 
Summary: Operational modal logic \Lambda
Sergei N. Artemov y
December, 1995
Abstract
Answers to two old questions are given in this paper.
1. Modal logic S4, which was informally specified by GĻodel in 1933 as a logic
for provability, meets its exact provability interpretation.
2. Brouwer Heyting Kolmogorov realizing operations (193132) for intu
itionistic logic Int also get exact interpretation as corresponding propositional
operations on proofs; both S4 and Int turn out to be complete with respect to
this proof realization.
These results are based on operational reading of S4, where a modality is split
into three operations. The logic of proofs with these operations is shown to be
arithmetically complete with respect to the intended provability semantics and
sufficient to realize every operation on proofs admitting propositional specification
in arithmetic.
1 Introduction
A provability reading of a modality 2F as
``F is provable''
was an intended informal semantics for the classical system S4 of propositional
