We come to the last of Peirce’s observations about the “number of” function in CP 3.76.
It is to be observed that
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 denotes the dyadic identity relation while the antique figure denotes what Peirce otherwise defines as
Collected Papers CP 3 gives which does not make sense. Chronological Edition CE 2 gives the 1’s in different styles of italics but reading the equation as 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 is defined as the average number of things related by the identity relation to one individual, and so it makes sense that where is the set of non-negative integers
With respect to the relative term in the syntactic domain and the number in the non-negative integers we have the following.
At long last, then, the “number of” mapping has another one of the properties required of an arrow from logical terms in to real numbers in
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term