## Peirce’s 1870 “Logic Of Relatives” • Selection 10

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

### The Signs for Multiplication (cont.)

The sum $x + x$ generally denotes no logical term.  But ${x,}_\infty + \, {x,}_\infty$ may be considered as denoting some two $x$’s.

It is natural to write:

 $x ~+~ x ~=~ \mathit{2}.x ~,$ and ${x,}_\infty + \, {x,}_\infty ~=~ \mathit{2}.{x,}_\infty ~,$

where the dot shows that this multiplication is invertible.

We may also use the antique figures so that:

 $\mathit{2}.{x,}_\infty ~=~ \mathfrak{2}x ~,$ just as $\mathit{1}_\infty ~=~ \mathfrak{1} ~.$

Then $\mathfrak{2}$ 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:

 $\mathit{l}\mathfrak{2}.\mathrm{w} ~\text{and}~ \mathfrak{2}.\mathit{l}\mathrm{w}$

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

 $\mathit{h}\mathfrak{2}.\mathrm{w} ~=~ \mathfrak{2}.\mathit{h}\mathrm{w} ~,$ and in general; $x,\!\mathfrak{2}.y ~=~ \mathfrak{2}.x,\!y ~.$

(Peirce, CP 3.75)

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