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

where the dot shows that this multiplication is invertible.  We may also use the antique figures so that

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,

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

(Peirce, CP 3.75)

References

• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

Resources

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