 
Summary: Proof realizations of typed –calculi
Sergei N. Artemov \Lambda
May, 1997
Abstract
The Logic of Proofs (LP) introduced in [2] provides a basic framework for the for
malization of reasoning about proofs. It incorporates proof terms into the propositional
language, using labeled logical operators ``t : '' with the intended reading of t : F being
``t is a proof of F''. LP is supplied with an exact provability semantics in Peano Arith
metic, a simple axiom system, and completeness and decidability theorems. LP naturally
expresses a number of constructions of logic involving the notion of proof, which have
previously been formulated and/or interpreted in an informal metalanguage, e.g. modal
logic, Intuitionistic logic with its BrouwerHeytingKolmogorov semantics, etc. ([2], [3]).
In the current paper we demonstrate how the typed –calculus and the modal –calculus
can be realized in the Logic of Proofs.
1 Introduction
The Logic of Proofs (LP) incorporates proof terms directly into the propositional language
using new logical operators t : labeled by special proof terms with the intended reading of t : F
being ``t is a proof of F'' (cf. [2]). Three basic operations on proofs are postulated: application,
proof checker, and choice. The language of LP has an exact intended semantics, where ``t is
a proof of F '' is interpreted as a corresponding arithmetical formula of provability in Peano
