Summary: Unified Semantics for Modality and –­terms
via Proof Polynomials \Lambda
Sergei N. Artemov y
Abstract
It is shown that the modal logic S4, simple –­calculus and modal –­calculus admit a
realization in a very simple propositional logical system LP , which has an exact provability
semantics. In LP both modality and –­terms become objects of the same nature, namely,
proof polynomials. The provability interpretation of modal –­terms presented here may
be regarded as a system­independent generalization of the Curry­Howard isomorphism of
proofs and –­terms.
1 Introduction
The Logic of Proofs (LP , see Section 2) is a system in the propositional language with an
extra basic proposition t : F for ``t is a proof of F ''. LP is supplied with a formal provability
semantics, completeness theorems and decidability algorithms ([3], [4], [5]).
In this paper it is shown that LP naturally encompasses –­calculi corresponding to intu­
itionistic and modal logics, and combinatory logic. In addition, LP is strictly more expressive
because it admits arbitrary combinations of ``:'' and propositional connectives.
The idea of logic of proofs can be found in G¨odel's lecture [14] (see also [20]) first published
in 1995, where a constructive version of the modal provability logic S4 was sketched. This
sketch does not contain formal definitions and lacks some important details, without which

Source: Artemov, Sergei N. - Faculty of Mechanics and Mathematics, Moscow State University

Collections: Computer Technologies and Information Sciences