Supervaluation on Trees for Kripke's Theory of Truth

A method of supervaluation for Kripke's theory of truth is presented. It di ers from Kripke's own method in that it employs trees; results in a compositional semantics; assigns the intuitively correct truth values to the sentences of a particularly tricky example of Gupta's; and it is argued is acceptable as an explication of the correspondence theory of truth. In his (1982) Gupta presents the following scenario in order to criticize Kripke's (1975) theory of truth: Assume that the sentences A1: Two plus two is three A2: Snow is always black A3: Everything B says is true A4: Ten is a prime number A5: Something B says is not true are all that is said by a person A , and the sentences B1: One plus one is two B2: My name is B B3: Snow is sometimes white B4: At most one thing A says is true are all that is said by a person B . The sentences A1, A2, and A4 are clearly false and B1, B2, and B3 are clearly true. So it seems unobjectionable to reason as follows: A3 and A5 contradict each other, so at most one of them can be true. Hence at most one thing A says is true. But that is what B says with his last sentence, so everything B says is true. This is again what A says with A3 and rejects with A5, so A3 is true and A5 false. But counterintuitively Kripke's theory in its strong Kleene scheme, minimal xed point version (with which I will assume familiarity and hereafter refer to as the basic version ) tells us that A3, A5, and B4 are all unde ned. The reason is that the evaluation of A3 and A5 awaits the determination of B4, which in turn cannot receive a truth value before A3 or A5 do. One way to obtain the intuitively correct truth values is to swear allegiance to one of the theories that assign truth values in a holistic manner. But that See for example (Walicki 2009), according to which truth values are not assigned in a