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

Peirce’s 1870 “Logic of Relatives” • Selection 2

Peirce’s remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse as usually to escape explicit mention, others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly slipped into the stream of thought that it’s all too easy to overlook their significance — that all I can do to highlight their impact is to dress them up in different words, whose main advantage is being more jarring to the mind’s sensibilities.

• Peirce’s mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what quantification theory is all about, definitely more to the point than the mere “innovation” of using distinctive symbols for the so-called quantifiers.
• The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as some kind of morphism or functor from a logical domain to a quantitative co-domain.
• Notice that Peirce follows the mathematician’s usual practice, then and now, of making the status of being an individual or a universal relative to a discourse in progress.
• It is worth noting that Peirce takes the plural denotation of terms for granted, or what’s the number of a term for, if it could not vary apart from being one or nil?
• I also observe that Peirce takes the individual objects of a particular universe of discourse in a generative way, as opposed to a totalizing way, and thus these contingent individuals afford us with a basis for talking freely about collections, constructions, properties, qualities, subsets, and higher types built on them.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)