Kripke’s (1975) theory in its different guises is based on partial logic. In order to obtain models for a theory in classical logic, the extension of the truth predicate in the partial model is used again as the extension of truth in the classical model. In the classical model false sentences and those without a truth value in the partial model are declared not true. KF is sound with respect to these classical models and thus incorporates two distinct logics. The first is the ‘internal’ logic of statements under the truth predicate and is formulated with the Strong Kleene valuation schema. The second is the ‘external’ logic which is full classical logic. An effect of formulating KF in classical logic is that the theory cannot be consistently closed under the truth-introduction rule 