 
Summary: Logic of Proofs \Lambda
Sergei Artšemov
Steklov Mathematical Institute,
Vavilov str. 42,
117966 Moscow, Russia
email: art@log.mian.su
y
August 10, 1993
Abstract
In this paper individual proofs are integrated into provability logic. Systems
of axioms for a logic with operators ``A is provable'' and ``p is a proof of A''
are introduced, provided with Kripke semantics and decision procedure. Com
pleteness theorems with respect to the arithmetical interpretation are proved.
1 Introduction
In [1] and [2] proofs were incorporated into propositional logic by means of
labeled modalities. The basic labeled modal logic contains the propositional
logic enriched by unary operators 2 p i , i = 0; 1; 2; : : : . This language helps to
provide a logical treatment of a rather general situation when we are interested
not only to know that a certain statement A is valid, but also have to keep track
on some evidences of its validness: 2 p A may stand for ``p is a proof of A'', ``p is
