EPSILON THEOREMS IN INTERMEDIATE LOGICS

• Published in: The Journal of symbolic logic Vol. 87; no. 2; pp. 682 - 720
• Main Authors: BAAZ, MATTHIAS, ZACH, RICHARD
• Format: Journal Article
• Language: English
• Published: New York, USA Cambridge University Press 01.06.2022
• Subjects: epsilon calculus; 03B20; 03B55; 03F05; intermediate logic; Herbrand’s theorem
• ISSN: 0022-4812; 1943-5886
• DOI: 10.1017/jsl.2021.103

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