[CG] Re: Precursors Of Category Theory

Precursors Of Category Theory • 6 • https://inquiryintoinquiry.com/2024/05/30/precursors-of-category-theory-6-a/ Hilbert and Ackermann • Principles of Mathematical Logic (1928) ❝For the intuitive interpretation on which we have hitherto based the predicate calculus, it was essential that the sentences and predicates should be sharply differentiated from the individuals, which occur as the argument values of the predicates. Now, however, there is nothing to prevent us from “considering the predicates and sentences themselves as individuals which may serve as arguments of predicates”. ❝Consider, for example, a logical expression of the form (x)(A → F(x)). This may be interpreted as a predicate P(A, F) whose first argument place is occupied by a sentence A, and whose second argument place is occupied by a monadic predicate F. ❝A false sentence A is related to every F by the relation P(A, F); a true sentence A only to those F for which (x)F(x) holds. ❝Further examples are given by the properties of “reflexivity”, “symmetry”, and “transitivity” of dyadic predicates. To these correspond three predicates: Ref(R), Sym(R), and Tr(R), whose argument R is a dyadic predicate. These three properties are expressed in symbols as follows: • Ref(R) : (x)R(x, x), • Sym(R) : (x)(y)(R(x, y) → R(y, x)), • Tr(R) : (x)(y)(z)(R(x, y) & R(y, z) → R(x, z)). ❝All three properties are possessed by the predicate ≡(x, y) (x is identical with y). The predicate <(x, y), on the other hand, possesses only the property of transitivity. Thus the formulas Ref(≡), Sym(≡), Tr(≡), and Tr(<) are true sentences, whereas Ref(<) and Sym(<) are false. ❝Such “predicates of predicates” will be called “predicates of second level”.❞ (p. 135). ❝We have, first, predicates of individuals, and these are classified into predicates of different categories, or types, according to the number of their argument places. Such predicates are called “predicates of first level”. ❝By a “predicate of second level”, we understand one whose argument places are occupied by names of individuals or by predicates of first level, where a predicate of first level must occur at least once as an argument. The categories, or types, of predicates second level are differentiated according to the number and kind of their argument places.❞ (p. 152). Reference — • Hilbert, D. and Ackermann, W., Principles of Mathematical Logic, Robert E. Luce (trans.), Chelsea Publishing Company, New York, 1950. 1st published, Grundzüge der Theoretischen Logik, 1928. 2nd edition, 1938. English translation with revisions, corrections, and added notes by Robert E. Luce, 1950. Resources — Precursors Of Category Theory • https://oeis.org/wiki/Precursors_Of_Category_Theory Propositions As Types Analogy • https://oeis.org/wiki/Propositions_As_Types_Analogy Survey of Precursors Of Category Theory • https://inquiryintoinquiry.com/2024/05/24/survey-of-precursors-of-category-… Regards, Jon cc: https://www.academia.edu/community/Lgrj0b cc: https://mathstodon.xyz/@Inquiry/112502692087155139