We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.

In this paper we give a new proof of the characterization of the closed fragment of the provability logic of Heyting's Arithmetic. We also provide a characterization of the closed fragment of the provability logic of Heyting's Arithmetic plus Markov's Principle and Heyting's Arithmetic plus Primitive Recursive Markov's Principle.

The language of the basic logic of proofs extends the usual propositional language by forming sentences of the sort x is a proof of F for any sentence F. In this paper a complete axiomatization for the basic logic of proofs in Heyting Arithmetic HA was found.

If the Visser rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of the Visser rules are admissible is not known. In this paper we give a brief overview of results on admissible rules in the context of intermediate logics. We apply these results to some well-known intermediate logics. We provide natural examples of logics for which the Visser (...) rule are derivable, admissible but nonderivable, or not admissible. (shrink)

In this paper we study the modal behavior of Σ‐preservativity, an extension of provability which is equivalent to interpretability for classical superarithmetical theories. We explain the connection between the principles of this logic and some well‐known properties of HA, like the disjunction property and its admissible rules. We show that the intuitionistic modal logic given by the preservativity principles of HA known so far, is complete with respect to a certain class of frames.

In this paper we study the modal behavior of Σ-preservativity, an extension of provability which is equivalent to interpretability for classical superarithmetical theories. We explain the connection between the principles of this logic and some well-known properties of HA, like the disjunction property and its admissible rules. We show that the intuitionistic modal logic given by the preservativity principles of HA known so far, is complete with respect to a certain class of frames.

Admissible rules of a logic are those rules under which the set of theorems of the logic is closed. In this paper, a Gentzen-style framework is introduced for analytic proof systems that derive admissible rules of non-classical logics. While Gentzen systems for derivability treat sequents as basic objects, for admissibility, the basic objects are sequent rules. Proof systems are defined here for admissible rules of classes of modal logics, including K4, S4, and GL, and also Intuitionistic Logic IPC. With minor (...) restrictions, proof search in these systems terminates, giving decision procedures for admissibility in the logics. (shrink)

Refutation systems are formal systems for inferring the falsity of formulae. These systems can, in particular, be used to syntactically characterise logics. In this paper, we explore the close connection between refutation systems and admissible rules. We develop technical machinery to construct refutation systems, employing techniques from the study of admissible rules. Concretely, we provide a refutation system for the intermediate logics of bounded branching, known as the Gabbay–de Jongh logics. We show that this gives a characterisation of these logics (...) in terms of their admissible rules. To illustrate the technique, we also provide a refutation system for Medvedev’s logic. (shrink)

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete $\Pi _{2}^{0}$. Similarly, we show that the predicate logic of HA*, i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete $\Pi _{2}^{0}$. These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove 'categoricity of interpretations' under certain assumptions.