Peirce’s 1870 “Logic of Relatives” • Comment 11.16

We now have enough material on morphisms 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 $\mathit{t}$ is indicated by writing the term in square brackets, as $[\mathit{t}].$ It is convenient to have a prefix notation for the function mapping a term $\mathit{t}$ to a number $[\mathit{t}]$ but Peirce previously reserved the letter $``n"$ for logical $\text{not},$ so let’s use $v(\mathit{t})$ as a variant for $[\mathit{t}].$

My plan will be nothing less plodding than to work through the statements 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 $[\mathit{t}].$

(Peirce, CP 3.65)

The role of the “number of” function may be formalized by assigning it a name and a type, in the present discussion $v : S \to \mathbb{R},$ 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 context, 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 $X$ and $Y$ contain all the teeth and all the people, respectively, 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 $Y.$

$\text{Figure 51. Dyadic Relation}~ T \subseteq X \times Y$

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

Resources

Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
Logic Syllabus • Relational Concepts • Relation Theory • Relative Term

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

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