Standard Gödel Modal Logics

We prove strong completeness of the -version and the 3-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both in…nitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the …rst logic is reducible to the class of frames having twovalued accessibility relation and this logic does not enjoy the …nite model property, while validity in the second logic requires truly fuzzy accessibility relations and it has the …nite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.