RELATIONAL AND PARTIAL VARIABLE SETS AND BASIC PREDICATE LOGIC

In this paper we study the logic of relational and partial variable se ts, seen as a generalization of set-valued presheaves, allowing transi tion functions to be arbitrary relations or arbitrary partial function s. We find that such a logic is the usual intuitionistic and co-intuit ionistic first order logic without Beck and Frobenius conditions relat ive to quantifiers along arbitrary terms. The important case of partia l variable sets is axiomatizable by means of the substitutivity schema for equality. Furthermore, completeness, incompleteness and independe nce results are obtained for different kinds of Beck and Frobenius con ditions.