Axiomatization of Crisp Godel Modal Logic

In this paper we consider the modal logic with both square and lozenge arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Godel algebra [0, 1](G). We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37-55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get lozenge as an abbreviation of square, nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.