Semantically closed intuitionistic abstract logics
Steffen Lewitzka · Journal of Logic and Computation · 2011
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 ∈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 modeltheoretic semantics. The resulting logic, called ∈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 ∈I (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 ∈I+ also predicates for validity and logical consequence. Applications of our construction to other non-classical abstract logics remain to be further investigated.