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 ε-calculus. The first and second ε-theorems for classical logic establish conservativity of the ε-calculus over its classical base logic. It is well known that the second ε-theorem fails for the intuitionistic ε-calculus, as prenexation is impossible. The paper investigates the effect of adding critical ε-and τ -formulas and using the translation of quantifiers into ε-and τ -terms to intermediate logics. It is shown that conservativity over the propositional base logic also holds for such intermediate ετ -calculi. The "extended" first ε-theorem holds if the base logic is finite-valued Gödel-Dummett logic, fails otherwise, but holds for certain provable formulas in infinite-valued Gödel logic. The second ε-theorem also holds for finitevalued first-order Gödel logics. The methods used to prove the extended first ε-theorem for infinite-valued Gödel logic suggest applications to theories of arithmetic.