Roy Cook has just argued that there are some unknowable propositions provided it is possible for the following situation to obtain. Ava, Brigitte and Dorothy are all inhabitants of Smullyan’s (1978) island occupied solely by knights and knaves. Knights are people who always speak truly, knaves are people who always speak falsely. Simultaneously, they utter the following:
Ava: What Brigitte is now saying cannot be known to be true
Brigitte: What Dorothy is now saying cannot be known to be true
Dorothy: What Ava is now saying cannot be known to be true
Cook shows that if this happens then at least two of Ava, Brigitte and Dorothy must be knights (see his p. 12), and goes on to show as a consequence of this that at least one of them – we do not know which one – has uttered a true but unknowable proposition. He assumes classical logic and certain facts about the behaviour of the knowability operator in order to get this result, all of which we can grant here for the sake of argument.
Cook stresses that his argument relies on the thought that the scenario he describes is free from ‘any paradox, failure of reference, or other pathology’ (p. 14). It seems to me, however, that the anti-realist who accepts the principle p --> Kp will think Cook’s scenario is paradoxical, because it presents a version of the paradox raised by what Cook would call the semantic open triple (sorry, no corner quotes):
(where T is the truth predicate). No assignment of (classical) truth values to these three propositions is consistent.
Anti-realists for Cook’s purposes are those who accept that if p then p is knowable, which Cook represents as p --> Kp, the contrapositive of which is ¬Kp --> ¬p. By this and the contrapositive of disquotation for the truth predicate, ¬p --> ¬Tp, these anti-realists will also accept that ¬Kp implies ¬Tp. Hence the three characters in Cook’s story are uttering claims which are either stronger than or identical in strength to the paradox-creating claims of the semantic open triple. In fact, anti-realists will presumably think the claims are identical in strength; for they are happy to accept ¬Kp --> ¬Tp, and ¬Tp --> ¬Kp is trivial.
One might respond that this is to beg the question; the only reason that is being offered for thinking that Cook’s situation is paradoxical is adherence to p --> Kp, which, of course, one does not accept unless one is an anti-realist.
But we need to be clear about where the burden of proof lies. Cook is (tentatively) offering an argument that should be capable of persuading anti-realists that there could be an unknown truth. To do this, as he acknowledges, the situation he describes should not be paradoxical. But the anti-realist will think it generates a Liar-like paradox, namely the semantic open triple. Therefore Cook’s argument will not be persuasive to an anti-realist.