Nominalism and conceptualism as predicative second-order theories of predication.

Abstract

There appears to be a growing consensus, even if not unanimity, that standard predicative second-order logic is the appropriate logical medium for the representation of a nominalist theory of predication.* We agree that this is indeed the case and formulate in this paper a model-theoretic approach which justifies that claim.* Because it is model-theoretic, our approach differs from the truth-value semantics approach of Leblanc and Weaver. Amongst other reasons, we prefer our model-theoretic approach so as to accommodate those nominalists for whom the assumption that there are potentially as many names as there are individuals is not acceptable. The models involved in our semantics, moreover, are precisely the same models as are already involved in standard first-order logic. Assignments of values (drawn from the domain of a given model) to the individual variables are extended, however, to what, relative to a given first-order language, we call nominalistic assignments to the π-place predicate variables (for each positive integer ή) these assign first-order formulas (wffs) of the language in question, relative to the free occurrences of n distinct individual variables occurring in those wffs, to the H-place predicate variables. The satisfaction by such an assignment of a second-order wff in a model is then defined by a double recursion on the logical structure of the wff and on the number of nested predicate quantifiers occurring therein.