We formulate Hilbert’s epsilon calculus in the context of expansion proofs. This leads to a simplified proof of the epsilon theorems by disposing of the need for prenexification, Skolemisation, and their respective inverse transformations. We observe that the natural notion of cut in the epsilon calculus is associative.

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.