Provability logics with quantifiers on proofs

We study here extensions of the Artemov’s logic of proofs in the language with quanti%ers on proof variables. Since the provability operator A could be expressed in this language by the formula ∃u[u]A, the corresponding logic naturally extends the well-known modal provability logic GL. Besides, the presence of quanti%ers on proofs allows us to study some properties of provability not covered by the propositional logics. In this paper we study the arithmetical complexity of the provability logic with quanti%ers on proofs qLPK(T ) for a given arithmetical theory T and a class K of proof predicates. In the last section we de%ne Kripke style semantics for the logics corresponding to the standard G6 odel proof predicate and its multiple conclusion version. c © 2002 Elsevier Science B.V. All rights reserved.