The Signs for Multiplication (cont.)

The sum generally denotes no logical term.  But may be considered as denoting some two ’s.

It is natural to write:

and

where the dot shows that this multiplication is invertible.

We may also use the antique figures so that:

just as

Then alone will denote some two things.

But this multiplication is not in general commutative, and only becomes so when it affects a relative which imparts a relation such that a thing only bears it to one thing, and one thing alone bears it to a thing.

For instance, the lovers of two women are not the same as two lovers of women, that is:

are unequal;  but the husbands of two women are the same as two husbands of women, that is:

and in general;

(Peirce, CP 3.75)