Abstract
AbstractAny 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.
Highlights
The ε-calculus was originally introduced by Hilbert as a formalization of classical first-order logic
As we’ll see in Section 4, the quantifier axioms of the intermediate predicate logics considered in the preceding section satisfy the condition that the ετ -translations of their additional quantifier axioms are derivable from critical formulas alone
We address the first challenge by considering the condition that suffices to overcome it: the existence of complete e-elimination sets for every ετ -term e. We show why this condition is satisfied in classical logic, so we can clarify the role of excluded middle in the proof for the classical case, as well as how the proof for classical logic and those for intermediate logics given later correspond to one another
Summary
The ε-calculus was originally introduced by Hilbert as a formalization of classical first-order logic. The system resulting from a propositional intermediate logic L by adding ε- and τ -terms and critical formulas is called the ετ -calculus for L. As we’ll see, the quantifier axioms of the intermediate predicate logics considered in the preceding section satisfy the condition that the ετ -translations of their additional quantifier axioms are derivable from critical formulas alone (i.e., already in Lετ ).
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