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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This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

One Response to Peirce’s 1870 “Logic Of Relatives” • Selection 10

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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