realism. But all that exists is infinitely determinate, and the infinitely determinate is the absolutely individual. This seems paradoxical, but the contradiction is easily resolved. That which exists is the object of a true conception. This conception may be made more determinate than any assignable conception; and therefore it is never so determinate that it is capable of no further determination. ^†P1

94. The old logics distinguish between individuum signatum and individuum vagum. "Julius Cæsar" is an example of the former; "a certain man," of the latter. The individuum vagum, in the days when such conceptions were exactly investigated, occasioned great difficulty from its having a certain generality, being capable, apparently, of logical division. If we include under the individuum vagum such a term as "any individual man," these difficulties appear in a strong light, for what is true of any individual man is true of all men. Such a term is in one sense not an individual term; for it represents every man. But it represents each man as capable of being denoted by a term Which is individual; and so, though it is not itself an individual term, it stands for any one of a class of individual terms. If we call a thought about a thing in so far as it is denoted by a term, a second intention, we may say that such a term as "any individual man" is individual by second intention. The letters which the mathematician uses (whether in algebra or in geometry) are such individuals by second intention. Such individuals are one in number, for any individual man is one man; they may also be regarded as incapable of logical division, for any individual man, though he may either be a Frenchman or not, is yet altogether a Frenchman or altogether not, and not some one and some the other. Thus, all the formal logical laws relating to individuals will hold good of such individuals by second intention, and at the same time a universal proposition may at any moment be substituted for a proposition about such an individual, for nothing can be predicated of such an individual which cannot be predicated of the whole class.

95. There are in the logic of relatives three kinds of terms which involve general suppositions of individual cases. The first are individual terms, which denote only individuals See 96ff. ^†1; the second are those relatives whose correlatives are individual: I term these infinitesimal relatives See 100ff. ^†2; the third are individual infinitesimal relatives, and these I term elementary relatives. See 121ff. ^†3

Individual Terms

96. The fundamental formulæ relating to individuality are two. Individuals are denoted by capitals.

(95)	If x > 0 x = X X' X'' X''' etc.
(96)	yX = yX.

We have also the following which are easily deducible from these two:

We have already seen that

1x = 0, provided that [x] > 𝟏.

97. As an example of the use of the formulæ we have thus far obtained, let us investigate the logical relations between "benefactor of a lover of every servant of every woman," "that which stands to every servant of some woman in the relation of benefactor of a lover of him," "benefactor of every lover of some servant of a woman," "benefactor of every lover of every servant of every woman," etc.

In the first place, then, we have by (95)

sw = s(W' W'' W''' etc.) = sW' sW'' sW''' etc.
sw = sW' W'' W''' etc. = sW' sW'' sW''' etc.

From the last equation we have by (96)

sw = (sW'),(sW''),(sW'''), etc.

Now by (31)	x' x'' etc. = x',x",x"', etc. etc.,
or
(101)	Π' ⤙ Σ',

where Π' and Σ' signify that the addition and multiplication with commas Π' signifies logical multiplication and Σ' signifies logical addition. ^†1 are to be used. From this it follows that

If w vanishes, this equation fails, because in that case (95) does not hold.

From (102) we have
(103)	(ls)w ⤙	lsw. A lover-of-a-servant of every woman is a lover-of-a-servant of a woman. †3
Since	a =	a,b etc.,
	b =	a,b etc.,
we have	la =	l(a,b etc.) = l(a,b) l (etc.),
	lb =	l(a,b etc.) = l(a,b) l (etc.).

Multiplying these two equations commutatively we have

(la),(lb) = l(a,b) etc.

or

(104)	lΠ' ⤙ Π'l. See Lewis' Survey of Symbolic Logic, p. 87n for a proof of this theorem. †1
Now	(ls)w = (s)W' W'' W''' etc. = Π'(ls)W = Π'lsW,
	lsw = lsW' W'' W''' etc. = lΠ'sW' = lΠ'sW.
Hence,
(105)	lsw ⤙ (ls)w,

or every lover of a servant of all women stands to every woman in the relation of lover of a servant of hers.

From (102) we have
(106)	lsw ⤙ lsw. A lover of every servant of all women is a lover of a servant of every woman. †2
By (95) and (96) we have
lsw =	ls(W' W'' W''' etc.) = lsW' lsW'' lsW''' etc.
	= lsW' lsW'' lsW''' + etc.
Now	sW = sW' W'' W''' etc. = sW',sW'',sW''', etc.
So that by (94)	sw ⤙ sW' ⤙ sW'.
Hence by (92)
	lsW' ⤙ lsw, lsW'' ⤙ lsw lsW''' ⤙ lsw.
Adding,	lsW' lsW'' lsW''' ⤙ lsw;
or
(107)	lsw ⤙ lsw.

That is, every lover of every servant of any particular woman is a lover of every servant of all women.

98. By similar reasoning we can easily make out the relations shown in the following table. It must be remembered
that the formulæ do not generally hold when exponents vanish. On Peirce's copy a line was drawn through the vinculum in each of these cases with the comment, "Crossed are not universally true." ^†1

99. It appears to me that the advantage of the algebraic notation already begins to be perceptible, although its powers are thus far very imperfectly made out. At any rate, it seems to me that such a prima facie case is made out that the reader who still denies the utility of the algebra ought not to be too indolent to attempt to write down the above twenty-two terms in ordinary language With logical precision. Having done that, he has only to disarrange them and then restore the arrangement by ordinary logic, in order to test the algebra so far as it is yet developed.

Infinitesimal Relatives

100. We have by the binomial theorem by (49) and by (47),

(1 +x)n = 1 + Σpxn-p + xn.

Now, if we suppose the number of individuals to which any one thing is x to be reduced to a smaller and smaller number, we reach as our limit

x𝟐 = 0,

Σpxn-p = [n].𝟏n-†𝟏,x†𝟏 = xn,

(1 + x)n = 1 + x n.

101. If, on account of the vanishing of its powers, we call x an infinitesimal here and denote it by i, and if we put

x n = i n = y,

our equation becomes

(110)

= (1 +i)1/i = 1 + 1.

102. In fact, this agrees With ordinary algebra better than it seems to do; for 1 is itself an infinitesimal, and is 1. If the higher powers of 1 did not vanish, we should get the ordinary development of .

103. Positive powers of are absurdities in our notation. For negative powers we have

(111)

-x = 1 - x.

104. There are two ways of raising -x to the yth power. In the first place, by the binomial theorem,

(1-x)y = 1-[y].1y-†𝟏,x†𝟏 + ([y].[y-𝟏]/𝟐).1y-‡𝟑,x‡𝟐—etc.;

and, in the second place, by (111) and (10).

-xy = 1 - xy. I.e., 1-(xy). ^†1

It thus appears that the sum of all the terms of the binomial development of (1-x)y, after the first, is -xy. I.e., -(xy). ^†2 The truth of this may be shown by an example. Suppose the number of y's are four, viz. Y', Y'', Y''', and Y''''. Let us use x', x'', x''', and x'''' in such senses that

Then the negatives of the different terms of the binomial developement are,

[y].1y-†𝟏,x†𝟏 = x' + x'' + x''' + x''''.
-(([y].[y-𝟏])/𝟐).1y-‡𝟐,x‡𝟐 = -x',x''-x',x'''-x',x''''-x'',x'''-x'',x''''-x''',x''''. I.e., -(x', x'') - (x', x''') etc. ^†3
+(([y].[y-𝟏][y-𝟐])/𝟐.𝟑).1-||𝟑x||𝟑 = x',x'',x'''+ x',x'',x'''' + x',x''',x'''' + x'',x''',x''''.
xy = -x', x'', x''''. I.e., -(x', x'', x''', x'''').^†4

Now, since this addition is invertible, in the first term, x' that is x'', is counted over twice, and so with every other pair. The second term subtracts each of these pairs, so that it is only counted once. But in the first term the x' that is x'' that is x''' is counted in three times only, while in the second term it is subtracted three times; namely, in (x',x''), in (x',x''') and in (x'',x'''). On the whole, therefore, a triplet would not be represented in the sum at all, were it not added by the third term. The whole quartette is included four times in the first term, is subtracted six times by the second term, and is added four times in the third term. The fourth term subtracts it once, and thus in the sum of these negative terms each combination occurs once, and once only; that is to say the sum is

x' x'' x''' x'''' = x(Y' Y'' Y''' Y'''') = xy.

105. If we write (a x)𝟑 for [x].[x-𝟏].[x-𝟐].1x-†𝟑,a†𝟑, that is for whatever is a to any three x's, regard being had for the order of the x's; and employ the modern numbers as exponents with this signification generally, then

1 - a x + (𝟏/𝟐!)(a x)2 - (𝟏/𝟑!)(a x)3 + etc.

is the development of (1 - a)x and consequently it reduces itself to 1 - a x. That is,

106. 1 - x denotes everything except x, that is, whatever is other than every x; so that - means "not." We shall take log x in such a sense that

log x = x. It makes another resemblance between 1 and infinity that log 0 = -1. ^†P1

107. I define the first difference of a function by the usual formula,

where Δx is an indefinite relative which never has a correlate in common with x. So that

(114)

x,(Δx) = 0

x + Δx = x Δx.

Higher differences may be defined by the formulæ

108. The exponents here affixed to Δ denote the number of times this operation is to be repeated, and thus have quite a different signification from that of the numerical coefficients in the binomial theorem. I have indicated the difference by putting a period after exponents significative of operational repetition. Thus, m2 may denote a mother of a certain pair, m2. a maternal grandmother.

109. Another circumstance to be observed is, that in taking the second difference of x, if we distinguish the two increments which x successively receives as Δ'x and Δ''x, then by (114)

(Δ'x),(Δ''x) = 0

If Δx is relative to so small a number of individuals that if the number were diminished by one Δ𝖓·φx would vanish, then I term these two corresponding differences differentials, and write them with 𝐝 instead of Δ.

110. The difference of the invertible sum of two functions is the sum of their differences; for by (113) and (18),

If a is a constant, we have

(118)	Δaφx = a(φx Δφx) - aφx = aΔφx - (aΔφx),aφx,
	Δ2.aφx = -Δaφx,aΔx, etc.
	Δ(φx)a = (Δφx)a - ((Δφx)a),φxa,
	Δ2.(φx)a = -Δ(φx)a, etc.
(119)	φ(a,φx) = a,Δφx.