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

Posted on June 8, 2014 by Jon Awbrey

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

We come to the last of Peirce’s observations about the “number of” function in CP 3.76.

NOF 4.4

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 text at this point.  To recall the conventions used in this transcription, the italic figure ``\mathit{1}" denotes the dyadic identity relation \mathit{1} while the antique figure ``\mathfrak{1}" denotes what Peirce otherwise defines as \mathit{1}_\infty = \text{something}.

Collected Papers CP 3 gives [\mathit{1}] = \mathfrak{1}, which does not make sense.  Chronological Edition CE 2 gives the 1’s in different styles of italics but reading the equation as [\mathit{1}] = 1 makes better sense if the latter “1” is the numeral denoting the natural number 1 and not the absolute term “1” denoting the universe of discourse.  The quantity [\mathit{1}] is defined as 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 the following.

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

At long last, then, the “number of” mapping v : S \to \mathbb{R} has another one of the properties required of an arrow from logical terms in S to real numbers in \mathbb{R}.

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