Epsilon Theorems in Intermediate Logics
Abstract
Any intermediate propositional logic (i.e., a logic including intuitionistic logic and contained in classical logic) can be extended to a calculus with epsilon- and tau-operators and critical formulas. For classical logic, this results in Hilbert's -calculus. The first and second -theorems for classical logic establish conservativity of the -calculus over its classical base logic. It is well known that the second -theorem fails for the intuitionistic -calculus, as prenexation is impossible. The paper investigates the effect of adding critical - and τ-formulas and using the translation of quantifiers into - and τ-terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate τ-calculi. The "extended" first -theorem holds if the base logic is finite-valued G\"odel-Dummett logic, fails otherwise, but holds for certain provable formulas in infinite-valued G\"odel logic. The second -theorem also holds for finite-valued first-order G\"odel logics. The methods used to prove the extended first -theorem for infinite-valued G\"odel logic suggest applications to theories of arithmetic.