I think some clarification is in order. The thesis in question is this:
(T1) If S believes that p, then S believes that the probability of p is >0.5.
It seems reasonable in many contexts. E.g. I believe that the Earth is round, and I believe that the probability that the Earth is round is >0.5. One might ponder what kind of probability that is: prior probability? Clearly not. So we ought to add a clause about what evidence we’re calculating, intuitively perhaps, the probability given
(T2) If S believes that p, then S believes that the probability of p is >0.5 given the evidence available to S.
This amendment seems to fix the unclearness about evidence and probabilities.
Now, as this is a categorical prop. we can look for counter-examples. First some further clarification is in order. The thesis is still ambiguous. The ambiguity is about what meaning is to be ascribed to the “if…then…” part. Is it a material or a logical implication? We can clarify it like this
(T3) S believes that p logically implies that S believes that the probability of p is >0.5 given the evidence available to S.
(T4) S believes that p materially implies that S believes that the probability of p is >0.5 given the evidence available to S.
Now obviously (T3) is much stronger than (T4) and so it is more probably false. The difference with counter-examples in relation to (T3) and (T4) is that counter-examples for (T3) only need to be logically possible to work, but they need to be actual against (T4).1 So, let’s first look for counter-examples for the stronger claim but note that if we can find one that works against (T4) then it also works against (T3) since actuality implies possibility.
One might find a counter-example in the vicinity of Pascal’s Wager. Suppose that someone manages to believe some proposition, p, because he thinks that it is a good idea because of the consequences of believing p irrespectively of what the probability of p is given that someone’s evidence. This theory that one can start believing something purely because of choice is called voluntarism about belief. It seems false when we think about it. Suppose for instance that you want to believe that you don’t exist, or that the Earth is flat. Can you? No. The voluntarism may respond that it’s just because you didn’t try hard enough. That seems to be an ad hoc reply. Now, supposing that it is false that if one tries to believe something, then one will do it. Thus, voluntarism is not a threat against (T4), that is, the weak version of the thesis.
But is voluntarism logically impossible? I seems not because I can find no contradiction in a possible world where it is true. Perhaps someone else can. Suppose that it is indeed logically possible. Then it follows that the strong thesis is false, that is, (T3) is false because (T3) implies that it is impossible that (S believes that p and it is not the case that S believes that the probability of p is >0.5 given his evidence). I ask of you to find a contradiction in this counter-example without begging the question.
Suppose the above objection to (T3) is successful, then we should ask ourselves: Can some version of the stronger thesis perhaps still be saved? Let’s consider another version of it
(T5) S believes that p is defined as S believes that the probability of p is >0.5 given the evidence available to S.
Now it is a definition. It’s not clear how it avoids the objection from before but let’s suppose that it somehow does. There is another objection that can be made against (T5), and that is that it is circular. Note that “belief” is defined via the same word as it is trying to define. I take this circularity to be vicious. Given the objection I conclude that the strong version of the thesis has been adequately refuted.
Inference to the best explanation
Now let’s return to the weak version (T4). Is it more defendable? Perhaps. However, consider a case where we know of only two theories of some phenomena (or -non). Suppose some person were to think about the phenomena and these two theories. The person happens to conclude for some reason that theory one is better than theory two, and on this basis he infers to the best explanation without having any belief about the probability of the theory being true given his evidence. He may even believe that the probability of the theory is <0.5. Are there any actual cases of this description? I would say yes. We sometimes infer to the best explanation in cases where we cannot find another theory that explains the data and still we accept the theory without believing it to be more probable than 0.5.
If there are counter-examples of the above type, then the weak thesis is also false, and I cannot seem to find any amendment to the thesis to make.
1Or physically possible, perhaps. Physically possible and actual are logically equivalent given a regularity theory of laws of physics.