Epistemic Extension of Godel Logic

In this paper, some epistemic extensions of Godel logic are introduced. We establish two proof systems and show that these logics are sound with respect to an appropriate Kripke semantics. Furthermore, we demonstrate weak completeness theorems, that is if a formula is valid then its double negation is provable. A fuzzy version of muddy children puzzle is given and using this, it is shown that the positive and negative introspections are not valid. We enrich the language of epistemic Godel logic with two connectives for group and common Knowledge and give corresponding semantics for them. We leave the problem of soundness and completeness in this general setting open. Finally, an action model approach is introduced to establish a dynamic extension of epistemic Godel logic.