Ryan Wasserman was here this week–what a great guy…and superb philosopher!–and we talked about knowability. He had an interesting idea. For standard knowability, you need to get from K(p&q) to Kp&Kq. Since nobody sophisticated about closure thinks that knowledge is closed under entailment, or that knowledge is closed under known entailment, he wondered how we’d get the argument to work. The most plausible closure principles are versions of competent deduction closure principles, and Ryan’s idea was that this might block the paradox. The idea is that some truths might be knowable only to incompetent deducers, and knowledge that a given claim is an unknown truth might be just such a knowable claim–only the logically challenged could have such knowledge.
Very interesting idea, and surely an advance beyond anything in the literature to this point. But conjunction seems special to me, so I’m not sure it works. To possess the concept of conjunction seems to be about as close to the Dummettian ideal as anything can be: to possess the concept is just to wield properly the intro- and elim-rules for &. If so, one wouldn’t be able to have any conjunctive beliefs without seeing that the belief either resulted from &-intro or was susceptible to &-elim. So conjunctive beliefs couldn’t be the kind that could only be known by incompetent deducers.
There’s also Williamson’s conjunctive knowledge bypass argument, but I won’t go into that here. Still, there are so few really interesting and new ideas about the knowability paradox that it is worth noting them when one finds them.