## Peirce’s 1870 “Logic Of Relatives” • Comment 11.16

We have enough material on morphisms now to go back and cast a more studied eye on what Peirce is doing with that “number of” function, whose application to a logical term $t$ is indicated by writing the term in square brackets, as $[t].$ It is convenient to have a prefix notation for the function that maps a term $t$ to a number $[t]$ but Peirce has previously reserved $\mathit{n}$ for logical $\text{not},$ so let’s use $v(t)$ as a variant for $[t].$

My plan will be nothing less plodding than to work through the statements that Peirce made in defining and explaining the “number of” function up to our present place in the paper, namely, the budget of points collected in Comment 11.2.

#### NOF 1

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 $[t].$

(Peirce, CP 3.65)

The role of the “number of” function may be formalized by assigning it a name and a type as $v : S \to \mathbb{R},$ where $S$ is a suitable set of signs, a syntactic domain, containing all the logical terms whose numbers we need to evaluate in a given discussion, and where $\mathbb{R}$ is the set of real numbers.

Transcribing Peirce’s example: $\begin{array}{ll} \text{Let} & \mathrm{m} ~=~ \text{man} \\[8pt] \text{and} & \mathit{t} ~=~ \text{tooth of}\,\underline{~~~~}. \\[8pt] \text{Then} & v(\mathit{t}) ~=~ [\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}. \end{array}$

To spell it out in words, the number of the relative term $\text{tooth of}\,\underline{~~~~}"$ in a universe of perfect human dentition is equal to the number of teeth of humans divided by the number of humans, that is, $32.$

The dyadic relative term $\mathit{t}$ determines a dyadic relation $T \subseteq X \times Y,$ where $X$ contains all the teeth and $Y$ contains all the people that happen to be under discussion.

A rough indication of the bigraph for $T$ might be drawn as follows, showing just the first few items in the toothy part of $X$ and the peoply part of $Y.$

Notice that the “number of” function $v : S \to \mathbb{R}$ needs the data represented by the entire bigraph for $T$ in order to compute the value $[\mathit{t}].$

Finally, one observes that this component of $T$ is a function in the direction $T : X \to Y,$ since we are counting only teeth that occupy exactly one mouth of a tooth-bearing creature.

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