Logic of Proofs \Lambda Sergei Artemov 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 Collections: Computer Technologies and Information Sciences