 
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
