Theoremhood Preserving Maps Characterising Cut Elimination for Modal Provability Logics

Propositional modal provability logics like G and Grz have arithmetical interpretations where 2φ can be read as “formula φ is provable in Peano Arithmetic”. These logics are decidable but are characterised 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 F(p) 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 L from this subset, we consider the (provability) logic LF obtained by the addition of an axiom based upon a formula generation map F(p) so that LF = L + F(p). The class of such logics includes G and Grz. By appropriately modifying the right introduction rules for 2, 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 Visit to A.R.P. supported by an Australian Research Council International Fellowship. On leave from Laboratoire LEIBNIZ, Grenoble, France. Supported by an Australian Research Council Queen Elizabeth II Fellowship.