  In this paper we consider the modal logic with both Box and Diamond arising
fromKripke models with a crisp accessibility and whose propositions are valued
over the stan-dard Godel algebra [0,1]G. We provide an axiomatic system
extending the one from [3]for models with a valued accessibility with Dunn
axiom from positive modal logics, andshow it is strongly complete with respect
to the intended semantics. The axiomatizationsof the most usual frame
restrictions are given too. We also prove that in the studied logicit is not
possible to get Box as an abbreviation of Diamond, nor vice-versa, showing that
indeedthe axiomatic system we present does not coincide with any ofthe
mono-modal fragmentspreviously axiomatized in the literature.
