Writings of Charles S. Peirce: A Chronological Edition, Volume 2. Charles S. Peirce

      XII

      Proof.—For call 0, Then by definition 3

      x belongs to the class zero.

      ∴ by definition 4

      Corollary 1.—The same reasoning applies to arithmetical multiplication.

      Corollary 2.—From Theorem x and the last corollary it follows that if , either .

      XIII

      XIV

      These do not hold with arithmetical operations.

      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. 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

      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.

       On Arithmetic

      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.

      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 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 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. 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.

      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 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 =.¹ 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, 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. 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 on page 55 (§12) of this volume, will, I hope, put the relations of logic and arithmetic in a somewhat clearer light than heretofore.

      1. 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) 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.

       Upon Logical Comprehension and Extension

       P 34: Presented 13 November 1867

      §1. That these Conceptions are not so Modern as has been represented

      The historical account usually given of comprehension and extension is this, “that the distinction, though taken in general terms by Aristotle, and explicitly announced with scientific precision by one, at least, of his Greek commentators, had escaped the marvellous acuteness of the schoolmen, and remained totally overlooked and forgotten till the publication of the Port-Royal Logic.”¹ I would offer the following considerations to show that this interpretation of history is not exactly true. In the first place, it is said that a distinction was taken between these attributes, as though they were previously confounded. Now there is not the least evidence of this. A German logician has, indeed, by a subtle misconception, considered extension as a species of comprehension, but, to a mind beginning to reflect, no notions seem more unlike. The mental achievement has been the bringing of them into relation to one another, and the conception of them as factors of the import of a term, and not the separation of them. In the second