Tarski taught us that we cannot have both, a classical, semantically closed language and an adequate definition of truth. Kripke taught us that we can define a truth predicate for a semantically closed language, if we are willing to embrace a subclassical logic (at least on the sentential level). Yet, Tarski’s ghost is still with us, as these subclassical logics do not possess an attractive conditional connective, that is, a conditional that enables us to `carry on sustained ordinary reasoning’. In this talk we show how conditionals can be added to subclassical logics, in particular strong Kleene logic K3 and investigate the costs and consequences of such an addition. This leads us to a discussion of hyperintensionality in logics and truth theories.