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?
6 comments:
" 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."
A better question would be "what is the causal connection between the facts of your stated disjunction"? Without a known causal connection, there could still just have been a coincidence making the statement true, and the truthful person could have just successfully found such a coincidence on purpose.
If I say I have no hands when pigs fly, making pigs fly won't remove my hands.
Even if it's just a coincidence, we can learn from it.
Another way to put this might be that the evidentialists' picture of evidence that is just "there", given, is often misleading. Rather than an investigator finding evidence, frequently the evidence finds the investigator. (Especially in cases of testimony.) And then one has to worry about selection effects in the evidence: why did just _this_ evidence come find me just _now_?
Testimony may be particularly special. :-) A testifier often testifies in order to produce a particular effect in you. In most other interactions in the world, the world doesn't care what you make of it.
Here is another way to state the point about protocols: to apply Bayes, it’s not enough to know what actually happened. You also have to know, or make assumptions about, all the relevant alternatives that might have happened. In fancy language, you are conditioning relative to an express or implied partition.
Example: a friend rolls a die and tells you that the outcome is even. What is the chance that it is 2? The obvious answer is 1/3. This is based on the implied partition {even, odd}. But maybe your friend had instructions say “even” iff the outcome was 2, and otherwise to say nothing. It is easy to make up variations.
In real life, it’s unusual to have the required knowledge.
Maybe the simplest thing to say is this. When a speaker informs you of something, you need to update not just on the content of what was said, but also on the fact that the speaker informed you of it. And lots of stuff can be learned from the latter fact.
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