Suppose you want to convince me that I have no hands but are unable to lie (and I know for sure you are unable to lie). However, you know a lot more than I about something, perhaps something completely irrelevant to the question. For instance, suppose you know the results of some very long sequence of die rolls that's completely irrelevant. It seems you can fool me with the truth. For you can find some true proposition p about the die rolls such that I assign an exceedingly low probability to p. You then reveal to me this disjunctive fact: p is true or I have no hands. Then: P(no hands | p or no hands) = P(no hands) / (P(p) + P(~p and no hands)) ≥ P(no hands) / (P(p) + P(no hands)). (Exercise: check the details.) If P(p) is sufficiently small, relative to my prior probability P(no hands) (which of course is non-zero--there is a tiny chance that I was in a terrible accident and superb prostheses have just been developed), this will be close to 1.
But of course, whether I have hands or not, if you know a lot more about something than I do, you will be able to find a truth that I assign a tiny probability to. So I really shouldn't be deceived by you. Rather, I should take myself to have learned p. Your disjunction is equivalent to the material conditional that if I have hands, then p. I know I have hands. So, p. But what about the Bayesian calculation, which is mathematically correct?
This is a protocol problem. If I happened to ask you whether the disjunction "p is true or I have no hands" was true, and you then revealed it to me that it was, the Bayesian calculation would have been correct. But the actual protocol was that you picked out a truth that I took to be unlikely, and disjoined it with a claim that I have no hands. If I knew for sure that this was your protocol, I would have learned two things: first that p is true, and second that p is true or I have no hands. The second would have been uninformative in light of the first, and so there would be no deceit. But of course if the above were to really happen, I wouldn't know for sure what your protocol was.
In real life, when someone tells us something odd out of the blue, we often ask: "Why are you telling me this?" The above case shows how epistemically important the answer to this question can be. If you tell me (remember that you are unable to lie) that you're telling me this to get me to think I have no hands, I will suspect that your protocol may be to find an unlikely truth and disjoin it with the claim that I have no hands. As long as I have significant suspicion that this is your protocol, your statement won't shake my near-certainty that I have hands. But if you tell me that you were telling me this because you decided, before finding out whether p was true, that you were going to tell me whether or not the disjunction is true, then my near-certainty that I have hands should be shaken. I wonder how often "Why are you telling me this?" involves a case of trying to find the protocol and thus to figure out how to update. Rarely? Often?
