Adequate conditionals and Kripke's theory of truth

In this paper we show how to introduce a conditional to Kripke's theory of truth that respects the deduction theorem for the consequence relation associated with the theory. To this effect we develop a novel supervaluational framework, called strong Kleene supervaluation, that we take to be a promising framework for handling the truth-conditions of non-monotone notion in the presence of semantic indeterminacy more generally.