Standard Godel Modal Logics

We prove strong completeness of the square-version and the lozenge-version of a Godel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Godel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite 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.