This special issue reports on some recent advances in the area of intuitionistic modal type theories and their application to Computer Science. It collects a selection of papers presented at the Logic in Computer Science (LICS'99) satellite workshop on Intuitionistic Modal Logics and Applications (IMLA'99) held at Trento, Italy in July 1999. All of the contributors to this one day workshop, which was widely attended, were invited to submit a full and revised version of their papers to this special issue of MSCS. The selection was based on a second round of peer reviewing.

We reconsider the foundations of modal logic, following Martin-Löf's methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility, which yields a simple and uniform system of natural deduction for intuitionistic modal logic that does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's monadic metalanguage.

In his 1903, Principles of Mathematics, Bertrand Russell mentioned possible definitions of conjunction, disjunction, negation and existential quantification in terms of implication and universal quantification, exploiting impredicative universal quantifiers over all propositions. In his 1965 Ph.D. thesis Dag Prawitz showed that these definitions hold in intuitionistic second order logic. More recently, these definitions have been used to represent logic in various impredicative type theories. This treatment of logic is distinct from the more standard Curry–Howard representation of logic in a dependent type theory.

The main aim of this paper is to compare, in a purely logical, non type-theoretic setting, this Russell–Prawitz representation of intuitionistic logic with other possible representations. It turns out that associated with the Russell–Prawitz representation is a lax modal operator, which we call the Russell–Prawitz modality, and that any lax modal operator can be used to give a translation of intuitionistic logic into itself that generalises both the double negation interpretation, double negation being a paradigm example of a lax modality, and the Russell–Prawitz representation.

This paper presents an extension of the simply typed λ-calculus that allows iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax. This calculus aims at being the kernel for a type theory in which the user will be able to formalize logics or formal systems using the LF methodology, while taking advantage of new induction and recursion principles, extending the principles available in a calculus such as the Calculus of Inductive Constructions. The key idea of our system is the use of modal logic S4. We present here the system, its typing rules and reduction rules. The system enjoys the decidability of typability, soundness of typed reduction with respect to the typing rules, the Church–Rosser and strong normalization properties and it is a conservative extension over the simply typed λ-calculus. These properties entail the preservation of the adequacy of encodings.

A Gentzen sequent calculus for Lax Logic is presented, the proofs in which correspond naturally in a 1–1 way to the normal natural deductions for the logic. The propositional fragment of this calculus is used as the basis for another calculus that uses a history mechanism in order to give a decision procedure for propositional Lax Logic.

In this abstract we will describe research in progress on the problem of extracting information from proofs. Here we will concentrate our attention on semiconstructive calculi, which is a kind of calculus that is of interest in the framework of program synthesis and formal verification. We will discuss the notion of uniformly semiconstructive calculus, introduce our information extraction mechanism and apply it to two calculi extending Intuitionistic Arithmetic.