Note on Carnap's relational asymptotic relative frequencies

Abstract

A (binary) relation is a collection of ordered couples. Two relations are isomorphic if there is a 1–1 correspondence between their fields which preserves the ordered couples. Isomorphism between relations is itself an equivalence relation, and a structure is an isomorphism class of binary relations.Carnap ([1], p. 124) asks certain questions concerning (both the exact and) the asymptotic value of the relative frequency that a relation on p objects satisfies certain properties. Among the most interesting special cases are the asymptotic value of the relative frequency that a binary relation on p objects be (a) symmetric, (b) reflexive, (c) transitive irreflexive anti-symmetric, and (d) symmetric irreflexive. These results already appear either in, or almost in, the literature, in various disguises. The object of this expository note is to bring them to light, especially since some of the references may not be generally known to logicians.Carnap ([1], p. 124) points out explicitly the correspondence between binary relations and linear graphs. Davis [2] and Harary [4] obtains precise results for the number of structures with certain properties using the language of relations and graphs respectively; but these do not supply the asymptotic values directly. The asymptotic results which are most useful here are either contained in, or are straight-forward extensions of, the formulas presented in Ford and Uhlenbeck [3], which in turn are based on unpublished work of G. Pólya. In addition, we utilize a result of L. Moser on transitive relations, which appears in Wine and Freund [5].