Display Calculi for Nominal Tense Logics

We define display calculi for nominal tense logics extending the minimal nominal tense logic ( ) by addition of primitive axioms. To do so, we use the natural translation of into the minimal tense logic of inequality ( ) which is known to be properly displayable by application of Kracht’s results. The rules of the display calculus Æ for mimic those of the display calculus Æ for . We show that every -valid formula admits a cut-free derivation in Æ . We also show that a restricted display calculus Æ , is not only complete for , but that it enjoys cut-elimination for arbitrary sequents. Finally, we give a weak Sahlqvist-type theorem for two semantically defined extensions of . Using Kracht’s techniques we obtain sound and complete display calculi for these two extensions based upon Æ and Æ respectively. The display calculi based upon Æ enjoy cut-elimination for valid formulae only, but those based upon Æ enjoy cut-elimination for arbitrary sequents.