In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #-calculus.

Pure type systems make use of domain-full λ-abstractions λx:D.M . We present a variant of pure type systems, which we call domain-free pure type systems, with domain-free λ-abstractions λx.M. Domain-free pure type systems have a number of advantages over both pure type systems and so-called type assignment systems (they also have some disadvantages) and have been used in theoretical developments as well as in implementations of proof-assistants. We study the basic properties of domain-free pure type systems, establish their formal relationship with pure type systems and type assignment systems, and give a number of applications of these correspondences.

Continuation passing style (CPS) translations of typed -calculi have numerous applications. However, the range of these applications has been conned by the fact that CPS translations are known for non-dependent type systems only, thus excluding well-known systems like the calculus of constructions (CC) and the logical frameworks (LF). This paper presents techniques for CPS translating systems with dependent types, with an emphasis on pure type-theoretical applications.

. The \Delta-calculus is a -calculus with a control-like operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce \Deltaexp, an explicit substitution calculus for \Delta, and study its properties. In particular, we show that \Deltaexp preserves strong normalisation, which provides us with the first example --moreover a very natural one indeed-- of explicit substitution calculus which is not structure-preserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of \Deltaexp is strongly normalising. In addition, we show that Plotkin's call-by-name continuation-passing style translation may be extended to \Deltaexp and that the extended translation preserves typing. This seems to be the first study of CPS translations for calculi of explicit substitutions. 1 Introduction Explicit substitutions were introduced by Abadi, Cardelli, Curien and L...

Abstract not found

The λΔ-calculus is a λ-calculus with a control-like operator whose reduction rules are closely related to normalisation procedures in classical logic. We introduce λΔexp, an explicit substitution calculus for λΔ, and study its properties. In particular, we show that λΔexp preserves strong normalisation, which provides us with the first example -- moreover a very natural one indeed -- of explicit substitution calculus which is not structure-preserving and has the preservation of strong normalisation property. One particular application of this result is to prove that the simply typed version of λΔexp is strongly normalising. In addition, we show that Plotkin's call-by-name continuation-passing style translation may be extended to λΔexp and that the extended translation preserves typing. This seems to be the first study of CPS translations for calculi of explicit substitutions.