Semantically closed intuitionistic abstract logics
Alfred Tarski called a language semantically closed if it contains its own truth predicate and technical means for (self-)reference. We consider the semantically closed non-Fregean logic is an element of(I), presented in (Lewitzka, 2009, Notre Dame Journal of Formal Logic, 50, 275-301), and introduce a predicate for validity by combining syntactical constructions with a model-theoretic semantics. The resulting logic, called is an element of(+)(I) (Epsilon-I plus), is able to express its own finitary consequence relation. We show that every intuitionistic (or classical) abstract logic extends to a logic which has the semantic features of is an element of(I) (of is an element of(+)(I)), respectively, namely a truth predicate that satisfies the Tarski biconditionals, a predicate for falsity (as intuitionistic negation), means for propositional (self-)reference, and in the case of is an element of(+)(I) also predicates for validity and logical consequence. Applications of our construction to other non-classical abstract logics remain to be further investigated.
intuitionistic abstract logic non-fregean logic truth theory propositional self-reference validity predicate