Negative operations on proofs and labels

Logic of proofs LP introduced by S. Artemov in 1995 describes properties of proof predicate “t is a proof of F ” in the propositional language extended by atoms of the form [t]F. Proof are represented by terms constructed by three elementary recursive operations on proofs. In order to make the language more expressive we introduce the additional storage predicate with the intended interpretation “x is a label for F”. It can play the role of syntax encoding, so it is useful for specification of operations that require formula arguments. In this paper we study operations on proofs and labels that can be specified in the extended language. First, we give a formal definition of an operation on proofs and labels. Then, for an arbitrary finite set of operations F the logic LPS(F) is defined. We provide LPS(F) with symbolic semantics and arithmetical semantics. The main result of the paper is the uniform completeness theorem for this family of logics with respect to the both types of semantics.

MOSCOW MATHEMATICAL JOURNAL Volume 1, Number 4, October–December 2001, Pages 475–490 ON FIRST ORDER LOGIC OF PROOFS

To the memory of I. G. Petrovskii on the occasion of his 100 th anniversary Abstract. The Logic of Proofs LP solved long standing Gödel’s problem concerning his provability calculus (cf. [4]). It also opened new lines of research in proof theory, modal logic, typed programming languages, knowledge representation, etc. The propositional logic of proofs is decidable and admits a complete axiomatization. In this paper we show that the first order logic of proofs is not recursively axiomatizable