We study arbitrary intermediate propositional logics extended with a collection of axioms from justification logics. For these, we introduce various semantics by combining either Heyting algebras or Kripke frames with the usual semantic machinery used by Mkrtychev’s, Fitting’s or Lehmann and Studer’s models for classical justification logics. We prove unified completeness theorems for all intermediate justification logics and their corresponding semantics using a respective propositional completeness theorem of the underlying intermediate logic. Further, by a modification of a method of Fitting, (...) we prove unified realization theorems for a large class of intermediate justification logics and accompanying intermediate modal logics. (shrink)

We study the topic of realization from classical justification logics in the context of the recently introduced Gödel justification logics. We show that the standard Gödel modal logics of Caicedo and Rodriguez are not realized by the Gödel justification logics and moreover, we study possible extensions of the Gödel justification logics, which are strong enough to realize the standard Gödel modal logics. On the other hand, we study the fragments of the standard Gödel modal logics, which are realized by the (...) usual Gödel justification logics. We prove the corresponding realization theorem by using Fitting’s merging of realizations as well as appropriate hypersequent calculi on the modal side, adapting the work by Metcalfe and Olivetti. For these hypersequent calculi, we also show a cut-elimination theorem. We provide natural semantical characterizations for all of these newly introduced logics. (shrink)