## Peirce’s 1870 “Logic Of Relatives” • Selection 4

Here is the next part of §3. Application of the Algebraic Signs to Logic.

The sign of addition is taken by Boole so that

$x + y$

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

Thus

$\mathrm{m} + \mathrm{w}$

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 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,

$\mathrm{f} + \mathrm{u}$

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, I preferred to take as the regular addition of logic a non-invertible process, such that

$\mathrm{m} ~+\!\!,~ \mathrm{b}$

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], 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.

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.  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

$[0] ~=~ 0.$

(Peirce, CP 3.67)

A wealth of issues arises here that I hope to take up in depth at a later point, but for the moment I shall be able to mention only the barest sample of them in passing.

The two papers that precede this one in CP 3 are Peirce’s papers of March and September 1867 in the Proceedings of the American Academy of Arts and Sciences, titled “On an Improvement in Boole’s Calculus of Logic” and “Upon the Logic of Mathematics”, respectively.  Among other things, these two papers provide us with further clues about the motivating considerations that brought Peirce to introduce the “number of a term” function, signified here by square brackets.  I have already quoted from the “Logic of Mathematics” paper in a related connection.  Here are the links to those excerpts:

In setting up a correspondence between “letters” and “numbers”, Peirce constructs a structure-preserving map from a logical domain to a numerical domain.  That he does this deliberately is evidenced by the care that he takes with the conditions under which the chosen aspects of structure are preserved, along with his recognition of the critical fact that zeroes are preserved by the mapping.

Incidentally, Peirce appears to have an inkling of the problems that would later be caused by using the plus sign for inclusive disjunction, but his advice was overridden by the dialects of applied logic that developed in various communities, retarding the exchange of information among engineering, mathematical, and philosophical specialties all throughout the subsequent century.

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