Replies to critics

Responding to the critical comments by the symposiasts Julien Dutant, Niccolò Rossi, Martin Smith, Daniel Waxman, and Yiwen Zhan, I further elaborate, refine, and defend the account of epistemic justification I advanced in my book Justification as Ignorance (Oxford: Oxford University Press, 2021). Central issues tackled here include logical omniscience and non-normal epistemic logics, the luminosity of epistemic justification, methods for telling what one is in no position to know, and the relation between doxastic and propositional justification.


Introduction
I am very grateful to Nikolaj Pedersen, editor of the Asian Journal of Philosophy, for organizing this symposium, as one of the journal's first, and to Julien Dutant, Niccolò Rossi, Martin Smith, Daniel Waxman, and Yiwen Zhan for their excellent and challenging comments that made me rethink central aspects of the account of justification presented in Justification as Ignorance (Rosenkranz, 2021;henceforth JAI) and, at the same time, gave me the opportunity to clarify or amend others. Lucky me. It is very rare to have such a formidable cast of thinkers dedicate so much time and energy in order to digest and criticize one's ideas. In what follows, I will do my best to answer their challenges.
Although I have conceived of my replies as self-standing pieces, they clearly hang together, which is why they frequently include cross-references. To the interested reader familiar with JAI, a brief roadmap might be helpful. In my reply to Rossi, I fur-ther clarify my misgivings about normal epistemic logics underlying much of chapters two, four, and five of JAI. My reply to Smith elaborates, clarifies, and refines the argument for the luminosity of propositional justification, on my construal of it, that I gave in chapter four of JAI. It enables me to stave off Smith's own anti-luminosity challenge, or so I argue. In different ways, Waxman, Dutant, and Zhan all challenge the pair of theses, put forward in chapter six of JAI, that being in no position to rule out that one is in a position to know p suffices for having propositional pro toto justification for p and that being in no position to rule out that one knows p suffices for having doxastic pro toto justification for p. Addressing Waxman's alleged counterexample to the first, I further refine and amend my conception of what it is to do one's best in order to decide a given question-a notion already at work in my reply to Smith. Dutant primarily targets the second thesis-more specifically its implication that one is always in a position to know either that one doesn't know p or that one doesn't know p's negation. In response, I clarify what I take to be the evidential force of misleading impressions that one knows the answer to a given question, in the process of doing one's best to find out that one actually doesn't. Like Waxman, Zhan presents what he takes to be a counterexample to the first of the aforementioned theses which is meant to improve upon an example I discuss in chapter six of JAI. My reply gives me the opportunity to say more about what I take to be the connection between propositional and doxastic justification.

Reply to Niccolò Rossi
In chapter five of JAI, I devised two distinct bimodal systems for both knowledge (k) and being in a position to know (K), one I called 'idealized', the other 'realistic' (JAI: 80-101). The realistic system is contained in the idealized system. Both systems are non-normal logics, rejecting the Kripke axiom for either modality, alongside a number of other principles that I had argued in earlier chapters were objectionable on general epistemological grounds. Unlike the realistic system, the idealized system nonetheless still contains implausibly strong assumptions-which is why I settled for the realistic system in the end. Instead of establishing the soundness of the realistic system directly, in terms of a semantics geared to its specific needs that allows for models that are not, at the same time, models of the idealized system, in JAI I did so indirectly by devising models for the idealized system. In this context, I also provided a class of models for the idealized system invalidating many of the unwanted principles earlier identified. However, as Niccolò Rossi observes, the models in that class still failed to invalidate either of the following: According to the first, if φ is a theorem, so is ˹kφ˺, implying that it is correct to partly characterize what the targeted epistemic agent knows by using any given logical truth. According to the second, whatever such an agent is in a position to know, they do know.
In his insightful contribution to the symposium, Rossi devises a better class of models for the idealized system that invalidate both RN k and K-k in addition to those invalidated by the ones I devised (Rossi, 2022). This is a welcome improvement. Invalidating K-k is of particular interest, as this principle would lead to a collapse of the two modalities, given only that its converse is valid in both systems. Failure to provide models that invalidate K-k is a clear shortcoming of JAI (even if such models can easily be given). According to Rossi's assessment, showing that the idealized system has models invalidating RN k is equally important, since RN k is one ingredient of the so-called thesis of logical omniscience, involving, as Rossi contends, 'a strong idealization, requiring that one knows any logical truth, even the most convoluted' (Rossi, 2022: 3).
Since the realistic system is strictly weaker than the idealized system, Rossi's result shows that the realistic system, too, has models invalidating all those principles at once. To invalidate those principles all at once has the benefit that, in Rossi's words, 'we don't need to assume one [unwanted principle] to invalidate another'. This is particularly important with respect to the purpose of invalidating K-k alongside other unwanted principles (and it is not as easy to come up with such models as it is to come up with models that invalidate K-k in isolation).
However, it transpires that the idealized system still has unwanted features that ultimately make the attempt to show that it has models invalidating RN k somewhat moot. To begin with, note that the following monotonicity rule is valid in the idealized system: As Rossi observes, with RM k in place, models invalidating RN k must involve worlds in which nothing is known (Rossi, 2022: 4). RM k straightforwardly entails the equivalence rule The logic for 'k' is intended as a logic of knowledge ascriptions. As it turns out, the validity of RE k already requires a conception of such ascriptions that, at the same time, assuages any worries about RN k .
To see this, compare two basic kinds of conceptions of knowledge ascriptions (JAI: 20-27). According to the first, the prejacents of such ascriptions give some indication, however indirectly (e.g. via translation), of the distinctive way the thing known is articulated in the mind of the subject to whom knowledge is being ascribed-a distinctive guise of the thing known. I say 'articulated in the subject's mind' rather than 'present to the subject's mind', intending a broad, non-Cartesian conception of the mind that encompasses inferential dispositions, as the knowledge ascribed may be implicit. 1 According to the second kind of conception, the prejacents of knowledge ascriptions give no such indication, even if it is agreed that, whenever such an ascription is true, there must be some way or other in which the thing known is articulated in the ascribee's mind-i.e. some guise of the thing known under which the ascribee knows that thing. In the semantics Rossi and I respectively provide for the idealized system, 'the thing known' is a coarse-grained proposition corresponding to a subset of possible worlds.
Plausibly, the thought that RN k involves too strong an idealization-one that requires subjects to know 'any logical truth, even the most convoluted' (Rossi, 2022: 3)-presupposes a conception of the former kind. For, plausibly, we shouldn't be worried by the idea that subjects know the universal proposition in one of its guises and, if knowledge ascriptions are silent on what the relevant guise might be, it doesn't follow from the validity of RN k that subjects know the universal proposition in all its guises, corresponding to all logical truths expressing that proposition.
However, a conception of the first kind casts equal doubt on the validity of RE k . For, if knowledge ascriptions are understood in accordance with such a conception, RE k , too, imposes overly strong demands on epistemic subjects, viz. that, whenever they know a given coarse-grained proposition under the guise indicated by one prejacent, they know it under the guise indicated by any other, logically equivalent prejacent. But, then, the idealized system ultimately presupposes a conception of the second kind-in which event the worries about RN k should likewise dissipate.
What here holds for knowledge ascriptions holds for ascriptions of epistemic states in general, including those that specify what subjects are in a position to know. The idealized system validates the luminosity thesis L implies that, in any world of any of the idealized system's models, the subject is in a position to know at least one proposition. The idealized system likewise validates Taken together, these principles allow us to derive If we presupposed a conception of truths of the form ˹Kφ˺ according to which the prejacent φ indicated the guise under which the subject in question is in a position to know the coarse-grained proposition assigned to φ as its semantic value, then RN K would likewise be problematic and almost as problematic as RN k . Thus, we may ask: if it is important to have a logical system with models invalidating RN k , 1 See, however, principle (1) in my reply to Dutant (2022) below.
because of legitimate worries about logical omniscience, why isn't it just as important to have a logical system with models invalidating RN K ? 2 All this goes to show that, on the first kind of conception of ascriptions of epistemic states, the idealized system proves inadequate. 3 Once a conception of the second kind is assumed, however, there is no longer any sense in which RN k , RE k , RN K , and RE K , are expressive of any controversial thesis of logical omniscience. Arguably, having the resources to express such a controversial thesis, if only to reject it, is itself a desideratum which, on a conception of the second kind, we simply cannot fulfil. Although models for the idealized system are models for the realistic system, by design they never invalidate all the principles that legitimate concerns about logical omniscience might identify as objectionable. Rossi is, of course, fully aware of this. However, the desire to find models for both systems that invalidate RN k , if driven by such concerns, is ultimately misplaced, because such models cannot but come at the cost of validating RN K , RE k , and RE K .
Things are in fact worse for the idealized system. For, even on a conception of the second kind, RM k and RM K are wildly implausible. Trivially, contingent coarsegrained propositions entail other, equally contingent ones. There is no guarantee that, if a subject knows, or is in a position to know, some such proposition under one guise or other, she knows, or is in a position to know, all the contingent propositions it entails under some such guise. For, if the entailing proposition is distinct from the proposition entailed, any guise of the former will differ from any guise of the latter and, while it may be deemed uncontroversial to think that we always know, or are in a position to know, the universal proposition in some of its guises, the same cannot be said about all the contingent propositions entailed by something we know or are in a position to know (JAI: 26, 31-32).
The lesson is that, on either conception of the ascriptions of epistemic states, the idealized system proves inadequate. The difficult task that lies ahead, at least for those sympathetic to the overall project, is to devise models for the realistic system that are not, at the same time, models for the idealized system. The task is difficult inter alia because the realistic system still contains principles that are instances of RM k and RM K and also contains L, which latter, as we've seen, ensures that subjects are always in a position to know something-indeed infinitely many things.

Reply to Martin Smith
One of the key theses defended, and exploited, in JAI was

L
If one is in no position to know that one is in no position to know p, one is in a position to know that this is so.
If we let 'K' be short for 'one is in a position to know' and conceive of cases as triples of worlds, subjects, and times, we can rephrase L as follows: for any case γ, if ˹¬K¬Kp˺ holds in γ, so does ˹K¬K¬Kp˺-where, throughout, 'one' refers to the subject at the centre of γ.
L implies that, for any p, ˹¬K¬Kp˺ encodes a luminous condition, i.e. a condition such that, if it obtains in one's situation, one is in a position to know that it obtains (Williamson, 2000: 13). As such, it is threatened by Timothy Williamson's argument to the conclusion that no non-trivial condition is ever luminous (Williamson, 2000: 126-29; see also Srinivasan, 2013). A condition will be non-trivial, if we can construct a series of cases running from cases in which the condition is clearly met to cases in which it clearly isn't met. There is no denying that, for any given p, we can construct such a series for ˹¬K¬Kp˺ (for a nice example, see Smith, 2022: 5). Accordingly, in JAI, I undertook to show that L can after all escape the threat posed by Williamson's argument. In doing so, I relied on a number of premises that I sought to independently justify.
In his exceptionally clear and illuminating comments, Martin Smith reconstructs and critically assesses my attempt to insulate L from this anti-luminosity challenge and then proceeds to formulate a challenge of his own (Smith, 2022). While he generously concedes that I succeed in defusing the first challenge, he argues that I didn't provide any resources to defuse the second. Below I rehearse, and further clarify, my defence of L and argue that it, or a natural extension of it, after all allows me to answer Smith's challenge.
In his reconstruction of my defence of L, Smith identifies the crucial premises on which I relied as PK 1 If ˹Kp˺ holds and one has done everything one is in a position to do in order to decide p, one knows p.
PK 2 If one does everything one is in a position to do in order to decide p, one does everything one is in a position to do in order to decide ˹¬p˺, and conversely. 4 PK 3 If one does everything one is in a position to do in order to decide ˹Kp˺, one does everything one is in a position to do in order to decide p.
While PK 2 is trivial, at least given a natural reading of 'whether' and classical logic, PK 3 is not. Smith is nonetheless willing to accept PK 3 for the sake of argument. Smith assumes that any characterization of the notion of being in a position to know will sanction PK 1 . Although closely related, these are not, in fact, the premises I ultimately used in arguing for L. With hindsight, it is best to first explain why. Smith writes that PK 1 'already gives us the result that being in a position to know is factive' (Smith, 2022: 2). This is incorrect. PK 1 only implies that being in a position to know and doing everything that one is in a position to do in order to decide p is factive. But, one may be in a position to know p without doing everything that one is in a position to do in order to decide p. PK 1 doesn't tell us anything about such cases. In this regard, it is better to replace PK 1 by: PK 1 ′ If ˹Kp˺ holds, then, were one to do everything one is actually in a position to do in order to decide p, one would know p.
Unlike PK 1 , PK 1 ′ has implications for cases in which one is in a position to know p without knowing p, viz. cases in which one does not, but could do everything that one is in a position to do in order to decide p. However, ˹Kp˺ may hold, even if it is impossible to do everything that one is actually in a position to do in order to decide p: implementation of some of the procedures that one is in a position to successfully implement to this end might preclude implementation of others (JAI: 41). This is why it is better to replace PK 1 ′ by PK 1 ″ If ˹Kp˺ holds, then, were one to do the best one is actually in a position to do in order to decide p, one would know p.
With hindsight, it is more convenient to rephrase PK 1 ″ in terms of cases as follows: where B(γ, p) is the set of possible cases, close to γ and centred on the same subject as γ, in which the subject of γ has done the best they are in γ in a position to do in order to decide p (and has done nothing they are not in γ in a position to do; see JAI: 40). 5 Correspondingly, PK 2 and PK 3 should be replaced by Plausibly, it's always possible to do the best that one is in a position to do in order to decide a given question-even if that may amount to doing very little-where some such possibility is furthermore close: if any possibility in which one Fs is remote, one isn't in a position to F. Thus, B(γ, p) ≠ ∅. Accordingly, P1 implies that, if ˹Kp˺ holds in γ, it is a close possibility that one knows p after having done the best that one is in γ in a position to do in order to decide p. To the extent that doing something in order to decide a given proposition leaves the truth-value of that proposition unaffected (Williamson, 2009), P1 will then also ensure the factivity of 'K'. These are not the only premises I need, however, and I will invoke two further principles below. But before we get to that, let me make two observations: • Since knowledge implies belief, P1 in turn implies that, if ˹Kp˺ holds in γ, one wouldn't do the best that one is in γ in a position to do in order to decide p, were one not to form a belief in p in the process.
Accordingly, if one is in a position to know p, one is in a position to believe p in the course of doing the best that one is in a position to do in order to decide p (JAI: 39).
• P1 does not imply the conditional that, if ˹Kp˺ holds in case γ, then, for any case γ′ centred on the same world and subject as γ and succeeding γ by very little, if B(γ, p) ≠ ∅, then, for some δ ∈ B(γ, p), δ ∈ B(γ′, p).
To use an analogy for illustration, it may be a close possibility in γ that I jump through the loop, and a close possibility in γ′ that I jump through the loop, where γ′ succeeds γ by very little, while it is not a close possibility in γ′ that I jump through the loop having jumped through the loop a little earlier. If so, then, even if I do in fact jump through the loop in γ, in γ′ it may still be a close possibility that I jump through the loop, albeit one in which I haven't jumped through the loop a little earlier (something, we assume, I might easily not have done).
Let us now assume a series of cases, centred on the same subject and world, running from cases in which ˹K¬Kp˺ clearly holds to cases in which ˹¬K¬Kp˺ clearly holds, where for any pair of adjacent cases in the series, their temporal distance can be arbitrarily small (this is the assumption Smith labels 'AL 1 '). Suppose that I am that subject, that α is the last case in the series in which ˹K¬Kp˺ holds, that α′ is the first case in the series in which ˹¬K¬Kp˺ holds, and that like any other pair of adjacent cases in the series, α and α′ are exactly one nanosecond apart.
In his reconstruction of Williamson's anti-luminosity argument, Smith assumes, as Williamson (2000), Srinivasan (2013), and others before him, that throughout the envisaged series of cases, I may in fact, in each case, do the best that I am, in that case, in a position to do in order to decide ˹¬K¬Kp˺. This implies that, in α′, it is an easy possibility that I do the best that I am in α′ in a position to do in order to decide ˹¬K¬Kp˺ after having done the best that I was in α in a position to do in this regard. Given that 'K' does not agglomerate over conjunction (Heylen, 2016), this assumption, though hardly ever questioned, should immediately raise suspicion.
The assumption is indeed far from innocent. Ex hypothesi, ˹K¬Kp˺ holds in α. Suppose that, for purely psychological reasons, whenever one knows a given proposition q, one couldn't easily believe, a nanosecond later, that one is in no position to know q. Call this thesis 'T'. Given P1, T implies that, if, in α, I have done the best that I am in α in a position to do in order to decide ˹¬Kp˺, I don't believe ˹¬K¬Kp˺ in α′ and hence do not know ˹¬K¬Kp˺ in α′. This alone doesn't show that ˹K¬K¬Kp˺ doesn't hold in α′. It at most implies that, if I do my best in α and ˹K¬K¬Kp˺ holds in α′, then, in α′, I don't do the best that I am in α′ in a position to do in order to decide ˹¬K¬Kp˺. For, there may then still be a case β′ close to α′ such that, in β′, I do the best that I am in α′ in a position to do in order to decide ˹¬K¬Kp˺ and come to know ˹¬K¬Kp˺ as a result-even if, given T, any such case would have to be one in which I haven't done the best that I was in a position to do to this end a nanosecond earlier. 7 To demand that, if ˹K¬K¬Kp˺ holds in α′, as L predicts, there should be some such β′ in which I have also done the best that I was in a position to do to this end a nanosecond earlier is to pitch T against L. Although I see in the end no compelling reason to accept T, 8 I don't see any compelling reason either why I should be committed to Smith's assumption which implies a conflict between T and L. 9 If ˹K¬K¬Kp˺ holds in α′, then, as we've observed, given P1, for any case γ′ close to α′ in which I do not believe (and hence do not know) ˹¬K¬Kp˺, I will not, in γ′, have done the best that I am in α′ in a position to do in order to decide ˹¬K¬Kp˺. Forming a belief in ˹¬K¬Kp˺ will then be a necessary condition on doing the best that I am in α′ in a position to do in order to decide ˹¬K¬Kp˺. The same is true, mutatis mutandis, of any case γ close to α in which I do not believe (and hence do not know) ˹¬Kp˺: in such a case γ, I will likewise not have done the best that I am in α in a position to do in order to decide ˹¬Kp˺, as ˹K¬Kp˺ holds in α. Forming a belief in ˹¬Kp˺ is a necessary condition on doing the best that I am in α in a position to do in order to decide ˹¬Kp˺.
Insofar as I am rational, as suitably idealized subjects are supposed to be, I will not believe ˹¬Kp˺ and believe ˹¬K¬Kp˺ on the same occasion. Let us follow Smith and define, for any case γ in the particular series of cases under consideration: δ is a near duplicate of γ iff (i) ˹¬K¬Kp˺ holds in γ iff it holds in δ, (ii) γ ∈ B(γ, ˹¬K¬Kp˺) iff δ ∈ B(δ, ˹¬K¬Kp˺), and (iii) γ and δ 'are very similar in all other respects' (Smith, 2022: 3-4). 10 Then, on assumption that, for some β′ ∈ B(α′, ˹¬K¬Kp˺), I believe ˹¬K¬Kp˺ in β′, the foregoing already falsifies the following anti-luminosity premise (corresponding to Smith's AL 2 ): 7 While the cases in the series immediately preceding or succeeding α′ are among the cases close to α′, the latter also include cases that are centred on a world distinct from that of α′. 8 Below, I'll argue that, if, in α′, I do the best that I am in α′ in a position to do in order to decide ˹¬K¬Kp˺ and believe ˹¬K¬Kp˺ as a result, α′ will be a case in which I believe ˹¬K¬Kp˺ inter alia because I believe p and, if, in α, I do the best that I am in α in a position to do in order to decide ˹¬K¬Kp˺ and believe ˹¬Kp˺ as a result, α will be a case in which I believe ˹¬Kp˺ inter alia because I don't believe p.
The assumption that I do both doesn't seem to be psychologically any more problematic than the idea that, while I continue to do the best that I am in a position to do in order to decide p, I at some point stop believing p as a result. 9 This may call for revision of some of the things I say about opportunities to know in JAI. It no longer strikes me as helpful to demand that, if I do not know p, but am in a position to know p, I have the opportunity to go on and do what I am so far merely in a position to do. Am I in a position to know that there is, for the flicker of a moment, a gnat on the rim of my glass, if I miss it because (a) I happen to blink, (b) I happen to have yawned a few seconds earlier so as to make my eyes water, or (c) I happen not to have started to pay attention in time? I think we would still want to say that the gnat's sitting on the rim of my glass in plain view is the kind of condition, however fleeting, that I am in a position to ascertain (borderline cases aside). Thanks to Julien Dutant for discussion. 10 All near duplicates of γ will be understood to be cases close to γ, but not vice versa.
For, given P2 and P3, any near duplicate of any β ∈ B(α, ˹¬K¬Kp˺) will be a case in which I believe ˹¬Kp˺ and so, being rational, don't believe ˹¬K¬Kp˺. For any q, if γ is a case near the borderline between q's being true and q's being false, then, in γ, one is in a position to believe q-and so there is a case γ′ close to γ in which one does believe q. Ex hypothesi, case α′ is near the borderline between ˹¬K¬Kp˺'s being true and ˹¬K¬Kp˺'s being false. Thus, it is here unproblematic to assume that I am, in α′, indeed in a position to believe ˹¬K¬Kp˺. As we've observed, in order for ˹K¬K¬Kp˺ to hold in α′, believing ˹¬K¬Kp˺ must be a necessary condition on doing the best that one is in α′ in a position to do in order to decide ˹¬K¬Kp˺. The question accordingly is whether, in α′, I am, furthermore, in a position to believe ˹¬K¬Kp˺ upon doing the best that I am in α′ in a position to do in order to decide ˹¬K¬Kp˺-and so whether there is a β′ ∈ B(α′, ˹¬K¬Kp˺) such that I believe ˹¬K¬Kp˺ in β′. Insisting that I am in no such position would seem to beg the question against the thesis, licensed by L, that ˹K¬K¬Kp˺ holds in α′.
But we can here say something more powerful in support of both this idea and L. Thus, consider the following condition:

C
For any γ and any δ ∈ B(γ, p), if p is false in γ, then, in δ, one doesn't believe p.
It's easy to check that, given P1 to P3 and suitable rationality assumptions, C is satisfied for the following choices of p: ˹¬Kp˺, ˹¬K¬Kp˺, etc. This already explains why I have no trouble accepting the following anti-luminosity premise (corresponding to Smith's AL 3 ): A3 For any β′ ∈ B(α′, ˹¬K¬Kp˺), if I know ˹¬K¬Kp˺ in β′, then, for any β ∈ B(α, ˹¬K¬Kp˺), there is no near duplicate of β in which I falsely believe ˹¬K¬Kp˺.
Against this backdrop, I take the following principle to be defensible: P4 If, for some δ ∈ B(γ, ˹¬Kp˺), one believes ˹¬Kp˺ in δ, then ˹K¬Kp˺ holds in γ. 11 P4 implies that, if one is in α′ in a position to believe ˹¬K¬Kp˺ upon doing the best that one is in α′ in a position to do in order to decide ˹¬K¬Kp˺, then ˹K¬K¬Kp˺ holds in α′. Similarly, P4 implies that, if ˹¬K¬Kp˺ holds in α′, as ex hypothesi it does, one is not in α′ in a position to believe ˹¬Kp˺ upon doing the best that one is in α′ in a position to do in order to decide ˹¬Kp˺. Given P2 and P3, this should render it sufficiently plausible that, if ˹¬K¬Kp˺ holds in α′, one is in α′ in a position to believe ˹¬K¬Kp˺ upon doing the best that one is in α′ in a position to do in order to decide ˹¬K¬Kp˺ (JAI: 72-74). By P4, it then follows that ˹K¬K¬Kp˺ holds in α′-just as L predicts. Note that the same cannot be said about ˹¬Kp˺. Even with P1 to P4 in place, we cannot derive that, if one is in γ in a position to believe p upon doing the best that one is in γ in a position to do in order to decide p, then ˹Kp˺ holds in γ. For, C only holds for certain choices of p and, if, throughout, we replaced '¬Kp' by 'p' in P4, the result wouldn't be defensible. For the same reason, we cannot derive the implausible conclusion that, if ˹¬Kp˺ holds in γ, one is not in γ in a position to believe p upon doing the best that one is in γ in a position to do in order to decide p (JAI: 73). 12 Note also that nothing yet implies that, for any case γ, if ˹¬K¬Kp˺ holds in γ and one believes ˹¬K¬Kp˺ upon doing the best that one is in γ in a position to do in order to decide ˹¬K¬Kp˺, one is rationally compelled to believe ˹¬K¬Kp˺. For all that has so far been said, such beliefs aren't rationally compelling in borderline cases.
Smith suggests that this type of defence of L assumes 'that there is a kind of constitutive connection between the obtaining of [ (Smith, 2022: 6-7). I find this talk slightly misleading (even if we here replace 'everything' by 'the best'). For, what one ends up believing, upon doing one's best, depends on what one's epistemic situation affords, rather than the other way around. Thus, what I'd rather want to say is that, if ˹K¬Kp˺ holds and I do the best that I am thereby in a position to do in order to decide ˹¬K¬Kp˺, then these facts, in conjunction with facts about my rationality, prevent that I believe ˹¬K¬Kp˺ as a result. Accordingly, if I rationally believe ˹¬K¬Kp˺, upon doing the best that I am in a position to do in order to decide ˹¬K¬Kp˺, that will be so partly because ˹¬K¬Kp˺ holds.
In formulating his own challenge to L, Smith invokes a principle-which he labels 'AL 4 '-whose presently relevant implication is this: A4′ If I do not believe ˹¬K¬Kp˺ in α, there is a near duplicate of α′ in which I do not believe ˹¬K¬Kp˺.
As noted earlier, Smith assumes that both α ∈ B(α, ˹¬K¬Kp˺) and α′ ∈ B(α′, ˹¬K¬Kp˺). In order to free ourselves from this unnecessarily controversial assumption, let us replace A4′ by A4 For any β ∈ B(α, ˹¬K¬Kp˺), if I do not believe ˹¬K¬Kp˺ in β, then, for some β′ ∈ B(α′, ˹¬K¬Kp˺), there is a near duplicate of β′ in which I do not believe ˹¬K¬Kp˺ either.
If acceptable, this principle would indeed spell trouble for L. The question is why we should accept it. Let's assume, as we may, that I do not believe ˹¬K¬Kp˺ in some β ∈ B(α, ˹¬K¬Kp˺). In the light of this and our earlier observations, A4 would simply beg the question against L. For, according to L, ˹K¬K¬Kp˺ holds in α′. From this, it straightforwardly follows, by the factivity of knowledge and P1, that ˹¬K¬Kp˺ holds in any β′ ∈ B(α′, ˹¬K¬Kp˺). From this, by Smith's definition of 'near duplicate', it in turn follows that, for any near duplicate γ′ of such a β′, both ˹¬K¬Kp˺ holds in γ′ and γ′ ∈ B(γ′, ˹¬K¬Kp˺). But, according to L, in any such near duplicate γ′, I do know ˹¬K¬Kp˺ and hence believe it.
To put it more succinctly, A4 alleges that, if I don't believe ˹¬K¬Kp˺ upon doing the best that I am, in α, in a position to do in order to decide ˹¬K¬Kp˺, which we assume, there should be some case in which (i) ˹¬K¬Kp˺ holds, (ii) I do the best that I am in a position to do in order to decide ˹¬K¬Kp˺, yet in which (iii) I don't believe ˹¬K¬Kp˺ as a result. But, according to L, no case in which one fails to believe ˹¬K¬Kp˺, yet in which ˹¬K¬Kp˺ is true, will be a case in which one does the best that one is in a position to do in order to decide ˹¬K¬Kp˺. This is not to deny that, even if ˹¬K¬Kp˺ holds in a given case, one may fail to believe ˹¬K¬Kp˺ upon doing slightly less than the best that one is, in that case, in a position to do in order to decide ˹¬K¬Kp˺. Thus, the proponent of L has no problem accommodating A4* For any β ∈ B(α, ˹¬K¬Kp˺), if I do not believe ˹¬K¬Kp˺ in β, then, for some β′ ∈ A(α′, ˹¬K¬Kp˺), there is a near duplicate of β′ in which I do not believe ˹¬K¬Kp˺, where, for any case γ and any proposition q, we let A(γ, q) be defined as the set of possible cases, close to γ, in which one does almost, but not quite the best that one is in γ in a position to do in order to decide q (and does nothing that one is not in γ in a position to do). As far as I can see, Smith provides no argument for A4 as opposed to merely A4*. If L holds, type II errors may still occur, but only if one behaves suboptimally (which, for all that has been said, one might easily do). Even if Smith's challenge might thus be defused in the end, the manoeuvre invites the friends of anti-luminosity to have another comeback. Thus, suppose they suggest combining A3 with A3* For any β′ ∈ B(α′, ˹¬K¬Kp˺), if I know ˹¬K¬Kp˺ in β′, then, for any β ∈ A(α, ˹¬K¬Kp˺), there is no near duplicate of β in which I falsely believe ˹¬K¬Kp˺.
Suppose that β′ ∈ B(α′, ˹¬K¬Kp˺) and, in β′, I believe ˹¬K¬Kp˺. Since ˹K¬Kp˺ holds in α, by P1 to P3, it follows that, for any β ∈ B(α, ˹¬K¬Kp˺), I know ˹¬Kp˺ in β-and hence, ˹K¬Kp˺ also holds in β. From this, it in turn follows that, for any near duplicate γ of such a case β, ˹K¬Kp˺ also holds in γ, γ ∈ B(γ, ˹¬K¬Kp˺), and I know ˹¬Kp˺, and hence do not falsely believe, ˹¬K¬Kp˺ in γ. But, for all that, it may then still be that, for some β ∈ A(α, ˹¬K¬Kp˺) and some near duplicate γ of β, I falsely believe ˹¬K¬Kp˺ in γ. Any case is a near duplicate of itself. For simplicity's sake, assume that α itself is such that, while α ∉ B(α, ˹¬K¬Kp˺), nonetheless α ∈ A(α, ˹¬K¬Kp˺) and that I falsely believe ˹¬K¬Kp˺ in α. It would then follow from A3* that I do not know ˹¬K¬Kp˺ in β′-and so, since ex hypothesi, in β′, I've done the best that I am in α′ in position to do in order to decide ˹¬K¬Kp˺, I am in no position to know ˹¬K¬Kp˺ in α′-in which event L fails. A3* is highly controversial, however, and I think there are good reasons to reject it. To see why, first note that A3* is meant to be a kind of safety principle and recall that such principles ultimately derive from a general idea about knowledge-conducive methods, viz. that, if one knows p on the basis of applying some method m, there is no close possibility in which one falsely believes p or a closely related proposition on the basis of applying any method relevantly similar to m. Here, closely related propositions are best conceived as propositions within the range of a method relevantly similar to m, where, trivially, m is relevantly similar to itself (cf. JAI: 27-28, 269-70).
Whether A3* is acceptable accordingly depends on whether doing the best that one is in a position to do in order to decide a given question, and doing almost, but not quite, the best in this sense, are guaranteed to involve use of relevantly similar methods. Methods that belong to different types aren't relevantly similar. (This coheres with the observation that, unlike identity of type, relevant similarity isn't transitive-to the extent, namely, that there may then still be three methods of the same type such that the first is relevantly similar to the second, and the second to the third, although the first isn't relevantly similar to the third.) If a method n is in some way constitutively geared to other methods M, then, plausibly, n′ will be of the same type as n only if there are methods M′ of the same type(s) as M such that n′ is constitutively geared to M′ in that same way. In line with P1 to P4, assume P5 The best that one is in a position to do in order to decide ˹Kp˺ is such that, upon doing it, one believes ˹¬Kp˺ iff one doesn't believe p, where this is so because, if one believes ˹¬Kp˺ upon doing the best that one is in a position to do in order to decide ˹Kp˺, one applies a method n for telling that ˹¬Kp˺ holds such that, among one's methods there are methods M for telling that p holds, where i. always, if ˹Kp˺ holds and one does the best that one is in a position to do in order to decide p, there is some m among M such that one applies m to p, ii.
always and constitutively, for any m among M, if m is applied to p and out puts p, n doesn't output ˹¬Kp˺, and iii.
always and constitutively, if n is applied to ˹¬Kp˺ and outputs ˹¬Kp˺, then, for any m among M, m doesn't output p, where, if a conditional holds constitutively, it sustains corresponding counterfactuals. P5 is a mouthful. Yet, complex as it is, upon reflection, P5 strikes me as rather plausible: the outputs of one's best method n for telling that one is in no position to know p shouldn't get ahead of, but rather be appropriately geared to, the outputs of one's best method m for telling that p holds. In the appendix to JAI, I gave some reasons in P5's favour that were of the same kind as those I gave for P3 in chapter 3 (see . But, for lack of space, I cannot here provide a more thorough argument for it (however, see my reply to Waxman, 2022, where I say more about methods).
With P2, P3, and P5 in place, it follows that, for any p, the best that one is in a position to do in order to decide ˹¬K¬Kp˺ is such that, upon doing it, one believes ˹¬K¬Kp˺ iff one doesn't believe ˹¬Kp˺ iff one believes p. Given that knowledge implies belief and given P1, it then likewise follows that, if one believes ˹¬Kp˺ upon doing the best that one is in a position to do in order to decide ˹¬Kp˺-and so, by P2 and P3, upon doing the best that one is in a position to do in order to decide p-one uses a method for telling that ˹¬Kp˺ holds such that one couldn't easily falsely believe ˹¬Kp˺ on its basis-a method, namely, that tracks those failures to believe p that wouldn't come to pass if ˹Kp˺ was true. 13 Similarly, if one believes ˹¬K¬Kp˺ upon doing the best that one is in a position to do in order to decide ˹¬K¬Kp˺-and so, by P2 and P3, upon doing the best that one is in a position to do in order to decide ˹¬Kp˺-one uses a method for telling that ˹¬K¬Kp˺ holds such that one couldn't easily falsely believe ˹¬K¬Kp˺ on its basis-a method, namely, that tracks those failures to believe ˹¬Kp˺ that wouldn't come to pass if ˹K¬Kp˺ was true (JAI: 271-72).
Doing almost, but not quite the best that one is in a position to do in order to decide ˹¬K¬Kp˺ may break these constitutive connections-with the effect that the method for telling that ˹¬K¬Kp˺ holds that one applied in the course of doing the suboptimal in this regard would no longer be of the same type. 14 This is the reason why I reject A3*. It thus remains the case that, even if L should ultimately be shown to fail, its failure anyway isn't obvious.

Reply to Daniel Waxman
One of the key theses of the account of justification given in JAI is the following: H1 One has propositional justification for believing p iff one is in no position to know that one is in no position to know p.
If we let 'J' be short for 'one has propositional justification for believing', and 'K' for 'one is in a position to know', and conceive of cases as triples of worlds, subjects, and times, we can rephrase H1 as follows: for any case γ, ˹Jp˺ holds in 13 Note that this does not imply that, whenever ˹¬Kp˺ holds, ˹K¬Kp˺ holds: whatever method n for telling that ˹¬Kp˺ holds might be, in the way described, constitutively geared to whatever methods M for telling that p holds, one might believe p on the basis of one of the methods M in circumstances in which ˹¬Kp˺ holds. This is another way in which it transpires that my defence of L is not, by extension, a defence of the S5 axiom for 'K'. 14 The same reasons will also defuse a remaining worry about P4, viz. that it unduly ignores beliefs in ˹¬Kp˺, formed upon doing the best that one is in a position to do in order to decide ˹¬Kp˺, that are unsafe because methods relevantly similar to those used may churn out propositions that do not satisfy C. If P5 is correct, there are no such relevantly similar methods. γ iff ˹¬K¬Kp˺ holds in γ-where, throughout, 'one' refers to the subject at the centre of γ. (Note that, by contrast, in JAI 'J' was just a metalinguistic abbreviation of '¬K¬K'.) H1 is a conjunction of two conditionals. Of these, the right-to-left conditional is the one most vulnerable to the threat of counterexample. The most obvious candidates for such counterexamples can be dismissed from the start because, as I made clear, H1 is only intended to hold for suitably idealized subjects (JAI: 108). Idealizations can protect theses from easy refutation. But they must not be carried too far, lest those theses lose much of their interest. For instance, crediting the subjects in question with unlimited epistemic powers would go too far. Theses exclusively concerned to capture general features of the epistemic lives of subjects with such unlimited resources won't tell us anything about what, in our best moments, we can aspire our epistemic lives to be. Thus, H1 is only meant to hold for subjects whose epistemic powers, while they outstrip our own, are still bounded-what we may call 'finitely improved versions of ourselves'.
Concerns about too strong idealizations were also on my mind when, in JAI, I was searching for a suitable logic for 'K'. For this reason, I refused to buy into the principle, characteristic of normal epistemic logics, according to which one is in a position to know whatever is logically implied by what one is in a position to know. Improved versions of ourselves, at least of the kind I envisaged, still are subject to computational and other limitations and so fail to live up to the standard set by that principle (JAI: 58-59; see also my reply to Rossi, 2022).
In his challenging comments, Daniel Waxman applauds the lessons I draw for epistemic logic, but goes on to devise what he takes to be a counterexample to the right-to-left direction across H1, in the sense in which it is intended, assuming that, throughout, 'one' denotes a single, suitably idealized epistemic agent of the requisite kind that still is subject to computational limitations (Waxman, 2022). Below, I clarify the structure of the challenge and, in addressing it, argue that the threat to H1 can after all be averted.
If one's computational powers are bounded, there will be mathematical proofs that are too computationally complex for one to construct or even go through. To fix ideas, and for the sake of simplicity alone, Waxman conceives of this computational limitation in terms of the length of proofs (Waxman, 2022: 3). There will accordingly be a maximal length a proof might have, if one is to be able to construct or go through it.
Waxman then envisages a case α in which a given mathematical claim t only admits of proofs, or disproofs, whose length exceeds that maximal length, in which one does not know, and has no independent justification to believe, t, its negation, or ˹¬Kt˺, and in which any candidate proofs that one is in a position to construct, or go through, will in fact leave t undecided-in the sense that one will then recognize that one neither knows t nor its negation (Waxman, 2022: 3-5).
To the extent that one has no propositional justification for t in α, as Waxman assumes (2022: 4), H1 forces us to conclude that ˹K¬Kt˺ holds at α. If ˹K¬Kt˺ holds at α, however, then there must be a method that one is in a position to apply in order to find out that one is in no position to know t. But, according to Waxman, no such method is forthcoming, even if it should be conceded that, in α, one has knowledge of the extent of one's computational limitations. For, according to Waxman, finding out that one is in no position to know t would require constructing, or going through, all candidate proofs for deciding t that one is in a position to construct or go through. This, however, is known to be too demanding a task for an agent with limited computational powers (Waxman, 2022: 4-5).
To bring Waxman's challenge into sharper relief, we can simplify his example by assuming that, while in α one neither knows nor has independent justification for t, ˹¬t˺, or ˹¬Kt˺ (e.g. by testimony), one knows in α that there are only two alternative proof strategies one is in a position to individually pursue-but which, given one's computational limitations, one cannot both pursue, simultaneously or successively-in order to decide t, and whose pursuit eventuates in either case β 1 or case β 2 . If we let ' ⃝ ' stand for a situation in which t remains undecided and '•' for a situation in which t is proved, then we can distinguish between (at least) three conceivable scenarios, 15 of which the first, ex hypothesi, is actual: Let 'k' be short for 'one knows'. We assume that, for any β ∈ {β 1 , β 2 }, if t remains undecided in β, then ˹k¬kt˺ holds at β and hence so does ˹K¬kt˺. Similarly, we assume that, for any β ∈ {β 1 , β 2 }, if t is proved in β, then ˹kt˺ holds at β. Given one's computational limitations, one is in fact in scenario (I). However, whichever proof strategy one pursues, in scenario (I), one will end up in a situation in which one cannot exclude both (II) and (III). More specifically, given that one is in fact in scenario (I), in β 1 , one neither knows nor is in a position to know that ˹¬kt˺ holds at β 2 -and so cannot exclude (II)-and, in β 2 , one neither knows nor is in a position to know that ˹¬kt˺ holds at β 1 -and so cannot exclude (III). Consequently, given that one is in scenario (I), one is not, in α, in a position to exclude that one is in either (II) or (III). But, importantly, according to Waxman, in scenarios (II) and (III), one is in α in a position to know t. From this, he concludes that, in scenario (I), one is not, in α, in a position to know that one is in no position to know t-i.e. that, in (I), ˹¬K¬Kt˺ holds at α. By H1, it would then follow that, in α, one has propositional justification for t-which seems intuitively unacceptable.
Indeed, the latter conclusion is also unacceptable on the account of justification given in JAI. For, that account implies H2 One has doxastic justification for p iff one is in no position to know that one doesn't know p.
Ex hypothesi, in scenario (I), ˹K¬kt˺ holds at both β 1 and β 2 . Consequently, given H2, in (I), one lacks doxastic justification for t in both β 1 and β 2 -and this is hard to reconcile with the assumption that, in (I), one has propositional justification for t in α (cf. my reply to Zhan, 2022). There is something initially puzzling about Waxman's confidence that, in (I), ˹¬K¬Kt˺ holds at α. To the extent, namely, that it is anyway known in α that after pursuing one of the two proof strategies one cannot go on to pursue the respective other, in scenario (I), it should hold at both β 1 and β 2 , not only that one knows one doesn't know t (i.e. ˹k¬kt˺), but also that one knows one is in no position to know t (i.e. ˹k¬Kt˺). 16 This implies that, in the present context, we anyway face no violation of the letter of one of the key principles of JAI that I already alluded to in my reply to Smith (2022), viz.

P1
For any δ ∈ B(γ, p), if ˹Kp˺ holds in γ, ˹kp˺ holds in δ, where, for any γ and any p, B(γ, p) is the set of possible cases, close to γ and centred on the same subject as γ, in which the subject of γ has done the best they are in γ in a position to do in order to decide p (and has done nothing they are not in γ in a position to do). In the present example, in scenario (I), B(α, t) = {β 1 , β 2 }. Given it follows that, in scenario (I), B(α, ˹¬Kt˺) ⊆ {β 1 , β 2 }. Accordingly, if he wants to spell trouble for the claim that, in (I), ˹K¬Kt˺ holds at α, Waxman would seem to need something stronger than P1, viz.
W1 For any β ∈ B(α, ˹¬Kt˺), if ˹K¬Kt˺ holds in α, then there is a term 'x' referring to α such that ˹k¬K x t˺ holds in β, where ˹K x ˺ abbreviates 'one is in x in a position to know˺. Note that W1 cannot be derived from P1 to P3 in conjunction with the factivity of 'K' alone. For, even if ˹K¬Kt˺ holds in α only provided that ˹¬K α t˺ holds, this alone does not entail that, for any β ∈ B(α, ˹¬Kt˺), ˹k¬Kt˺ holds in β only if ˹k¬K x t˺ holds in β, for some 'x' referring to α. 16 In Waxman's original example, such knowledge will only be reached if one has exhausted one's resources to conduct any further proof searches and knows that one has done so. Waxman wouldn't seem opposed to the idea that, insofar as one knows the extent of one's computational limitations and is a suitably idealized agent, such a point can indeed be reached (see Waxman 2022: 4; cf. also Waxman forthcoming).
Accordingly, a first line of response would be to reject W1. However, even if W1 may in the end be motivated on intuitive grounds, there is another assumption in play that we aren't forced to accept-viz. Waxman's assumption that, in scenarios (II) and (III), ˹Kt˺ holds at α. Given P1, this assumption implies that, in scenario (II), B(α, t) = {β 2 } and, in scenario (III), B(α, t) = {β 1 }. If, on the contrary, in all scenarios, including (II) and (III), B(α, t) = {β 1 , β 2 }, then, to the extent that the latter is also known in α, in scenario (I), one's knowledge of P1 will allow one to know, in both β 1 and β 2 , that ˹¬K α t˺ holds-in which event, even with W1 in place, there is no reason to deny that, in scenario (I), ˹K¬Kt˺ holds at α and Waxman's challenge can successfully be defused. This is the second line of response I wish to pursue. But, before motivating its key assumption-i.e. that, in all scenarios, B(α, t) = {β 1 , β 2 } -, it is best first to tease out the implications this line of response has beyond the confines of Waxman's particular example. In this context, it is important to realize that Waxman's challenge is of a far more general kind than his focus on computational limitations and the failure of logical omniscience might lead one to expect. 17 Below I will therefore devise another example that exhibits a more general structure than Waxman's own and that, at the same time, will provide for a good test of the viability of the line of response just sketched (see also Waxman forthcoming).
Throughout, I shall continue to assume P1 to P3 and likewise that the inferences from ˹K¬Kp˺ to ˹K¬kp˺ and from ˹k¬Kp˺ to ˹K¬Kp˺ are in good standing (for the latter two inferences, see JAI: 37-38, 86-89). Since we are here concerned with challenges to the right-to-left conditional across H1, we also need a dialectically neutral criterion that helps to determine if subjects lack propositional justification. Reliable pre-theoretic intuitions about the lack of propositional justification are hard to come by, if only because propositional justification is a mere potential for doxastic justification and so need not consist in any piece of evidence currently in one's possession (see my reply to Zhan, 2022). Dialectic neutrality doesn't mean theoretical neutrality. Thus, for the purposes of Waxman's challenge and in line with the result he intends, I suggest that the following criterion will do: crit For all β ∈ B(α, ˹Kp˺), if ˹K¬kp˺ holds in β (and so, according to H2, one lacks doxastic justification for p in β), 18 then one lacks propositional justification for p in α. 19 Let me now turn to the advertised example (cf. also Waxman forthcoming). Suppose that, in case α, one knows-and so it is true-that there are two distinct boxes, box 1 and box 2 , such that, in α, one has no indication or clue what either 17 This point was made during the symposium, at least in basic outline, by Julien Dutant. 18 Note that this only exploits the left-to-right conditional across H2, which is far less controversial (cf. JAI: 118). 19 For more on the link between propositional and doxastic justification, see my reply to Zhan (2022).
box contains, such that one is in a position to open either one of the boxes, but not both, thereby advancing to either β 1 (in which case one opens box 1 ) or β 2 (in which case one opens box 2 ), and such that, given what one presently knows or is presently justified in believing, all of the following conceivable scenarios are equally likely so that, upon opening one of the boxes and finding it either empty or non-empty, one has no indication or clue as to whether the other box is empty or not (where '□ i ' indicates that box i is empty and '⊠ i ' indicates that box i is non-empty): Unlike in Waxman's original example, we make no assumption about which of these scenarios is actual. Let q be the proposition that at least one of the two boxes is non-empty. We assume that, for any β ∈ {β 1 , β 2 }, ˹kq˺ holds at β iff the box one opens in β is non-empty. In scenario (2), q is false. A fortiori, in (2), ˹¬Kq˺ holds at α. In fact, (2) is the only scenario in which q is false. Given the setup, we may assume that one already knows in α that, no matter which box one opens and no matter what's inside, one will at most be able to exclude two of the four scenarios. Consequently, in all four scenarios, ˹K¬K¬q˺ holds at α. For the time being, let us suppose, with Waxman, that, in all the other scenarios, (1), (3), and (4), ˹Kq˺ holds at α. (This corresponds to Waxman's earlier assumption that, in scenarios (II) and (III), ˹Kt˺ holds at α.) Then, in scenarios (1), (3), and (4), ˹¬K¬Kq˺ holds at α, and by P1 to P3, in (3), both β 2 ∉ B(α, q) and β 2 ∉ B(α, ˹¬Kq˺), while in (4), both β 1 ∉ B(α, q) and β 1 ∉ B(α, ˹¬Kq˺). Given this supposition, it is then no problem to maintain that, in those three scenarios, one has propositional justification for q in α. For, then, in those scenarios, if one does the best that one is in α in a position to do in order to decide ˹¬Kq˺, one will know q and so be in no position to know that one doesn't know q (and hence, according to H2, will have doxastic justification for q).
Given Waxman's supposition, the only potentially problematic scenario will then be (2) (which corresponds to scenario (I) above). In (2), ˹K¬kq˺ holds at both β 1 and β 2 . Therefore, by crit, P2 and P3, one has, at α, no propositional justification for q. Applied to the present example, Waxman's diagnosis then is that, in (2), ˹¬K¬Kq˺ nonetheless holds at α-in which event H1 will fail.
We have in effect already seen that, for this diagnosis to be forced upon us, we would in addition need a principle of the kind already exemplified by W1, viz.
W2 For any β ∈ B(α, ˹¬Kp˺), if ˹K¬Kp˺ holds in α, then there is a term 'x' referring to α such that ˹k¬K x p˺ holds in β.
Let us not here take issue with W2. Instead, we can respond by insisting that, pace Waxman, in each of the four scenarios, B(α, q) = {β 1 , β 2 }-thereby rejecting the aforementioned supposition that ˹Kq˺ holds at α in scenarios (3) and (4). 20 We can now also see more clearly that rejection of this supposition is independently called for, as long as we wish to maintain that, in (2), ˹K¬Kq˺ holds at α. For, if (2) is indeed the only scenario in which ˹¬Kq˺ holds at α, and one knows this much about the setup, then, since (2) is also known to be the only scenario in which q is false, ˹K(¬Kq ⊃ ¬q)˺ should hold at α in all four scenarios. Accordingly, if ˹K¬Kq˺ holds at α in (2), then, given the availability of a harmless closure step, in scenario (2), one would after all in α be in a position to know that one is in that scenario-which, given the setup, is impossible.
In α, one already knows that, no matter which box one opens and no matter what's inside, one will at most be able to exclude two of the four scenarios. If, in all four scenarios, B(α, q) = {β 1 , β 2 }, then, given P1, (1) will be the only scenario in which ˹Kq˺ holds at α. To the extent that this is a further thing one knows in α, consequently, in all four scenarios, not only does ˹K¬K¬q˺ hold at α, but so does ˹K¬KKq˺. But, if ˹K¬KKq˺ holds at α, we cannot similarly exploit ˹K(Kq ⊃ r)˺ in order to arrive at ˹Kr˺ by a single closure step, where r is the proposition that none of the boxes is empty. Equally, then, we may assume, in line with W2, that, for any β ∈ {β 1 , β 2 }, ˹k¬Kq˺ will hold at β iff ˹k¬K α q˺ holds at β (i.e. iff the box one opens in β is empty). Accordingly, given the inference from ˹k¬Kq˺ to ˹K¬kq˺, scenario (2) no longer presents a problem for H1, all the while (1) continues to present no such problem either.
Given our rejection of Waxman's supposition, the only potentially troublesome scenarios now are (3) and (4). For, if, in both (3) and (4), B(α, ˹Kq˺) = {β 1 , β 2 }, P1 and P2 will imply that, in those scenarios, ˹¬K¬Kq˺ holds at α, because, upon opening a non-empty box, one will not know that one is in no position to know p. Given H1, crit is equivalent to P6 For any δ ∈ B(γ, ˹Kp˺), if ˹K¬kp˺ holds in δ, then ˹K¬Kp˺ holds in γ. 20 This implies rejection of the converse of P1, i.e.
CP1For any δ ∈ B(γ, p), if ˹kp˺ holds in δ, then ˹Kp˺ holds in γ, which I had relied on in my argument for L (see JAI: 41-42, 72). However, the instance of CP1 that I needed for the latter purpose was in fact For any δ ∈ B(γ, ˹¬Kp˺), if ˹k¬Kp˺ holds in δ, then ˹K¬Kp˺ holds in γ, which can be derived from P2, ˹k¬Kp ⊃ K¬kp˺, and principle P6 below, which latter I argue remains acceptable (cf. principle (8) in JAI: 72).
Given P6, that ˹¬K¬Kq˺ holds at α will then imply that ˹K¬kq˺ holds at neither β 1 nor β 2 -which is patently false, because, upon opening an empty box, one will know that one is in no position to know q.
However, as long as it can in addition be maintained, in line with P1 to P3, that, in (3), β 1 ∉ B(α, ˹¬Kq˺), while in (4), β 2 ∉ B(α, ˹¬Kq˺), any looming trouble can be averted. 21 We can derive this additional assumption as follows. Recall another principle that I invoked in the context of my reply to Smith (2022): P4 If, for some δ ∈ B(γ, ˹¬Kp˺), one believes ˹¬Kp˺ in δ, then ˹K¬Kp˺ holds in γ.
One might think that the combination of W2 and P1 puts pressure on P4. For, given W2 and P1, ˹K¬Kq˺ will hold in α, only if, for all β ∈ B(α, ˹¬Kq˺), one knows, and hence believes, ˹¬K α q˺ in β, for some 'x' referring to α. Yet, given our assumption that it is already known in α that ˹Kq˺ holds only if ˹kq˺, and hence ˹Kq˺, holds at both β 1 and β 2 , this alone is insufficient to refute P4. For, given this assumption, plausibly, for any β ∈ B(α, q), if one believes ˹¬Kq˺ in β, one also believes ˹¬K α q˺ in β.

Consequently, in
What, on this line of response, still stands in need of an explanation is the key assumption that, in some scenarios, viz. those in which box 1 is empty iff box 2 isn't, it is the case that B(α, q) = {β 1 , β 2 }-although ˹¬kq˺ holds in β 1 iff ˹kq˺ holds in β 2 . Any such explanation must cohere with the finding that, nevertheless, in those very scenarios, for any β ∈ {β 1 , β 2 }, β ∈ B(α, ˹¬Kq˺) iff ˹¬kq˺ holds in β (i.e. iff the box one opens in β is empty)-with the consequence that {β 1 , β 2 } ⊈ B(α, ˹¬Kq˺). The sought-after explanation will have to say something suitably instructive about what it is to do the best that one is in a position to do in order to decide a given question.
Let us say that a given method m is among one's ex ante best methods for telling whether p holds iff before, and independently from the results furnished by, the use of any of one's methods for telling whether p holds, one has propositional justification for thinking that m is no less promising a method to use to this end than any other of one's methods. 22 Let a method's conditions of application to a given proposition p within its range be those conditions under which alone the method yields any output with regard to p-including, as the case may be, yes, no, or nought. Now consider: P7 If, among one's ex ante best methods for telling whether p holds, there are some whose conditions of application to p one is in a position to bring about, then one does the best that one is in a position to do in order to decide p iff for some such method, one brings about its conditions of application to p and applies it to p.
It follows that, if, among one's ex ante best methods for telling whether p holds, there is one method such that one brings about its conditions of application to p and applies it to p, then one does the best that one is initially in a position to do in order to decide p. It likewise follows that, if one does the best that one is in a position to do in order to decide p, then, if, among one's ex ante best methods for telling whether p holds, there is indeed one whose conditions of application to p one is in a position to bring about, one brings about the conditions of application to p of one such method and applies it to p. Given a suitably realist conception of the individuation of methods and their application conditions-a conception that allows us to clearly distinguish between methods and their individual applications, as well as between circumstances in which a given method yields the output nought and circumstances in which it isn't applied at all-we can now respond to the explanatory challenge as follows. (What I am about to say chimes with what I say towards the end of my reply to Smith, 2022.) Given our choice for q, in α, one already knows that one will be unable to tell that ˹¬q˺ holds, never mind what one does, and knows that one will be unable to tell that q holds upon opening an empty box, but able to tell that q holds upon opening a non-empty box. So, there is, in α, no method for telling whether q holds that one is in a position to apply and that would ever output no. There is, in α, only one ex ante best method m for telling whether q holds: looking inside one of the boxes and telling whether the box in question is empty. If the box in question is non-empty, m will output yes, and if the box in question is empty, m will output nought. To pretend to be able to identify two methods here-looking inside an empty box and looking inside a non-empty box-would be to individuate methods too finely and to mistake methods for their individual applications. Accordingly, m's conditions of application to q are the conditions under which one looks into one of the boxes. To bring about those conditions is to open any one of the boxes (to the exclusion of the respective other). Ex hypothesi, in α, one is in a position to do so. So, by P7, in α, the best one is in a position to do in order to decide q is to open one of the boxes, look inside, and then tell whether the box is empty. Accordingly, in scenarios (3) and (4), B(α, q) = {β 1 , β 2 } and so, by P1, one is in a position to know q in α only if one knows q in both β 1 and β 2 (i.e. only in scenario (1)).
Given our choice for q, in α, one already knows that one will be unable to tell that ˹Kq˺ holds, never mind what one does, and knows that one will be unable to tell that ˹¬Kq˺ holds upon opening a non-empty box, but able to tell that ˹¬Kq˺ holds upon opening an empty box. So, there is, in α, no method for telling whether ˹Kq˺ holds that one is in a position to use and that would ever output yes. There is, in α, only one ex ante best method n for telling whether ˹Kq˺ holds: concluding from m's output of nought (in combination with what else one knows about one's situation) that one is in no position to know q (see principle P5 in my reply to Smith, 2022). If m outputs nought, n outputs yes. But, if m doesn't output nought (and so, assuming m is applied, outputs yes instead), then n isn't even applied and so doesn't output anything, not even nought: that m outputs nought is part of n's conditions of application to ˹¬Kq˺. To bring about these conditions is to open an empty box and apply m. Ex hypothesi, in α, one is in a position to do so. So, by P7, in α, the best one can do in order to decide ˹Kq˺ is to open an empty box, apply m, and then conclude that one is in no position to know q. Accordingly, in scenario (3), B(α, ˹¬Kq˺) = {β 2 } and so, by P1, one is in a position to know ˹¬Kq˺ in α only if one knows ˹¬Kq˺ in β 2 , even if one doesn't know ˹¬Kq˺ in β 1 . Similarly, in scenario (4), B(α, ˹¬Kq˺) = {β 1 } and so, by P1, one is in a position to know ˹¬Kq˺ in α only if one knows ˹¬Kq˺ in β 1 , even if one doesn't know ˹¬Kq˺ in β 2 .
In a nutshell, then, my response to Waxman is this. Waxman contends that (i) when none of the routes one is in a position to take in order to decide a given p leads to a decision, while one is in no position to take them all, one is initially in no position to know that one is in no position to know p, and that (ii) when one of those routes does lead to one's knowledge of p, one is initially in a position to know p. I reject both these assumptions. Instead, I contend that (iii) when some but not all those routes lead to one's knowledge of p, one is initially in no position to know p (that is, even if there then is some route such that, were one to take it, one would end up knowing p), and that (iv), because of that, one is in a position to recognize that one was not initially in a position to know p by taking one such route that leaves p undecided.
The smaller B(γ, p) is, the weaker are the demands that P1 imposes on being in a position to know p in γ. The foregoing thus meshes well with the contention, implicit in much of what I say in JAI, that it is never harder to be in a position know one's ignorance than it is to be in a position to know the condition one is ignorant of. Even so, I expect that, unless more is said about the nature of methods-and about the nature of methods for telling if one is in no position to know, in particular-few will be swayed by these considerations. So, more work needs to be done.

Reply to Julien Dutant
The two key theses of the account of justification I gave in JAI are H1 One has propositional justification for believing p iff one is in no position to know that one is in no position to know p.
H2 One has doxastic justification for believing p iff one is in no position to know that one is in no position to know p.
(See JAI: 108.) If we let 'J' be short for 'one has propositional justification for believing', 'D' for 'one has doxastic justification for believing', 'k' for 'one knows', and 'K' for 'one is in a position to know', and conceive of cases as triples of worlds, subjects, and times, we can rephrase H1 and H2 as follows: for any case α, ˹Jp˺ holds in α iff ˹¬K¬Kp˺ holds in α, and for any case α, ˹Dp˺ holds in α iff ˹¬K¬kp˺ holds in α-where, throughout, 'one' refers to the subject at the centre of α. (Note that, by contrast, in JAI, 'J' and 'D' were just metalinguistic abbreviations of '¬K¬K' and '¬K¬k', respectively; see JAI: 83.) Both types of justification were conceived of as being pro toto (all things considered). For this reason, it was essential to establish both of the following For, plausibly, a proposition and its negation are never both pro toto justified. D J is the so-called Geach axiom for 'K'. In chapter 5 of JAI, I derived D J from the following two principles which I had argued for in chapter 4: JAI: 62,[68][69][70][71][72][73][74][75][76]. L is a controversial luminosity principle for which, however, I gave arguments (JAI: 68-79, 269-74; see also my reply to Smith, 2022). A6 was motivated by the factivity of 'K' (i.e. the validity of ˹Kp ⊃ p˺) and instances of normal modal reasoning of the kind that, despite my rejection of the idea that 'K' behaves normally throughout, I found unobjectionable, given the idealizations in play (JAI: 62, 86). D D was then derived from D J and

D-J K¬Kp ⊃ K¬kp
which latter can be motivated by the validity of ˹kp ⊃ Kp˺ and similarly unobjectionable normal modal reasoning (JAI: 86-87, 89). In his challenging review, Julien Dutant (2022) argues that D D has counterexamples which, given D-J, are also counterexamples to D J and, to the extent that A6 remains acceptable, in turn counterexamples to L. Worse still, the alleged counterexamples are cases in which, for some proposition, one has neither doxastic nor propositional justification for that proposition or its negation-so that H1 and H2, too, would be refuted. Accordingly, Dutant can be seen as mounting a broadside attack on the account developed in JAI.
To this end, he invites us to consider cases in which one is in no position to know, and a fortiori does not know, the answer to a given yes-or-no question, and yet, once the question is raised and understood, one nonetheless has the impression that one knows the answer-an impression that, misleading though it is, persists throughout one's best efforts to decide the question, without ever giving way to any belief in a positive or negative answer to that question. Dutant's preferred examples involve rather resilient tip-of-the-tongue phenomena. He contends that one's persistent impression that one has the answer 'on the tip of one's tongue', and thus that one knows it, debars one both from being in a position to know that one doesn't know p and from being in a position to know that one doesn't know the negation of p, where the question at hand is whether p holds (Dutant, 2022: 4-5). If so, we would have (#) ¬K¬kp ∧ ¬K¬k¬p and thus a counterexample to D D (and hence, modulo D-J and A6, to D J and L). Plausibly, in the cases as described, one has no doxastic or propositional justification for p or its negation and, accordingly, to the extent that (#) holds, H1 and H2 would founder, too.
In what follows, I wish to cast doubt on Dutant's contention that the persistent tip-of-the-tongue impression suffices for (#). Note that, in order to do so, I am under no obligation to postulate any epistemic asymmetry between p and its negation (which, given that one is ex hypothesi neither in a position to know p nor in a position to know its negation, would anyway be a tall order). For, the account proposed in JAI clearly allows for cases in which both ˹K¬kp˺ and ˹K¬k¬p˺ hold (i.e. is consistent with the failure of the converse of D D ).
Knowledge implies belief where such a belief need not be occurrent. Still, in line with the idealizations I put in place (JAI: 108), I shall assume that, just as one will know p if one is in a position to know p and does the best that one is in a position to do in order to decide p (see P1 in my reply to Smith, 2022), the following holds: (1) If one knows the answer to a yes-or-no question that has been raised and understood, and one has made one's best efforts to come up with that answer in response to that question, then one will come up with that answer, where coming up with an answer here implies making that answer explicit (if only in one's mind).
(1) rejects the suggestion that one's knowledgeable beliefs may remain implicit forever despite one's best efforts to bring them to mind, although one has all the conceptual capacities necessary to do so. For all that (1) implies, there may be a notion of implicit belief that has this feature, but (1) anyway rejects the thought that knowledge implies a belief of this sort. Should Dutant insist that some of our knowledge implies a belief of that sort, and be able to make a successful case for this assumption, then, as a last resort, I still have the option to concede the point, restrict the scope of 'k', and ask readers to understand the account of justification devised in JAI accordingly (and as it was intended; see JAI: 22).
Let me now make a number of observations on which I will rely in my response to Dutant's challenge: (i) Impressions, or seemings, may persist although one knows better or knows something that is evidence for their misleading nature.
(ii) One may have evidence of a given kind even if one doesn't know that one has evidence of that kind.
This is plausible on a conception on which one's evidence consists in what one knows: ˹kq ⊃ kkq˺ is arguably invalid and, similarly, what one's evidence is evidence for, and how strong such evidence it is, need not always be known (even, that is, if one knows that one has that evidence) (cf. Williamson, 2000). This is also plausible on a conception on which one's evidence is what one is in a position to know (JAI: 204-205): ˹Kq ⊃ KKq˺ is arguably invalid and, similarly, what one's evidence is evidence for, and how strong such evidence it is, isn't luminous (even, that is, if one is in a position to know that one has that evidence) (JAI: 66, 227).
(iii) To heed evidence of a given kind, one needn't have evidence that one has evidence of that kind.
This is plausible on a conception on which one's evidence consists in what one knows: to heed one's evidence p is to act as it would be rational to act were one to know p, yet to act as it would be rational to act were one to know p isn't the same as to act as it would be rational to act were one to know that one knows p. On a conception on which one's evidence consists in what one is in a position to know, plausibly, in order to heed one's evidence p one must know p, so that, again, to heed one's evidence p is to act as it would be rational to act were one to know p. Yet, to act as it would be rational to act were one to know p isn't the same as to act as it would be rational to act were one to know, or at least to be in a position to know, that one knows p.
(iv) We are familiar with there being propositions p, features F, and potentially co-referring concepts a and b such that, first, a refers iff a and b co-refer iff some favourable condition obtains, and secondly, if knowing <a has F> would put one in a position to know p, then, all things being equal, knowing <b has F> would put one in that same position. If knowledge of <a implies that the pressure is above threshold t> would put me in a position to know that the pressure is too high, then, to the extent that my calculation is in fact error-free, knowledge of <b implies that the pressure is above t> would, all things being equal, put me in that same position. Examples can be multiplied. Against the backdrop of these observations and in the light of (1), I take the following to be eminently plausible: (2) If a yes-or-no question has been raised and understood and if, even after prolonged efforts to come up with an answer in response to that question, one doesn't come up with any such answer, then the observation that one doesn't come up with any such answer even after those efforts are made will eventually provide one with inductive evidence for the thought that one doesn't know the answer to that question.
(2) doesn't say how long one must have unsuccessfully tried to come up with an answer, before observation that one fails to do so in spite of such attempts turns into inductive evidence for the negative conclusion that one doesn't know the answer to the question at hand. It merely claims that some such point will eventually be reached-a point at which whatever evidential weight one's tip-of-the-tongue impression had will be outweighed (cf. JAI: 203). Beyond that point, the longer the observed unsuccessful efforts last, the stronger the inductive evidence for the negative conclusion. Note that, given (ii), having inductive evidence of this kind doesn't require having evidence that one has evidence of this kind. Given (iii), nevertheless, one may then still heed that evidence. If it still seems to one that one knows the answer to the question at hand, then inductive evidence for thinking that one doesn't know the answer to that question is also evidence for thinking that those seemings are misleading. Accordingly, given (2), to the extent that one should heed one's evidence, in the case described by Dutant, as she continues to rake her brain, the subject should give less and less weight to her tip-of-the-tongue impression and become more and more inclined to conclude that it is misleading, even if, phenomenally, that impression might persist. Subjects who observe that they don't come up with an answer to the question at hand even after prolonged efforts to do so-and continue making observations of the same kind as they continue making such unsuccessful efforts-will fail to behave in the rational ways in which idealized agents were stipulated to behave, if they nonetheless continue to give as much weight to their tip-of-the-tongue impression as before. For, this would be an evidential bias of the kind the idealization was designed to exclude (see, e.g. JAI: 108).
I think it is telling that Dutant nowhere assumes that the subject in question believes that they know the answer to the question; for, their rational failing would then be more evident. But even the refusal to acknowledge, after so many observed unsuccessful attempts to come up with an answer, that the tip-of-the-tongue impression ceases to be a reason to continue trying is a rational failing. So-called epistemic seemings, unlike phenomenal seemings, 'go away in the presence of a rebutting defeater, if the agent is rational' (Brogaard, 2013: 274). So, if the tip-of-the-tongue impression persists with equal force, as Dutant assumes it does and as (i) acknowledges it may, then, if the subject is fully rational, the impression in question is phenomenal rather than epistemic. As such, it does not have the force Dutant assumes it to have, viz. to debar the subject from acknowledging that they do not know the answer to the question. This also casts doubt on Dutant's diagnosis that his case refutes what I say about the availability of responsive beliefs in the context of my argument for L (Dutant, 2022: XX;JAI: 73-74). The notion of responsive belief was explicated in terms of epistemic seemings rather than merely phenomenal ones (JAI: 51-53).
Plausibly, if one has made one's best efforts to come up with an answer to the question at hand, and those efforts prove unsuccessful, then one will have tried long enough in order for one's observation that one's efforts remain unsuccessful to constitute pretty strong inductive evidence for the negative conclusion that one doesn't know the answer after all. In the light of this and (ii), I suggest the following: (3) If a question has been raised and understood and, even after one's best efforts to come up with an answer in response to that question, one doesn't come up with any such answer, then the observation that one doesn't come up with any such answer even after those efforts are made puts one in a position to know that one doesn't know the answer to that question.
(3) doesn't say how long one must have tried to come up with an answer to the question, in order for one to have made one's best efforts to come up with such an answer. It merely presupposes that there is a point at which one has done one's best in this regard.
To the extent that one knows (1) and one is sufficiently logically astute, it isn't hard to see that, if one has made one's best efforts to come up with an answer but has failed, observing that this is so would put one in a position to know that one doesn't know the answer. But, given (iv), this needn't be the only observation that may put one in a position to know this: if the efforts one has made are de facto one's best, then, plausibly, the observation that one has made the efforts one has made yet without success will likewise put one in that position. (Notice that the formulation of (3) deliberately remains neutral on which of these two kinds of observations is made.) By (ii), in order for one's observation to put one in a position to know that one doesn't know the answer, one needn't have evidence for thinking that it does so. 23 The following principle seems rather plausible: (4) If, for any case β ∈ B(α, p), one is in a position to know p in β, then one is already in a position to know p in α, where, as before, B(α, p) is the set of possible cases, close to α and centred on the same subject as α, in which the subject of α has done the best they are in α in a 23 Note that, on the account devised in JAI, ˹K¬kp˺ doesn't encode a luminous condition while ˹K¬Kp˺ does. Futhermore note that, on that account, the converse of D-J fails. Accordingly, could Dutant now concede that ˹K¬kp˺ and ˹K¬k¬p˺ both hold, but nonetheless try to make a case for ˹¬K¬Kp ∧ ¬K¬K¬p˺ (and hence directly against D J ) by exploiting the fact that, if ˹K¬Kp˺ held, then, by my own lights, so would ˹KK¬Kp˺? He might, although the devil is, as always, in the detail. But, in any event, in order for ˹KK¬Kp˺ to hold in either scenario-as I am anyway committed to saying since one lacks propositional justification for p in either of them-it still needn't be the case that one be in a position to know that one's obervation puts one in a position to know that one is in no position to know p.
position to do in order to decide p, and has done nothing they are not in α in a position to do. For, if, for all β ∈ B(α, p), whatever one ends up knowing in β was something that one was already in a position to know in α, if there is any such knowledge that puts one in a position to know p in β, plausibly, one was already in that position in α. 24 Unless Dutant can make a case that there are, besides raking one's brain and continuing to observe that one doesn't come up with any answer, other equally good or even better things one is in a position to do in order to decide whether one knows the answer and that don't put one in a position to know one doesn't know the answer, we may now use (3) and (4) to conclude that, in the kind of case Dutant invites us to consider, both ˹K¬kp˺ and ˹K¬k¬p˺ hold and hence (#) fails.

Reply to Yiwen Zhan
In his thought-provoking comments (Zhan, 2022), Yiwen Zhan, like Daniel Waxman (2022), targets one of the key theses I advanced in JAI, viz.

H1
One has propositional justification for believing p iff one is in no position to know that one is in no position to know p. 25 Zhan contrasts H1 with the following thesis: H1* One has propositional justification for believing p iff one might be in a position to know p, where, according to Zhan, 'might' encodes some kind of 'epistemic opportunity' (Zhan, 2022: 2). 26 24 In JAI, I had argued for something stronger than (4) which, in light of the strategy I pursue in my reply to Waxman (2022), now seems too strong (cf. principle (1c) in JAI: 42). 25 Zhan contends that my account of justification implies that, if I am in no position to know 'whether the fact that I am in a position to know p obtains, the second best thing is to remain agnostic about whether the opposite fact, i.e. that I am not in a position to know p, obtains' (Zhan 2022: 1). On a natural reading of 'whether', this is to say that the account implies that, if ˹¬KKp ∧ ¬K¬Kp˺ holds, one should remain agnostic about whether ˹Kp˺ holds. This implication is plausible quite independently from my account. If Zhan instead intends ˹¬KKp ⊃ ¬K¬Kp˺, then his contention is false: by the factivity of 'K', ˹K¬Kp˺ implies ˹¬KKp˺, and so, by the transitivity of the conditional, ˹¬KKp ⊃ ¬K¬Kp˺ would imply that ˹¬K¬Kp˺ holds, for arbitrary propositions p. This is clearly not implied by the account I gave in JAI. 26 One of Zhan's reasons for claiming that 'it might be that' should not be construed as ˹¬K¬˺ is this: 'If [I assert] that the train might arrive at 5pm, and if my assertion is interpreted simply as an ¬K¬-statement about my own informational state at the time, it can neither explain the linguistic data that you may still [disagree with] my statement (should you be convinced that the train will not arrive at 5pm), nor explain the perceived need for me to retract my previous assertion upon learning that the train is delayed' (Zhan 2022: 2n2). In Rosenkranz (2007), I tried to show that, on the contrary, such an explanation is after all to be had, arguing that, in reasoned debates, participants who make a disputed claim are committed to denying that any party to the debate is in a position to know that they are in no position to know that claim-where, plausibly, any such party will be in a position to learn by testimony what any other party is in a position to know (cf. also JAI: 127n5).
Given what he says about his first toy model discussed below, Zhan would seem to think of such epistemic opportunities in terms of epistemic states to which one is in a position to advance by acquiring more evidence-evidence that would eliminate some of the actually accessible epistemic possibilities and thereby put one in a position to know something that one wasn't in a position to know before. Since ˹¬K¬Kp˺ and ˹¬Kp˺ may hold while knowing p is impossible, it should initially be clear that, on the account of justification H1 provides, propositional justification does not entail such epistemic opportunities. Surprisingly, however, Zhan writes that even on 'the ¬K¬K Rule's construal' of justification'-i.e. even given H1-'a non-trivial case of justification [for φ, i.e. a case in which one is in no position to know the truth of φ] corresponds to an epistemic state with two kinds of potentials: the subject could either get to know the truth of φ (or ˹¬φ˺) or get to know that it is actually something [they are] in no position to know' (Zhan, 2022: 4).
This doesn't seem to be correct-at least if 'could get to know' here implies 'is in a position to know'. For, if ˹¬K¬Kp ∧ ¬Kp˺ implied ˹Kp ∨ K¬Kp˺, there would simply be no 'non-trivial cases of justification', in which event we would be saddled with the hopeless S5 principle for 'K', i.e. ˹¬Kp ⊃ K¬Kp˺. Given ˹K¬p ⊃ K¬Kp˺ (see JAI: 62), we would get the same unpalatable result, if ˹¬K¬Kp ∧ ¬Kp˺ implied ˹(Kp ∨ K¬p) ∨ K¬Kp˺. 27 Clearly, H1 has no such unpleasant consequences. 28 If, instead, 'could get to know' here just means 'for all that one is in a position to know, one is in a position to know', then, since the disjunction ˹(¬K¬Kp ∨ ¬K¬K¬p) ∨ ¬K¬K¬Kp˺ trivially follows from ˹¬K¬Kp˺ alone, we don't learn anything illuminating about non-trivial cases of justification. It is unclear what other pertinent readings of 'could get to know' there are.
In fact, what one would have expected Zhan to say instead is that, according to H1, cases in which one lacks justification for p are cases in which one could get to know, if not ˹¬p˺, then at least ˹¬Kp˺. That's of course true on the first reading of 'could get to know'. 29 But such cases are not in any way illustrated by his first toy model (or his second toy model, for that matter; see below). This first model consists of three worlds, w 1 to w 3 -with w 1 serving as the actual world-where • p is true at w 1 and w 3 , but false at w 2 • {w 1 , w 2 , w 3 } is the set of worlds accessible from w 1 and w 2 , and {w 1 , w 3 } is the set of worlds accessible from w 3 27 If ˹¬K¬Kp ∧ ¬Kp˺ implied ˹K¬Kp˺, it would imply a contradiction, and so ˹¬Kp ⊃ K¬Kp˺ would follow. 28 To the extent that 'one might be in a position to know p' entails ˹¬K¬Kp˺, but not the other way round, if H1* should have these consequences, this would be a good reason to reject it. Although I think of H1* as being equivalent to H1, if a case can be made that it isn't, I'm happy to discard H1* and stick to H1. 29 Note that it is a matter of controversy whether, on the first reading, ˹one could get to know p˺ implies ˹one has the opportunity to come to know p˺. For my more recent worries about the alleged link between being in a position to know and opportunities to come to know, see footnote 9 of my reply to Smith (2022). and where • ˹Kφ˺ is true at a world w of the model iff φ holds in all worlds accessible from w. 30 It follows that ˹Kp˺ is true at w 3 but false at both w 1 and w 2 and that ˹¬K¬Kp˺ is true at all worlds of the model and, hence, so is ˹K¬K¬Kp˺ (just as the luminosity of ˹¬K¬Kp˺ requires) (Zhan, 2022: 4). According to H1, then, this model is of cases in which the subject has propositional justification for p. Zhan is, of course, quite right to point out that the model itself doesn't show that it models cases in which the subject has such justification. But, equally clearly, such formal models do not aspire to show this. To cause trouble for H1, Zhan needs to argue that there are cases, adequately captured by this model (or a related model preserving all its relevant features), in which one has no justification for p. Devising models alone doesn't do the job.
Setting out to construct such a case, Zhan nonetheless sees the need to devise a more complex model, even if it should be clear by now that even such a more complex model, taken on its own, will remain silent on what subjects at the centre of the cases that it illustrates have, or do not have, justification for. The second model is a model of four worlds, w 1 to w 4 -with w 1 again serving as the actual world-where • p is true at w 1 , w 3 , and w 4 , but false at w 2 • q is true at w 1 , w 2 , and w 4 , but false at w 3 • {w 1 , w 2 , w 3 , w 4 } is the set of worlds accessible from w 1 and w 2 , {w 1 , w 3 , w 4 } is the set of worlds accessible from w 3 , and {w 1 , w 4 } is the set of worlds accessible from w 4 and where, again, • ˹Kφ˺ is true at a world w of the model iff φ holds in all worlds accessible from w.
It follows that ˹Kp˺ is true at both w 3 and w 4 , but false at both w 1 and w 2 , that ˹Kq˺ is only true at w 4 , and that all of ˹¬KKp˺, ˹¬K¬Kp˺, and ˹¬K¬Kq˺ are true at all worlds of the model and, hence, so are both ˹K¬K¬Kp˺ and ˹K¬K¬Kq˺ (Zhan, 2022: 6-7). According to H1, then, all the cases for which this model serves as a model are cases in which one has justification for both p and q.
Zhan contends that this model is a model of cases in which one's failure to be in a position to know q explains why one is in no position to know ˹¬Kp˺-i.e. cases that exhibit what he calls p's 'second-order structural unknowability'-where, he claims, these cases importantly include some in which one lacks justification for p (Zhan, 2022: 6).
The first thing to note is that not only does ˹K¬K¬Kq˺ hold in the model, but so does ˹K(Kq ⊃ Kp)˺ and, hence, ˹K(Kq ⊃ ¬K¬Kp)˺. Accordingly, one is in a position to recognize that one cannot rule out a situation in which ˹Kq˺ holds-a situation which one is in a position to know would suffice for the truth of ˹¬K¬Kp˺. Given factivity, such a situation would indeed so suffice. This puts some pressure on the idea that the model adequately captures p's second-order structural unknowability, in the way the latter was defined. At least, for all that the model tells us, a situation in which both ˹Kq˺ and ˹¬K¬Kp˺ hold (i.e. one corresponding to w 4 ) is not merely an epistemic but a real possibility. If so, to what extent is the truth of ˹¬K¬Kp˺ explained by the truth of ˹¬Kq˺, if it then doesn't even require it? In fact, the model validates ˹K((Kq ∨ ¬Kq) ≡ ¬K¬Kp)˺ and ˹K¬K¬Kp˺. So, one is in a position to know that, whether or not one is in a position to know q, one is anyway in no position to rule out that one is in a position to know p. So, one is in a position to recognize-and so it is the case-that the truth of ˹¬Kq˺ doesn't make any difference on this score. Accordingly, at best some of the cases the model serves to model will be cases in which p is second-order structurally unknowable-viz. cases in which, while one is in a position to know that ˹¬K¬Kp˺ would hold even if ˹¬Kq˺ did not, it is then the truth of ˹¬Kq˺, together with some hitherto unspecified features of the actual world, that explains why ˹¬K¬Kp˺ holds. Consequently, the model is not itself a model of secondorder structural unknowability.
Given that ˹K¬K¬Kp˺ holds in the model, Zhan will now also have to argue that there are counterexamples to (*) If one is in a position to know that one is in no position to know that one is in no position to know p, then one has propositional justification for believing p.
On the account developed in JAI, the right-to-left conditional across H1 is equivalent to ( *), since, on that account, we have But, L remains controversial and so (*) is arguably weaker than the right-to-left conditional across H1. Zhan's focus on cases for which his second model serves as a model therefore inadvertently makes for a heavier dialectical burden on his part. Moreover, as noted, ˹K¬K¬Kq˺, too, holds in the model. Accordingly, lest he be charged with begging the question, he must either allow for the assumption that q is justified or else give arguments for thinking that, in the cases he envisages, not only does p present a counterexample to (H1 and) (*), but so does q. The model itself is silent on the matter. Let us now at last turn to Zhan's example of Professor Moore who selects one of two candidates for a vacancy according to criteria whose application she knows has led to successful choices in the past, all the while the success of those choices was in fact, but unbeknownst to her, pure chance-insofar, namely, as past candidates, like the one she presently chooses, merely happened to meet other criteria which she is in no position to recognize are in fact more important than the ones she applies. 31 Let the chosen candidate be Ben, let C M be the criteria used by Professor Moore, let p be <Ben is the better candidate>, and let q be <C M are not the most important criteria to use>. According to Zhan, while Moore is in no position to know p, she is in no position to know that she is in no such position because she is in no position to know q. Let that indeed be so.
Zhan contends that his second toy model adequately reflects the structure of this case. However, how plausible is it to assume that Professor Moore is in a position to know that ˹Kq ⊃ Kp˺ holds, all the while she is in a position to know that she is in no position to rule out that ˹Kq˺, and hence q, holds? It is hard to see how she might be in the former position as long as she is in the latter. For, what reason does she possibly have for assuming that, if she was in a position to know (and so it was true) that her criteria are not the most important ones-as she is in a position to know she might-she would then still be in a position to know that Ben is the better candidate? This problem is in no way defused by observing that ˹Kq ⊃ Kp˺ is a material conditional, and so equivalent to ˹¬Kq ∨ Kp˺. For, ex hypothesi, she is in no position to know ˹Kp˺ either. And, if she really was in a position to know that she is in no position to rule out that she is in a position to know q, why wouldn't she thereby have justification for q, just as H1 then predicts? We may ask a similar question about p.
Zhan takes great care to describe the case of Professor Moore in such a way as to make intelligible why the professor selects Ben. The professor's selection criteria may not be best, but she has evidence that their past application has always had the desired results. She is, of course, mistaken about the reason why this has been the case, but she possesses that evidence nonetheless. Its possession-combined with her lack of any evidence for thinking that there is after all no real connection between meeting C M and being a good candidate-already in part explains why she is in no position to recognize that she is in no position to know p. The recognition that Ben meets C M also plays its part, and so does the fact that the professor is in no position to know q. For, plausibly, if she were in a position to know q, she would also be in a position to recognize that she is in no position to know p on the basis of her own criteria. But, or so I contend, this combination of factors likewise explains why the professor has justification for believing p. I see nothing in the description of the case that would refute this claim. 32 In his closing section, Zhan anticipates this line of response, but argues that it falls short of guaranteeing that no other cases can be constructed to which his toy model applies but in which the subject lacks justification for the relevant p. While I don't think that this observation alone, though accurate, is good enough as a challenge to H1, I agree that more needs to be done to defend H1. In the remainder, I will accordingly try to say something more principled in response to the allegation that the right-to-left direction of H1 is threatened by this type of counterexample.
My opponent considers a case α in which ˹¬K¬Kp˺ holds, yet in which the subject at the centre of α has no propositional justification for p. It might seem easy enough to find a p and a subject who has no propositional justification for p-quite independently from whether or not she is in a position to know ˹¬Kp˺. But one shouldn't underestimate what this involves. It may often be straightforward to point to a piece of information, or evidence, present in α, in order to identify what the subject's propositional justification for a given p is in α. It doesn't follow, conversely, that the absence of such a piece of information, or evidence, in α implies that the subject has no such propositional justification for p in α. Here is one reason.
Commonly, propositional justification for a given p is understood as a potential for doxastic justification, viz. as a potential of one's epistemic situation that one is in a position to realize so as to convert it into the basis for a justified belief in p. To the extent that such a basis for belief is taken to consist in a piece of information or evidence-and there are legitimate doubts that this must always be the case-what anyway ultimately matters is its presence once α's potential has been realized. The question then is what it takes for this potential to be realized. Here is one plausible answer: (LINK) For any β ∈ B(α, ˹Kp˺), one has propositional justification for p in α iff one has doxastic justification for p in β, where, as before, for any q and γ, we let B(γ, q) be the set of possible cases close to γ, and centred on the same subject as γ, in which that subject has done the best they are in γ in a position to do in order to decide q (and has done nothing they are not in γ in a position to do; see JAI: 40). Since one is always in a position to do the best that one is in a position to do to decide a given question, even if this may amount to doing very little, we can safely assume that, for any q and γ, B(γ, q) ≠ ∅. According to (LINK), one has propositional justification for p just in case one would end up having doxastic justification for p, were one to do one's best to find out whether one is in a position to know p. With (LINK) in place, my opponent will accordingly be committed to claiming that, with respect to the p and α in question, for some β ∈ B(α, ˹Kp˺), one has no doxastic justification for p in β. The account of doxastic justification proposed in my book (JAI: 108) implies H2 One has doxastic justification for believing p iff one is in no position to know that one doesn't know p.
We can derive (LINK H ) as follows. From the fact that knowing implies being in a position to know, we get (1) If ˹k¬Kp˺ holds in α, ˹K¬Kp˺ holds in α.
As I argued in JAI, we can and should assume that, at least for reasonably idealized subjects, the following likewise holds (JAI: 89): (2) If ˹K¬Kp˺ holds in α, ˹K¬kp˺ holds in α.
Recall the following two principles that I already invoked in the context of my reply to Martin Smith (2022): P1 For any β ∈ B(α, p), if ˹Kp˺ holds in α, ˹kp˺ holds in β P2 B(α, p) = B(α, ˹¬p˺).
This establishes one half of (LINK H ). The other half of (LINK H ) is equivalent to the following principle that I invoked in my reply to Daniel Waxman (2022): P6 For any β ∈ B(α, ˹¬Kp˺), if ˹K¬kp˺ holds in β, then ˹K¬Kp˺ holds in α.
There, P6 was derived, from a criterion for the lack of propositional justification, by means of H1 (cf. crit in my reply to Waxman, 2022). But, unless my opponent has an independent argument against H1, I see no reason why I cannot here appeal to P6. In fact, P6 may be defensible irrespective of whether or not H1 is assumed. So, pending any independent argument against P6, with (LINK) and (LINK H ) in place, my opponent is now forced to say that, with respect to the p and α in question, for some β ∈ B(α, p), in β, the subject at the centre of β has no doxastic justification for believing p although she is in no position to rule out that she in fact knows p. Yet, what might prevent her from ruling out that she in fact knows p that does not, at the same time, make her belief in p (if any) justified? For starters, any such belief would be epistemically blameless. As I argued in JAI, there is also a clear sense in which such a belief would be epistemically permissible (JAI: 126-37). There are, of course, plenty of competing accounts of doxastic justification that might allow prising these factors apart. But that would be an altogether different discussion to have. My opponent might nonetheless want to insist that structurally the same argument can now simply be rerun with doxastic justification instead of propositional justification. What if there is a fact whose obtaining the subject is in no position to recognize and that, at the same time, debars her from ever advancing to an epistemic state in which she recognizes that she doesn't currently know p, all the while she has no doxastic justification for p? We are going in circles. For, why shouldn't this fact itself serve as a ground for the subject's doxastic justification for p? Evidently, this fact itself will not be a piece of information, or evidence, supporting p that is in the subject's possession. For, ex hypothesi, the subject is in no position to know that this fact obtains. But, as I argued in JAI, grounds for doxastic justification needn't be luminous, even if doxastic justification is. Given the connection between the truth of ˹¬K¬kp˺ and one's epistemic blamelessness in, and epistemic permissibility of, believing p (if one does), this just goes to show that doxastic justification for p doesn't require possession of any such distinct piece of information or evidence from which one could infer that one is in no position to rule out that one doesn't know p.