Peirce’s 1870 “Logic Of Relatives” • Comment 11.19

Up to this point in the 1870 Logic of Relatives, Peirce has introduced the “number of” function on logical terms and discussed the extent to which its use as a measure, $v : S \to \mathbb{R}$ such that $v : s \mapsto [s],$ satisfies the relevant measure-theoretic principles, for starters, these two:

The “number of” map exhibits a certain type of uniformity property, whereby the value of the measure on a uniformly qualified population is in fact actualized by each member of the population.
The “number of” map satisfies an order morphism principle, whereby the illative partial ordering of logical terms is reflected to a degree by the arithmetical linear ordering of their measures.

Peirce next takes up the action of the “number of” map on the two types of, loosely speaking, additive operations that we normally consider in logic.

NOF 3.1

It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions.

(Peirce, CP 3.67)

The sign $+\!\!,$ denotes what Peirce calls “the regular non-invertible addition”, corresponding to the inclusive disjunction of logical terms or the union of their extensions as sets.

The sign $+$ denotes what Peirce calls “the invertible addition”, corresponding to the exclusive disjunction of logical terms or the symmetric difference of their extensions as sets.

NOF 3.2

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.

(Peirce, CP 3.67)

A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word collection, and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he’s saying relative to the present frame of discussion.

The “number of” map $v : S \to \mathbb{R}$ evidently induces some sort of morphism with respect to logical sums. If this were straightforwardly true, we could write:

$\begin{matrix} ? & v(x ~+\!\!,~ y) & = & v(x) ~+~ v(y) & ? \end{matrix}$

Equivalently:

$\begin{matrix} ? & [x ~+\!\!,~ y] & = & [x] ~+~ [y] & ? \end{matrix}$

Of course, things are not quite that simple when it comes to inclusive disjunctions and set-theoretic unions, so it is usual to introduce the concept of a sub-additive measure to describe the principle that does hold here, namely, the following:

$\begin{matrix} v(x ~+\!\!,~ y) & \le & v(x) ~+~ v(y) \end{matrix}$

Equivalently:

$\begin{matrix} [x ~+\!\!,~ y] & \le & [x] ~+~ [y] \end{matrix}$

This is why Peirce trims his discussion of this point with the following hedge:

NOF 3.3

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.

(Peirce, CP 3.67)

Finally, a morphism with respect to addition, even a contingently qualified one, must do the right thing on behalf of the additive identity element:

NOF 3.4

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)

With respect to the nullity $0$ in $S$ and the number $0$ in $\mathbb{R},$ we have:

$v(0) ~=~ [0] ~=~ 0.$

In sum, therefore, it can be said: It also serves that only preserves a due respect for the function of a vacuum in nature.

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