Knowledge, stakes, and closure

I've been sitting in on, and enjoying, Carrie Jenkins's grad seminar in epistemology. Today, one of our grad students, Kousaku Yui, brought up a pretty interesting suggestion in response to Jason Stanley's stakes-relative approach to knowledge. I didn't recognize the point as one that I've seen discussed before -- if there is a literature on it, I'd be very interested to see it.

The worry is this. Jason thinks that when the stakes are high, it's harder to know. But stakes aren't just a feature of an individual at a time; stakes are high for certain propositions when the truth or falsehood of those propositions make a big difference. It's possible to be such that the stakes for p are high, but the stakes for q are low. For example, it may be very important to Hannah and her wife Sarah whether the bank is open tomorrow, but not at all important to them whether it will rain tomorrow. In such a case, they would need to meet more exacting 'standards' in order to know about the bank than they would to know about the rain. That's a little bit counterintuitive, but only in the way that pragmatic encroachment is generally a little bit counterintuitive.

But here's what might be a deeper problem. Suppose someone is in a situation like the one just mentioned -- the stakes for p are high, but the stakes for q are low -- but where the subject knows that if q, then p. If so, then it's easy to know q, but hard to know p; but it looks like anyone who knows q could easily infer p. Closure plus the possibility of a case with this structure looks like they entail that the stakes-sensitive view can't be right.

Do we have to say such cases are possible? I don't see anything that forces us to, but certain cases are very naturally described in that way. Suppose Hannah and Sarah have an important bill, as per the standard high-stakes bank case; it's very important to them whether the bank will be open on Saturday. Suppose also that they have a friend Franklin who is a bank teller, and they have some small interest in whether he will be at the bank on Saturday. Here, however, the stakes are low -- nothing much hangs on whether they're correct about Franklin's location on Saturday. Assume that they have a good enough position for arbitrary strong knowledge standards for the proposition that Franklin will be at the bank only if it is open. So we have:

• p: The bank is open Saturday


• q: Franklin is at the bank Saturday


• The stakes for p are high


• The stakes for q are low


• Everyone knows that if q, then p.

If Hannah and Sarah have a middling epistemic position with respect to q, then it looks like they're in a position to know q, but not to know p. But this violates closure.

Might Jason say that in such a case, the high stakes for p force the stakes up for q as well? He might, but it seems like a pretty strange thing to say. Intuitively, it doesn't matter to them much at all whether Franklin is at work on Saturday. Their bill situation has nothing to do with Franklin. Maybe we can wrap our heads around the idea that the bill makes it harder to know that the bank is open -- but can it really make it harder to know where their friends are?

The Knowledge Norm of Practical Reasoning

For reasons exactly like the ones outlined in the previous post, these two claims are importantly distinct:

(1) If S knows p, then S can appropriately rely on p in practical reasoning.

(2) If S knows p, then p is warranted enough to justify S in phi-ing, for any phi.

I argued a couple of weeks ago that (1) is not strong enough to establish pragmatic encroachment. I suggested then that this was a problem for Fantl and McGrath; the discussion with Jeremy in the comments thread is part of what helped me to see the distinction between (1) and (2) more clearly. If their argument proceeded from (2), rather than (1), my argument doesn't apply. However, in light of the important distinction between these two claims linking knowledge and action -- I won't speak for anybody else, but this is a distinction I certainly hadn't been thinking clearly about before recently -- we should, if relying on claims like (2), proceed carefully, distinguishing arguments for (2) from arguments for (1).

I gave in my most recent post, linked at the top, an argument against a strictly weaker principle than (2). If that argument was right, then (2) is false. Let phi be an action that, for accidental reasons concerning the background environment, is only justified if S has some super-knowledge access to p. The example from that post was, let phi be the assertion that p, and let the background be such that S has promised, in a morally weighty way, not to assert p unless S knows that she knows that she is absolutely certain that p; let p be known, but the higher condition not be met. Then S is not justified in phi-ing, under the circumstances, even though she knows p, and even though, were p better warranted, she would be.

But maybe that argument went a little too quick. For maybe it's question-begging to assume, as I did, that one can know p without being in the super-epistemic position, under the circumstances described. Maybe the act of promising collapses that distinction. If so, then my argument against (2) can be resisted. Indeed, it's sort of the point of (2) that the 'standards' for knowledge raise to as high a level as one might need in any given circumstance.

But -- and here's the main point of this post -- one can retain (1) without collapsing that distinction. That's another respect in which (1) is interestingly different from (2).

Knowledge Norms and Pragmatic Encroachment

I'm thinking a bit more today about the point I made in a post yesterday about the use of intuitions about cases to evaluate knowledge norms. That point was basically that facts about whether S knows p and whether S is well-enough situated epistemically in order appropriately to X don't by themselves say anything about the knowledge norm of practical reasoning; S may know p without X's being appropriate just by virtue of p's not being a good enough reason to X. Yesterday I used this observation to rebut a certain kind of argument against knowledge norms.

I now think that, in addition to that use, this observation undercuts certain implications sometimes drawn from knowledge norms. In particular, I think it points to a lacuna in one of the central arguments of Fantl and McGrath's defense of pragmatic encroachment. I think this is a pretty fair reconstruction of their §4.3:

1. KJ: If you know that p, then p is warranted enough to justify you in phi-ing, for any phi.


2. Consider some low-stakes action, X, for which LOW has q as a sufficient reason; LOW can appropriately perform X. (Their example: Matt knows the train is a local; this justifies his boarding it.)


3. There is a possible counterpart of LOW in the same epistemic position but with higher stakes, HIGH, such that HIGH cannot appropriately perform X. (Their example: Jeremy really needs to get off at a local stop; the stakes are too high for him to risk boarding.)


4. So q is not well-enough warranted to justify X in phi-ing.


5. So HIGH does not know q.


6. So two subjects in the same epistemic position can differ with respect to knowledge of q.


7. So purism is false.

(For maximum faithfulness to F&M's broader intentions, we should understand this entire argument, including its conclusion, as being offered under the conditional assumption that fallibilism is true -- that it's possible to know that p, even though there is some epistemic possibility that not-p.)

I'm concerned with the move from (3) to (4). It has pretty much the same form as the arguments sketched above, except that it's holding different bits fixed. Jessica Brown argued from what she saw as intuitive verdicts about knowledge and appropriate action against the knowledge norm; Fantl and McGrath argue from intuitive verdicts about appropriate action and the principle of the knowledge norm against the knowledge verdict Brown found intuitive. My argument shows the flaw in both of these arguments. (In a sense, they are instances of the same argument, run in different directions.)

The move from (3) to (4) relies, like Brown's surgeon case argument, on the assumption that the knowledge of q, the non-actionability of X, and the knowledge norm are incompatible. But as I've shown, they're not. They're incompatible only on the assumption that q is, if possessed as a reason, a good enough reason to X. And there just aren't obvious intuitive verdicts about facts like these; neither are there clear theories that dictate which claims like these to accept. It's obvious enough whether S knows q, and whether S would be justified in Xing -- but whether q itself is well-enough justified to be among the reasons S has for Xing is an esoteric question on which naive intuitions are silent.

Everybody who accepts (2) and (3) has to think there's some important difference that derives from a change in the stakes that bears on actionability. But you can think this without giving up on KJ, fallibilism, or purism, if you want to. You can say that what propositions are good enough reasons to X depends in part on the stakes. That is: whether p is known is stakes-independent; so too is whether p is warranted enough to be a reason for phi-ing, for any phi. What varies by stakes is whether p, supposing that it is a reason, is by itself a good enough reason to phi. When the stakes are high, p, though still genuinely a reason, isn't a good enough reason. In lower stakes, p is a good enough reason. Insensitive knowledge; insensitive reasons for action; sensitive needs of actions for reasons.

This does look to me like a significant and substantive gap in F&M's argument. I'm generally pretty sympathetic to F&M-style views -- I'm a different sort of contextualist than those who are motivated by 'intellectualism' -- but this does look to me to be a potentially promising avenue for resisting pragmatic encroachment from F&M-style arguments.

It will not obviously help, however, with the knowledge intuitions about bank cases and the like. Lots of pragmatic encroachment people seem to be turning their emphasis away from these cases recently, in favor of broader theoretical arguments. Certainly, this seems to be one of F&M's aims. I rather suspect, though, that the pragmatic encroachment theorist may end up needing to rely on judgements about cases more than they think. (I don't know that that's a problem.)

Knowledge Norms and Intuitions about Cases

Here's a boring thought experiment that doesn't demonstrate anything.

Smith burgled the house last night; Detective Stanley is investigating the crime scene. He acquires evidence sufficient for knowledge that the burglar came in through the window, but finds very little evidence about whether it was Smith or someone else who committed the crime.

Here are two intuitive verdicts that aren't in any tension at all:

1. Stanley knows that the burglar came in through the window.


2. Stanley would need to have more evidence in order for it to be appropriate for him to arrest Smith.

Everybody can accept these obvious claims. In particular, these obvious claims are in no tension with the knowledge norm of practical reasoning, which claims that p can be an appropriate reason for action for S if and only if S knows that p. It would be an anemic objection to the knowledge norm to point out that Stanley knows that the burglar used the window, but needs more evidence in order for it to be appropriate to arrest Smith. That the burglar used the window just isn't a strong enough reason to arrest Smith, so this case doesn't tell us anything about what is and is not a reason. So it doesn't tell us anything about knowledge norms.

The moral of the story is that claims about who knows what, and about what actions are inappropriate, are in general insufficient to refute the knowledge norm of practical reasoning. (So, mutatis mutandis, for the knowledge norm of assertion.)

When you look at the case given above, this moral is really obvious. But sometimes, I think, it's neglected. Jessica Brown, for instance, argues against the knowledge norm of practical reasoning by citing this case:

A student is spending the day shadowing a surgeon. In the morning he observes her in clinic examining patient A who has a diseased left kidney. The decision is taken to remove it that afternoon. Later, the student observes the surgeon in theatre where patient A is lying anaesthetised on the operating table. The operation hasn’t started as the surgeon is consulting the patient’s notes. The student is puzzled and asks one of the nurses what’s going on:

Student: I don’t understand. Why is she looking at the patient’s records? She was in clinic with the patient this morning. Doesn’t she even know which kidney it is? Nurse: Of course, she knows which kidney it is. But, imagine what it would be like if she removed the wrong kidney. She shouldn't operate before checking the patient’s records.

We have, as before, a pair of intuitive verdicts: one attributing knowledge, and another denying appropriateness of action. Brown considers this to be a counterexample to the knowledge norm of practical reasoning, but the case of the burglar shows that this cannot not enough. Just as the burglar argument was transparently invalid, because the burglar's use of the window wouldn't be sufficient reason for arresting Smith, Brown's argument is valid only on the assumption that the disease in the left kidney would be a sufficient reason for operating without checking the charts. But Brown has given us no reason to think that is so. It's entirely open to the defender of the knowledge norm to argue that knowledge of p is sufficient for p to be a reason, but that in this case, p isn't a good enough reason for action.

This strategy is always available. I think this shows that trading in intuitions about who knows what, and who ought to do what, is not a helpful strategy for evaluating knowledge norms.

'Significant Possibilities' and Concessive Knowledge Attributions

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.

Justification and Action

Fantl and McGrath argue that the combination of the following two views is problematic:

(JJ) If you are justified in believing that p, then p is warranted enough to justify you in phi-ing, for any phi. (Quoted from p. 99)

(Moderate Externalism about Justification) Justification does not supervene on the subject's internal states. In particular, external properties like reliability and Gettierizedness can make a difference in whether one is justified in a particular belief. (Paraphrased from p. 107)

Fantl and McGrath argue that (JJ) implies that 'purist fallibilism' about justification cannot be true. Now as I wrote a little while ago, I don't really buy into the notion of purism. And to be honest, I have some problems with the notion of fallibilism, too -- I'll try to write them up sometime soon. But set all that aside. The basic idea is that, if you accept (JJ), then you think that there might be two subjects that differ only in, for example, how important p is to each subject, such that one is justified in believing that p and the other is not. I guess I think that's right, although I'm thinking of things in a way different from the way Fantl and McGrath do.

Fantl and McGrath think that people who accept this and are also moderate externalists (hereafter 'externalists') "commit themselves to counterintuitive claims about action." First, Fantl and McGrath observe the familiar point that externalists think there could be intrinsic duplicates who differ in their justification facts; externalists, therefore they face the New Evil Demon problem. That's familiar stuff, and, as Fantl and McGrath say, there are many possible responses. But they think things get worse once you also accept (JJ). They write:

Moderate externalists who accept JJ not only have to say that two subjects who differ only in how reliable they are can differ in what they are justified in believing. They also have to say that the subjects can differ in what they are justified in doing. This is counterintuitive. (108)

It's not really clear to me that this is a counterintuitive verdict. But more to the point, I just don't see why Fantl and McGrath think externalists who accept JJ are thereby committed to it. They don't, as far as I can see, explain why they think this result should obtain. It plainly doesn't follow in any direct way from externalism and JJ; externalism says that external properties can influence belief-justification facts, and JJ gives one link between belief-justification and action-justification, but it's just nowhere near strong enough to imply, as Fantl and McGrath seem to think it implies, that external properties can shift action-justification facts.

Take subject LOW who is justified in believing p, and for whom p justifies Xing. Externalists are committed to the possibility, in at least some cases, of another subject, HIGH, intrinsically identical to LOW, who is not justified in believing p. Fantl and McGrath seem to think that externalists are committed by (JJ) to think it possible, consistent with these stipulations, that HIGH is not justified in Xing, but (JJ) just doesn't get them anywhere near that commitment. Indeed, (JJ) is silent about HIGH. This principle tells you about what happens when a subject is justified in believing p; it entails nothing about what happens when a subject is not justified in believing p. For all (JJ) says, p may justify HIGH in Xing, too. (Consider this coherent principle that entails (JJ): If anyone intrinsically identical to you is justified in believing p, then p is warranted enough to justify you in phi-ing, for any phi.)

Suppose we considered a stronger, biconditional, principle:

(JJ*) If and only if you are justified in believing that p, then p is warranted enough to justify you in phi-ing, for any phi.

I don't know whether (JJ*) is plausible or not; it's strictly stronger than the principle Fantl and McGrath defend. It gets around the problem I just raised for their charge against the externalist. But even (JJ*) isn't strong enough to deliver an entailment from externalism to a difference in what actions are justified between LOW and HIGH. Externalism and (JJ*) commit one to the verdict that p cannot justify HIGH in Xing, even though it can justify LOW in Xing. That's a far cry from the stated claim that nothing justifies HIGH in Xing. And I just don't see any plausible argument that this could be the case. It may be, for all (JJ*) says, that HIGH and LOW must be  justified in performing all the same actions, but that they have divergent propositions justifying those same actions. (The plausible way to develop this line, I think, is that HIGH's reasons are a proper subset of LOW's.)

So I don't think that externalists who like (JJ), or even those who accept (JJ*), are committed to the allegedly counterintuitive claims about action that Fantl and McGrath charge.

What is purism? What are epistemic standards?

I'm reading Fantl and McGrath's new knowledge book. An important thesis of the book is that of Impurism. Impurism is defined in chapter one as the denial of Purism, given thus:

(Purism about Knowledge) For any subjects S1 and S2, if S1 and S2 are just alike in their strength of epistemic position with respect to p, then S1 and S2 are just alike in whether they are in a position to know that p.

Impurism is also defined, a bit differently, in chapter two:

(Impurism) How strong your epistemic position must be -- which purely epistemic standards you must meet -- in order for a knowledge-attributing sentence, with a fixed content in a fixed context of use, to be true of you varies with your circumstances. (35)

Impurism, Fantl and McGrath think, is a counterintuitive claim; adopting it is, according to the central argument of the book, the heavy cost that is worth paying for fallibilism. I have a hard time understanding why, if it really is so counterintuitive, people find it so. I'm also far from convinced that people do find it so. Purism and impurism are technical notions that require a fairly sophisticated background in epistemology to understand. Furthermore, they are given here in terms of the far from explicit notion of 'purely epistemic standards' and 'strength of epistemic position'. What factors influence the 'strength of one's epistemic position'? Intuitively, what one knows is of great relevance to the strength of one's epistemic position, but Fantl and McGrath cannot be using the term in a way that licenses this intuitive verdict; otherwise purism would be trivially true. They must have some notion other than the intuitive one in mind. What is it? And do we really have intuitions about it?

At points, Fantl and McGrath describe the factors that count towards strength of epistemic position as 'truth-relevant' factors. (I think that DeRose also used this gloss in his characterization of 'intellectualism', which I think is just meant to be the same thing as F&M's 'purism'.)  This is meant to rule in facts about the actual or probable truth of the indicated belief, and to rule out facts like what is salient to the subject or how much is at stake for her. That's some progress -- but is it clear enough? It is, I think, meant to be consistent with purism that whether a subject knows depends on whether she is proceeding responsibly in forming her belief. (It'd better be, because that's a traditional view, and purism is supposed to include that tradition.) Is this factor 'truth-relevant'? I guess it's supposed to be. We could rely on a principle like this: if a belief is responsibly formed, then it is likely to be true.

Similarly, it's meant to be consistent with purism that whether a subject knows depends on features of her environment -- even those that don't affect the truth value of her belief. For example, whether a subject knows that there is a barn in front of her depends in part on whether there are barn façades nearby. Presumably, this is roped in under truth-relevance by the effect of such circumstances on the reliability of (a certain specification of) the subject's belief-forming process, which is correlated with truth.

But if a connection that weak is sufficient to count responsibility and environmental features as truth-relevant, then it's hard to see why it shouldn't also count in knowledge, and thus, if the views developed by e.g. Fantl & McGrath, Stanley, etc. are right, make practical situations truth-relevant. Your stakes, like your environment, play a role in determining what you know, and knowledge, like responsibility and reliability, is strongly connected to truth. In what sense is such a stakes-sensitive view 'impurist'? In what sense are stakes disconnected from 'purely epistemic standards'?

I don't even understand what purism amounts to, if it's not the triviality that this line of reasoning suggests. And I certainly don't have intuitions about purism. Therefore, I have a hard time seeing what all the fuss is about. (Interesting sidenote: it looks like common ground among at least many SSI types and at least many contextualists that one of the motivations for contextualism is to maintain purism. I'm a contextualist who has never had anything like that motivation; indeed, it looks pretty incomprehensible to me.)

Contextualism, Intellectualism, and Ignorant Third Persons

It's a little bit natural to think that 'knows' contextualism and the shifty kind of invariantist that's sometimes called an 'SSI theorist' or an 'IRI theorist' come to a bit of an intuitive draw considering two kinds of third-person knowledge attributions. High Howie has whatever features you think makes it harder to know, or makes 'know' express a stronger relation: he's thinking about skeptical possibilities, it's really important to him whether p, or whatever. Low Louie is just the opposite: to Louie, p's no big deal, he's not worried about it, whatever. High Howie says things like "I don't know that p," while Low Louie says things like "I know that p," and both utterances look pretty good, even though in some sense Howie and Louie look to be in identical epistemic situations -- they have the same evidence, or something like that.

(Now I happen to think that it's not at all clear how to make sense of that last stipulation. This basically amounts to a worry whether there is any correct generalization characterizing the difference between the shifty SSI-like views from 'classical invariantism'. But I'm setting that aside for now, assuming, as is usual in this discussion, that the sense in which Howie and Louie are in the same 'epistemic position' is tractable -- and does not at least really trivially entail that they're alike with respect to knowledge. I'll here use 'epistemic position' technically to mean the stuff that traditional invariantists affirm, but shifty people deny, comprise a supervenience base for knowledge.)

DeRose on 'knowledge' norm of assertion

In his 2002 paper "Assertion, Knowledge, and Context," Keith DeRose gave an argument for contextualism about 'knows' that took basically this form: knowledge is the norm of assertion; assertability varies according to context; therefore, knowledge varies according to context.

This was a pretty confused argument -- though of course this is much clearer in retrospect, with the advantage of years of engagement with SSI. The problem is that contextualism is a thesis about the word 'knows', not about knowledge, while 'knowledge is the norm of assertion' seems like it must be a thesis about knowledge, not about English. In fact, something like a knowledge norm for assertion, combined with the observation that what you're allowed to assert depends on your situation, provides a pretty good argument for SSI; I take it to be exactly parallel to the main argument for SSI that Stanley and Fantl and McGrath give.

In chapter 3 of his new book The Case for Contextualism, DeRose essentially reproduces the content of that 2002 paper, but he does add about two new pages of material designed to correct this aspect of the original. Now, in contrast to earlier, he recognizes the need to clarify the statement of the knowledge norm of assertion, if it is to be understood in contextualist terms. He gives us:

The Relativized Knowledge Account of Assertion (KAA-R): A speaker, S, is well-enough positioned with respect to p to be able to properly assert that p if and only if S knows that p according to the standards for knowledge that are in place as S makes her assertion. (99)

Allegedly inconsistent knowledge principles

Matt Weiner argues that 'our use of the word "know" is best captured by' an inconsistent set of inference rules. His setup strikes me as strange. He writes:

These are the Knowledge Principles:

(Disquotational Principle)  An utterance of “S knows that p” at time t is true iff at time t S knows-tenseless that p.
(Practical Environment Principle)  S’s evidence concerning p is good enough for knowledge iff S’s evidence for p is good enough to make it epistemically rational for her to act on the assumption that p.
(Parity of Evidence Principle) If the evidence concerning p for S and T is the same, then S’s evidence is good enough for knowledge iff T’s evidence is good enough for knowledge.

The Knowledge Principles are inconsistent, given only the truism that different people can have different practical stakes. Take a Bank Case (DeRose 1992), in which Hanna and Leila each have the same rather good evidence that the bank is open Saturday, but acting on a mistaken belief would harm Hannah much more than Leila.  Hannah is in a high-stakes context, Leila in a low-stakes context.  The Practical Environment Principle, which entails that Leila knows that the bank is open and Hannah does not, here generates an inconsistency with the Parity of Evidence Principle, which entails that Leila knows if and only if Hannah does.

Two things strike me as really strange about this claim, even setting aside the question of whether these principles are plausibly constitutive of the meaning of 'knows'.