Peirce’s 1870 “Logic Of Relatives” • Comment 11.24

We come to the end of the “number of” examples that we noted at this point in the text.

NOF 4.5

It is to be observed that

$[\mathit{1}] ~=~ 1.$

Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation.

Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.

(Peirce, CP 3.76 and CE 2, 376)

There are problems with the printing of the text at this point. Let us first recall the conventions we are using in this transcription, in particular, $\mathit{1}$ for the italic 1 that signifies the dyadic identity relation and $\mathfrak{1}$ for the “antique figure one” that Peirce defines as $\mathit{1}_\infty = \text{something}.$

CP 3 gives $[\mathit{1}] = \mathfrak{1},$ which I cannot make sense of. CE 2 gives the 1’s in different styles of italics, but reading the equation as $[\mathit{1}] = 1,$ makes the best sense if the “1” on the right hand side is read as the numeral “1” that denotes the natural number 1, and not as the absolute term “1” that denotes the universe of discourse. In this reading, $[\mathit{1}]$ is the average number of things related by the identity relation $\mathit{1}$ to one individual, and so it makes sense that $[\mathit{1}] = 1 \in \mathbb{N},$ where $\mathbb{N}$ is the set of non-negative integers $\{ 0, 1, 2, \ldots \}.$

With respect to the relative term $``\mathit{1}"$ in the syntactic domain $S$ and the number $1$ in the non-negative integers $\mathbb{N} \subset \mathbb{R},$ we have:

$v(\mathit{1}) ~=~ [\mathit{1}] ~=~ 1.$

And so the “number of” mapping $v : S \to \mathbb{R}$ has another one of the properties that would be required of an arrow $S \to \mathbb{R}.$

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