Home Philosophy Advances in Proof-Theoretic Semantics

## Unknown StatementsI have said that the assertibility of ¬ Let me observe first that “K proof of K
It should be noticed that Dummett's remark—that intuitionistic truth, when it is equated with the actual possession of a proof, does not commute with negation—is certainly correct when it is understood as referring to strong negation. For example, the observation that what one is presented with is not a proof that it is raining is not the same thing as the observation that what one is presented with is a proof that it is not raining. As a consequence, Dummett's objection to the validity of the (T) schema as a criterion for being a truth operator seems to cause trouble in this case. However, in this case the argument (9) is no longer valid: the second step is an application of contraposition, but contraposition is not valid for strong negation. As a consequence, the fact that strong negation does not commute with truth does not entail the invalidity of the (T) schema. We can therefore conclude that, even when we add to intuitionism strong negation, the knowledge operator K is a truth operator. 25A discussion of this assumption is beyond the limits of this paper. 26Reference [24]. 27In general empirical sentences have (non-conclusive) justifications. A definition of the notion of justification for the sentences of suggested such a definition in [23]. It seems to me that, if the existence of undecided statements can be expressed at all in an intuitionistic language, it should be expressible in (15) Goldbach's conjecture is undecided (unknown) the following formula seems to be a plausible formalization in (16) ∼ K Williamson argues against this formalization: if ∼ is to count intuitionistically as any sort of negation at all, ∼ In other words, the schema (17) ∼ should be valid; then, from (16) one could derive ¬ K (17) is valid (16) is inconsistent, and, on the other hand, the motivation for it seems insufficient. Notice that (17) (18) ∼ K it asserts the existence of a function However, if we look at the interplay between intuitionistic logical constants, strong negation and K from the standpoint of Kripke semantics, the assertibility of (18) seems to be out of the question. A 28Reference [26], p. 139. 29Reference [9].
• If if • For every For every The notion F F F F F F F F Now, if we add the operator K to (19) F F Call any Kripke model for
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∼ K ∼ K ∼ K ∼ K ¬ On the other hand, F |

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