## Peirce’s 1870 “Logic Of Relatives” • Selection 11

We continue with §3. Application of the Algebraic Signs to Logic.

### 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 $2 \times 3$ is a pair of triplets, and $3 \times 2$ 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:

$[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]$

holds arithmetically.

So if men are just as apt to be black as things in general:

$[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}]$

where the difference between $[\mathrm{m}]$ and $[\mathrm{m,}]$ must not be overlooked.

It is to be observed that:

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

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)

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