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We design an intuitionistic predicate logic that supports a limited amount of classical reasoning, just enough to prove a variant of Markov's principle suited for predicate logic. At the computational level, the extraction of an existential witness out of a proof of its double negation is done by using a form of statically-bound exception mechanism, what can be seen as a direct-style variant of Friedman's...
A unified approach is developed for constructing first-order cut-free sequent calculi without equality representing two classes of modal logics in dependence of whether classical logic or intuitionistic one is taken as a basis. It uses the original notions of admissibility and compatibility, which, in general, permits to avoid skolemization being a forbidden operation for the logics under consideration...
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