At some point during Q&A, the KK principle came up, namely the principle that to know p entails to know that you know p (Kp --> KKp). Williamson was arguing against it (as he must have done during the lecture) on conceptual grounds, suggesting that there are situations where it simply does not seem to hold (some convoluted thought-experiment). His interlocutor at that point, Allard Tamminga, insisted that the KK principle is fundamentally correct, and it wasn't clear how the debate could be carried on any further.
So I asked Williamson whether he thought that a debate on the KK principle might benefit from attention to empirical data. He first thought that I meant carrying out surveys and asking people around whether they thought that the KK principle holds -- which sort of beats the point, as they may know the principle and yet not know that they know it! (duh...) I then clarified that I actually had psychological experiments in mind, in particular the kind of thing that has been investigated under the heading of meta-cognition in recent years (very cool stuff!). He was still not very enthusiastic about my idea, arguing that, just as psychological phenomena are not reliable guides for the truth of mathematical statements, they cannot be reliable guides for the truth of logical principles, and according to him KK is a logical principle.
But is it really? I admit that I tend to think that an awful lot of questions are ultimately empirical questions, but I take KK to be essentially about cognition, and thus not a logical principle as such (unless one wants to take the Kantian transcendental road to cognition and infer everything a priori). In fact, many of the results in the meta-cognition tradition seem to suggest that we often 'know' without knowing that we know (and similarly, that we often think we know when we actually do not know). In other words, the meta-cognition literature investigates, among other things, the accuracy with which we judge our own epistemic capacities. My hunch is that the results basically support the view that KK does not hold, but a more serious investigation would have to be carried out for more definitive claims. In any case, I think it is pretty obvious that it could be very interesting to look into the meta-cognition material from the point of view of KK.
I must say that I was surprised to see Williamson so dismissive of the possible relevance of empirical data for this issue. After all, in The Philosophy of Philosophy, he does enlist the mental models theory of reasoning to argue against the inferentialist program (something that Ole still owes us a paper on!). (Basically, his argument doesn't work, but at least it departs from the premise that empirical data could be relevant for such philosophical debates.) But today, he was not in any way sympathetic to the idea -- a shame, if you ask me.