Epistemic possibility and the Liar

Here’s a fun Liar paradox involving epistemic possibility. Say that a proposition p is epistemically possible if it is consistent with all you know.

Construct a sentence G such that:

0. G is true if and only if G is not epistemically possible.

E.g., “The proposition expressed by the first sentence in this post found in quotation marks is not epistemically possible.”

Now, you only know truths, and truth is consistent with truth. Thus:

1. If G is true, then it is consistent with everything you know.

But G is true if and only if it is not epistemically possible. So:

2. If G is true, then it is not consistent with everything you know.

Hence:

3. G is not true.

But now that you’ve seen this argument, you surely are in a position to know G not to be true. Suppose you exploit this and indeed come to know G not to be true. But then we have a contradiction. For if you know G not to be true, then G is not epistemically possible, and hence by (0), it must be that G is true.

Might well

It’s occurred to me that the “might well happen that” operator makes for an interesting modality. It divides into an epistemic and a metaphysical version. In both cases, if it might well happen that p, then p is possible (in the respective sense). In both cases, there is a tempting paraphrase of the operator into a probability: on the epistemic side, one might say that it might well happen that p if and only if p has a sufficiently high epistemic probability, and on the metaphysical side, one might say that it might well happen that p if and only if p has a sufficiently high chance given the contextually relevant background. In both cases, it is not clear that the probabilistic paraphrase is correct—there may be (might well be!) cases of “might well happen that” where numerical probabilities have no place. And in both cases, “might well happen that” seems context-sensitive and vague. It might well be that thinking about this operator could lead to progress on something interesting.