Knowability and bivalence: intuitionistic solutions to the Paradox of Knowability

Abstract

In this paper, I focus on some intuitionistic solutions to the Paradox of Knowability. I first consider the relatively little discussed idea that, on an intuitionistic interpretation of the conditional, there is no paradox to start with. I show that this proposal only works if proofs are thought of as tokens, and suggest that anti-realists themselves have good reasons for thinking of proofs as types. In then turn to more standard intuitionistic treatments, as proposed by Timothy Williamson and, most recently, Michael Dummett. Intuitionists can either point out the intuitionistc invalidity of the inference from the claim that all truths are knowable to the insane conclusion that all truths are known, or they can outright demur from asserting the existence of forever-unknown truths, perhaps questioning-as Dummett now suggests-the applicability of the Principle of Bivalence to a certain class of empirical statements. I argue that if intuitionists reject strict finitism-the view that all truths are knowable by beings just like us-the prospects for either proposal look bleak.

The Paradox of Knowability from an Intuitionistic Standpoint

“…The former claim is perfectly compatible with the idea, embraced above, that the validity of (7) is not the definition, but the material adequacy condition of the definition, of O. 23 See [16], p. 275. Now, if we observe that there are statements p that the intuitionist acknowledges as being decidable (i.e.…”

“…In conclusion, when the truth of a (mathematical) statement α is equated with the actual possession of a proof of α, truth does commute with intuitionistic negation. 16 …”

“…Hence, the material adequacy condition 'does justice' to the intuitive notion of truth as correspondence. The question whether the intuitive notion of truth as correspondence is bivalent is not explicitly addressed by Tarski; an affirmative answer from him is suggested by the fact that in [20] he derives the principle of bivalence from the (materially adequate) definition of 16 Notice that truth commutes with intuitionistic disjunction as well, since the principle…”

“…There is an argumentcalled by [16] "The Standard Argument"-that purports to show that it is not. 23 It consists in the following derivation of ∃α(α ∧ ¬ K α) from the assumptions p ∨ ¬p and ¬ K p ∧ ¬ K ¬ p:…”

Conditionals in Interaction

“…So, rather than thinking of the making of an assertion as expressing a fully formed propositional content, which may be thought of as true or false, we rather think of it as playing the statement as a kind of token in a game. 24 At this point, the statement may be treated hypothetically, and can be challenged and tested by other agents. It may, for example, be defended by the provision of reasons, and it may be contested by other reasons and counterexamples.…”