Suppose you think that it's possible to know that p, even though your epistemic position vis-a-vis p is weak enough for 'it might be that not-p', in its epistemic reading, to be true. I don't really see why you'd want to think this myself, but I guess some people think that (a) this is a good reading of 'fallibilism' and (b) fallibilism is true. If you think this, then you face the problem to explain the infelicity of concessive knowledge attributions. Why's it sound so bad to say "I know that p but p might be false"?
The obvious explanation is that it's a contradiction: according to standard epistemic modal logic, 'might', in its epistemic reading, is just the dual of 'know'. But the fallibilist of this stripe has closed off that response. What's he say instead? Dougherty and Rysiew propose a pragmatic line: "p might be false," they say, implicates but does not entail that there is a significant chance of not-p. And while a chance of not-p is consistent with knowledge that p, a significant chance of not-p is not. Fantl and McGrath supplement the story by suggesting that the significance of various chances can be a stakes-sensitive matter; the same possibility, with the same likelihood, can be significant if the stakes are high, and insignificant if the stakes are low.
Now I get nervous when Gricean pragmatic stories are asked to do work like this. Too often, the data don't generalize the right ways. Here's one problem: the pragmatic effect doesn't seem appropriately cancelable. Consider:
It's possible that it will rain today, but I know it won't rain today.
The badness of this sentence is explained, on the view in question, by suggesting that the first conjunct pragmatically implicates that there is a significant chance that it will rain today. It predicts, then, that if we cancel the implication, we're left with felicity. But this prediction is not borne out; this is still bad:
It's possible that it will rain today, but there's no significant chance that it will rain today, so I know it won't rain today.
Also, there's a point that Derek Ball raised in Jason Stanley's seminar last week, inspired by Seth Yalcin: the infelicity of concessive knowledge attributions persists in non-assertoric contexts. "Suppose that you know it will rain today and it might not rain today." "If you know it will rain today and it might not rain today, then you know something that might not happen." Etc. The Gricean story is peculiar to assertions, and therefore insufficiently general.
I think there's a better view in the same spirit. (Well, maybe in the same spirit; I'm not quite sure what the intuitive motivation behind this project is. My suggestion won't vindicate the coherence of concessive knowledge propositions. But like I said, I'm not sure I see why anyone would want to do that.) The line we've been considering is one in which "there is some possibility of p" pragmatically implicates that there is some significant possibility of p. But the existential quantifier is going to have a context-sensitive domain restriction anyway. We could suppose that in the relevant contexts, we're only quantifying only significant possibilities. Then "there is some possibility of p" would, in the relevant context, entail that there is some significant possibility of p.
On this approach, you can still get a lot of the stuff that Fantl and McGrath want. On this view, whether there is a possibility of p will depend on the stakes, since all possibilities are significant possibilities, and whether a possibility is significant depends on stakes. So their 'impurism' would infect 'possibility' talk too. (This is not a result of the view they actually offer, which I'm criticizing: they have 'pure' possibilities, where talk of them implicates results about 'impure' significant possibilities.) But the concessive knowledge attributions will be genuine contradictions.