Linear logic is a refinement of classical and intuitionistic logic. Instead of emphasizing truth, as in classical logic, or proof, as in intuitionistic logic, linear logic emphasizes the role of formulas as resources. To achieve this focus, linear logic does not allow the usual structural rules of contraction and weakening to apply to all formulas but only those formulas marked with certain mod...

Intuitionistic logic is an important subsystem of the classical one. In intuitionistic logic the effective interpretation and mechanical extraction of programs from proofs are allowed. The programming interpretation of intuitionistic propositional logic by simply typed lambda-calculus is a significant part of interest of computer scientists. The intuitionistic natural deductions are actually th...

A unified view over well-known interpretations of intuitionistic logic, such as Gödel’s dialectica interpretation [6], Diller-Nahm interpretation [1] and Kreisel’s modified realizability [7] was achieved through a parametrised interpretation in the intuitionistic logic context [8] but also, very elegantly, in the linear logic setting (see [10], [9] and [3]). In this talk we report on work in pr...

We give Hennessy-Milner classes for intuitionistic, dual-intuitionistic and bi-intuitionistic logic interpreted in intuitionistic Kripke models, generalise these results to modal (dual- bi-)intuitionistic logics. Our main technical tools are image-compact pre-image-compact relations that provide a semantical description of saturation properties.

Craig interpolation theorem (which holds for intuitionistic logic) implies that the derivability of X,X’-¿Y’ implies existence of an interpolant I in the common language of X and X’-¿Y’ such that both X-¿I and I,X’-¿Y’ are derivable. For classical logic this extends to X,X’-¿Y,Y’, but for intuitionistic logic there are counterexamples. There is a version true for intuitionistic propositional (b...

Logic L was introduced by Lewitzka [8] as a modal system that combines intuitionistic and classical logic: L is a conservative extension of CPC and it contains a copy of IPC via embedding φ 7→ φ. In this article, we consider L3, i.e. L augmented with S3 modal axioms, define basic epistemic extensions and prove completeness w.r.t. algebraic semantics. Our systems involve, in adapted form, some o...

Dialogue games are a two-player semantics for a variety of logics, including intuitionistic and classical logic. Dialogues can be viewed as a kind of analytic calculus not unlike tableaux. Can dialogue games be an effective foundation for proof search in intuitionistic logic (both first-order and propositional)? We announce Kuno, an automated theorem prover for intuitionistic first-order logic ...