On Bimodal Logics of Provability

We investigate the bimodal logics sound and complete under the interpretation of modal operators as the provability predicates in certain natural pairs of arithmetical theories (F’, ?2). Carlson characterized the provability logic for essentially reflexive extensions of theories, i.e. for pairs similar to (PA, ZF). Here we study pairs of theories (F’, %!) such that the gap between F and Q is not so wide. In view of some general results concerning the problem of classification of the bimodal provability logics we are particularly interested in such pairs (F, 42) that % is axiomatized over T by L’,-sentences only, and, for each n > 1, Q proves the n-times iterated consistency of LT. A complete axiomatization, along with the appropriate Kripke semantics and decision procedures, is found for the two principal cases: finitely axiomatizable extensions of this sort, like e.g. (PA, PA + Con(ZF)), (IC,, ZC, + Con(Z,Y,)), etc., and reflexive extensions, like (PRA, PRA + {Con(lC,) 1 n 2 l}), etc. We show that the first logic, ICP, is the minimal and the second one, RP, is the maximal within the class of the provability logics for such pairs of theories. We also show that there are some provability logics lying strictly between these two. As an application of the results of this paper, in the last section the polymodal provability logics for natural recursive progressions of theories based on iteration of consistency are characterized. We construct a system of ordinal notation 2, which gives exactly one notation to each constructive ordinal, such that the logic corresponding to any progression along %” coincides with that along natural Kalmar elementary well-orderings.