Theoremhood-preserving Maps Characterizing Cut Elimination for Modal Provability Logics

Propositional modal provability logics like and have arithmetical interpretations where can be read as ‘formula is provable in Peano Arithmetic’. These logics are decidable but are characterized by classes of Kripke frames which are not first-order definable. By abstracting the aspects common to their characteristic axioms we define the notion of a formula generation map in one propositional variable. We then focus our attention on the properly displayable subset of all (first-order definable) Sahlqvist modal logics. For any logic from this subset, we consider the (provability) logic obtained by the addition of an axiom based upon a formula generation map so that . The class of such logics includes and . By appropriately modifying the right introduction rules for , we give (not necessarily cut-free) display calculi for every such logic. We define the pseudo-displayable subset of these logics as those whose display calculi enjoy cut-elimination for sequents of the form for any formula . We then show that for any provability logic having a conservative tense extension, there is a map on formulae such that is pseudo-displayable if and only if maps theorems of to theorems of the underlying logic and vice versa. By using a standard renaming technique we can guarantee that there is a polynomial-time translation from into . All proofs are purely syntactic and show the versatility of display calculi since similar results using traditional Gentzen calculi are not possible for as broad a range of logics and require further conditions. Our maps generalize previously known maps from into . An application of our results gives an translation from the (‘second order’) provability logic into a decidable subset of first-order logic. Since each of our logics is a Sahlqvist logic, it is first-order definable, and hence each has a translation into first-order logic. Our results therefore show that all pseudo-displayable logics are ‘essentially first-order’ even though their characteristic axiom may not be first-order definable.