We continue with §3. Application of the Algebraic Signs to Logic.
Numbers Corresponding to Letters
I propose to use the term “universe” to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan’s, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains.
I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (men), the number of “tooth of” would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus,
(Peirce, CP 3.65)
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.
- This 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 up thereon.