Post : Peirce's 1870 “Logic Of Relatives” • Selection 11
http://inquiryintoinquiry.com/2014/04/29/peirces-1870-logic-of-relatives-%e2%80%a2-selection-11/
Posted : April 29, 2014 at 12:30 am
Author : Jon Awbrey
Peircers,
Among other things, we may note in this section the
roots of a connection between logic and measurement.
We continue with §3. Application of the Algebraic Signs to Logic.
<quote>
The Signs for Multiplication (concl.)
=====================================
The conception of multiplication we have adopted is that of the application of one relation to
another. So, a quaternion being the relation of one vector to another, the multiplication of
quaternions is the application of one such relation to a second.
Even ordinary numerical multiplication involves the same idea, for 2 × 3 is a pair of triplets, and
3 × 2 is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives.
If we have an equation of the form:
xy = z
and there are just as many x’s per y as there are, ''per'' things, things of the universe, then we
have also the arithmetical equation:
[x][y] = [z].
For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect
understood) as there are to any one of the universe, then:
[t][f] = [tf]
holds arithmetically.
So if men are just as apt to be black as things in general:
[m,][b] = [m,b]
where the difference between [m] and [m,] must not be overlooked.
It is to be observed that:
[_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.
</quote>(Peirce, CP 3.76)
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