Yes, I take it the ‘middle ground’ sort of approach you describe is a lot like the one I intended to be making available in my previous clarification. I want to say that the infelicity of (2) is explained by contradiction; this is consistent with there being contexts in which (2) is neither infelicitous nor contradictory.

Notice that, although (1) entails (2), it’s easy to get confused here, on the approach I suggest, since the proposition (1) expresses in typical contexts for (1) does not entail the proposition (2) expresses in typical contexts for (2). So it’s easy tacitly to equivocate when thinking about such arguments.

Generally, I like the strategy you’re exploring. But I’d like to think (1) is true, or at least consistent:

(1) Even though it’s strictly speaking possible that all my seventy intro logic students are going to get 100% on their logic final, I know that won’t happen.

But (1) entails (2)

(2) It is possible that all my sixty logic students are going to get 100% on their logic final, but I know that won’t happen.

And if there is this entailment, this would help explain why, if I start by thinking about (1), I feel pressure to say 2 is true – though I feel I want to explain 2 further and to remove misleading implications. One way to do that is to say: yes, it’s possible but it’s outlandish; I know that won’t happen.

I wonder if there might be a middleground between your view and the one Fantl and I mentioned in our first chapter. One might try to say that without an elaborate conversational background, a bare CKA - one in which ‘possible’ is left bare and unqualified - will be false. This would be because, without such a set-up, ‘it’s possible’ is semantically associated only with significant possibilities. Consistently with that, one might think that with the necessary conversational background – which might include the assertion of a non-bare CKA – ‘possible’ can extend to insignificant possibilities, so that the bare CKAs may be true.

Some other analogues might be:

“Do I have a reason to eat this piece of metal? Well, it has iron. But even though I strictly speaking have a reason to eat this piece of metal, it would be completely irrational to do so, because of all the far more powerful reasons against eating it.”

Here we heavily qualify “I have a reason to…” Without a pretty elaborate set-up like this stressing the overall weakness of the reason, ‘I have a reason to eat this piece of metal’ will seem clearly wrong. Moreover, without an elaborate set-up like this, even when it is in embedded contexts, ‘I have a reason’ will be understood as being about minimally significant reasons. Mark Schroeder uses an example something like this in his book *Slaves of the Passions*.

Or maybe think about Achinstein on evidence. Achinstein would say that the fact that Michael Phelps is taking his daily swim is not evidence he will drown, even though it raises the probability of his drowning. So, Achinstein concludes probability-raising is not sufficient for evidence. But I like the following reply. Without an extremely elaborate set-up stressing the weakness of the evidence – just how little it affects our overall reason to believe, ‘there is evidence Michael Phelps will drown’ – even in unasserted contexts – will be interpreted as concerning only meaningful or significant evidence. However, suppose we start by saying, “The only way you can drown is by being in the water, and even the best swimmers could always drown, and since Phelps is swimming right now, there is a teeny tiny bit of evidence that he will drown.” This seems right, and it seems to entail ‘there is some evidence Phelps will drown’.

So, clarificatory questions: are you comfortable saying that the proposition I normally express by an utterance of 2 is true?

I mean, to be honest, I do think it seems pretty bad. So if you’re really asking me to tell you what I think is correct, I do think that (2), in typical contexts, will be a contradiction. But the point of my comment was that the view given in the post doesn’t mandate that. Someone who thinks (2) typically expresses a consistent proposition can still have my view about CKAs. So to the end of demonstrating this flexibility, for the purpose of argument, I’m willing to allow that (2) is typically true. (And it is my considered view that there can be contexts in which it is true.)

And are you comfortable saying that there is no relevant contextual shift from the assertion of the first conjunct to the assertion of the second?

Again, there are different answers to give for the purpose of telling what my view is, and for demonstrating what the suggestion of this post can handle. My own considered view is a Lewisean infallibilist contextualist approach to ‘knows’ — so my considered view is that true readings of (2) need the contextual parameter to shift mid-sentence. But that’s not mandatory for the view laid out about CKAs. You could also think — and this is much closer to what you think — that ‘knows’ is invariant, and requires no significant possibilities, regardless of what possibilities are quantified over later in the sentence. Then you could think that in some contexts without any shifts, (2) expresses a truth.

1) All concessive knowledge attributions are genuine contradictions.

Or is that you do want to be saying this, but also saying that claims like,

2) I know that at least one strike will be thrown in the upcoming baseball season, though of course there’s a non-zero but insignificant chance there won’t be.

are not genuinely concessive knowledge attributions (because, perhaps, the relevant features of the context change during the course of the utterance)?

If you’re comfortable saying that 2 is both literally true and literally a concessive knowledge attribution, then I think I can get on board. But, if that’s the case, then I don’t see how possibilities end up inevitably impure (though if contextualism about ‘possibility’ is true, ‘possibility’-talk might be).

So, clarificatory questions: are you comfortable saying that the proposition I normally express by an utterance of 2 is true? And are you comfortable saying that there is no relevant contextual shift from the assertion of the first conjunct to the assertion of the second?