Summary: Logic of Proofs \Lambda
Steklov Mathematical Institute,
Vavilov str. 42,
117966 Moscow, Russia
August 10, 1993
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.
In  and  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