We present an overview of the unintended interpretations of intuitionistic logic that arose after Heyting formalized the “observed regularities” in the use of formal parts of language, in particular, first-order logic and Heyting Arithmetic. We include unintended interpretations of some mild variations on “official” intuitionism, such as intuitionistic type theories with full comprehension and ...

The proof search method is a traditionally established way to prove the completeness theorem for various logics. The purpose of this paper is to show that this method can be adapted to linear logic. First we prove the completeness theorem for a certain fragment of intuitionistic linear logic, called naive linear logic, with respect to naive phase semantics, i.e., phase semantics without any clo...

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M→) is PSPACE-complete. He showed a polynomialy bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACEcompleteness of S4, proved by Ladner, and the Gödel translation from S4 into Intuitionistic Logic, the PSPACE-completeness...

In [2] an alternative skolemization method called eskolemization was introduced that is sound and complete for existence logic with respect to existential quantifiers. Existence logic is a conservative extension of intuitionistic logic by an existence predicate. Therefore eskolemization provides a skolemization method for intuitionistic logic as well. All proofs in [2] were semantical. In this ...

Bi-intuitionistic logic is a conservative extension of intuitionistic logic with a connective dual to implication, called exclusion. We present a sound and complete cut-free labelled sequent calculus for bi-intuitionistic propositional logic, BiInt, following S. Negri’s general method for devising sequent calculi for normal modal logics. Although it arises as a natural formalization of the Krip...

We study the relationship between classical phase semantics for classical linear logic (LL) and intuitionistic phase semantics for intuitionistic linear logic (ILL). We prove that (i) every intuitionistic phase space is a subspace of a classical phase space, and (ii) every intuitionistic phase space is phase isomorphic to an “almost classical” phase space. Here, by an “almost classical” phase s...

There are a number of cut-free sequent calculus proof systems known that are complete for first-order intuitionistic logic. Proofs in these different systems can vary a great deal from one another. We are interested in providing a flexible and unifying framework that can collect together important aspects of many of these proof systems. First, we suggest that one way to unify these proof system...

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M→) is PSPACEcomplete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Gödel translation from S4 into Intuitionistic Logic, the PSPACE-completenes...

In intuitionistic sequent calculi, detecting that a sequent is unprovable is often used to direct proof search. This is for instance seen in backward chaining, where an unprovable subgoal means that the proof search must backtrack. In undecidable logics, however, proof search may continue indefinitely, finding neither a proof nor a disproof of a given subgoal. In this paper we characterize a fa...

We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is shown to be a decision procedure for a new syntactically described decidable class of intuitionistic ...