Here's some chitchat about one of his cases, since I can't help it. There are lots of big problems with Hazlett's account, not least his use of Grice.
(1) Alice knew that stealing is a crime.
(2) Alice knew that stealing isn't a crime.
(3) Alice didn't know that stealing is a crime.
(4) Alice wasn't aware that stealing is a crime.
Hazlett points out that both (1) and (3) implicate (either imply or entail) that stealing is a crime, supporting, he thinks, the case that "stealing is a crime" is a (conversational, i.e., non-conventional?) implicature of (1). But what it supports, if anything, is that "stealing is a crime" is a presupposition of (1) and (3). Presupposition is not the same thing as conversational implicature. (On Strawson's account, roughly, where "The present king of France is bald" and "The present king of France is not bald" both presuppose that there is a present king of France.)
Every philosophy neophyte learns to distinguish (2) from (3), and that (2) is not the negation of (1), but at the same time learns that (3) is ambiguous between (2) and (4). The version of (3) that aligns with (2) could be expanded to
(3') Alice didn't know that stealing is a crime because it is isn't.
Now we're in the territory of something else that might look like conversational (rather than conventional) implicature, because this looks like
cancellation, just as one might say
(5) I haven't stopped beating my spouse, because I never started beating my spouse.
But similarly, we might say
(6) The present king of France is not bald, because there is no present king of France.
And that's Russell's account, which disambiguates the scope as
(7) It is not the case that the present king of France is bald, because there is no present king of France.
But Russell's account is not based on conversational, non-conventional implicature, but simply entailment. On Russell's account,
(8) The present king of France is bald.
has the logical form
(9) There is a unique entity such that it is the present king of France, and that entity is bald.
Can we apply a similar analysis to Alice's knowledge of the criminality of theft? On the one hand, (3) could have the form:
(10) Stealing is a crime but Alice didn't know that.
or
(11) Stealing is not a crime, so Alice could not know that it is, and therefore did not know that it is.
((Or, "what's more, she didn't know," etc. There are options here.))
(10) makes a simple claim about Alice's epistemic state. (11) makes a claim about what Alice's epistemic state could or could not possibly be, and then infers what it was. Both make simple claims about the criminality of theft, which allow us to negate them by negating that claim, without reference to Alice, as with Russell's analysis.
(10) is noncommittal on whether knowledge entails truth, as it simply states two facts, one about stealing and one about Alice; (11) is not only consistent with a claim that knowledge entails truth, but relies on it.
Where does that leave the question of conversational implicature?
Grice claims that conversational implicature is "triggered" by an apparent violation of a maxim of conversation, which suggests that what you mean by uttering p must be different from the plain meaning of p, in order to preserve the assumption that you are cooperative (and not after all violating a maxim).
It does seem that the most natural reading of
(3) Alice didn't know that stealing is a crime.
is
(10) Stealing is a crime but Alice didn't know that.
rather than (11), and if you mean (11), you need to say so explicitly. Why should that be? And is this indication of how you expect (3) to be understood a case of implicature?
One reason (10) might be the more natural reading is because we expect the clause governed by "know" to be true or to be asserted to be true, so it is surprising bordering on misuse to place after "know" a proposition you assert to be false, just because you intend also to deny that this is a case of knowledge, precisely because its object is false. To speak in such a way would be a rhetorical flourish. ("I know no such thing, because it is not so!")
There may be other points in favor of (10): it is simpler, and more to the point, suggesting compliance with other maxims to be relevant and concise. But what we're looking for, as evidence of implicature, is apparent maxim violation, not compliance.
I'm also tempted to wonder whether (10) is more natural because it is "common knowledge" that stealing is a crime, but that's not (to my memory) part of Grice's account.
I haven't resolved the implicature issue but I still see nothing to support "knows" not being factive.
+++
To clarify: the presupposition analysis relies on a pair of entailments, not implicature; neither of those entails that Alice knows something that is not the case.
(10) says stealing is wrong and she doesn't know it; (11) is perhaps most simply taken as the negation of (2):
(12) Alice did not know that stealing is not a crime.
But then we have ambiguity again, so that's no help, hence (11).
******
Actually there's no need to stress over (11) and its relation to (10). (Or about implicature, since his usage has other issues anyway.)
What Hazlett is interested in is the straightforward (10), because then we have both
Kp → p and
~Kp → p. That's the point of his argument. That's supposed to undercut the unique entailment from
Kp to
p. But that's because he gets there by (10), rather than (11), which doesn't even lead there.
And (10) interprets "Alice didn't know that stealing is a crime" as "Stealing is a crime, and Alice didn't know that," which of course entails that stealing is a crime.
The issue here is how we justify the (10) interpretation of (3). We would not treat all content this way; we would not, for instance, render
(B1) Harry thinks today is Sunday.
as
(B2) Today is Sunday and Harry thinks that.
Why not?
The simplest answer is that "believes" is not factive, but "knows" is. It allows us to rewrite
(K1)
S knows that
p
as
(K2)
p and
S knows that.
Another Russellian move would be to look at the scope: we're taking
(3) Alice didn't know that stealing is a crime.
as
(3') It is not the case that: Alice knew that stealing is a crime.
That means we have all of (1) embedded, and it's form should come out that same as before, without negation in front of it. If that's as above, we have a negated conjunction, and our ambiguity is a matter of which conjunct is negated.
(3'') Not both (i) stealing is a crime, and (ii) Alice knew that.
And, again, that analysis only comes off if we have the rewrite rule (K2).