2. Two Williamsonian arguments against introspection principles

As mentioned in the introduction, the most plausible source of contemporary distaste for introspection principles may be traced to the rising popularity of epistemic externalist intuitions and the sympathy such intuitions lend to the margin-for-error principle for knowledge – MAR – formulated by Timothy Williamson. Williamson’s arguments against the introspection principles universally employ MAR and, in turn, his opposition towards them is often monolithically derived from support for the said principle. Yet throughout his writings, Williamson offers at least two logically distinct arguments varying in their scope: the general argument against luminosity of the mental, utilizing the unrestricted form of the margin-for-error principle (as employed in [Williamson Reference Williamson1996] and chapter 4. of Knowledge and Its Limits [2000]) and a more specific anti-KK argument based the agent’s knowledge of this principle (as presented canonically in [2000]’s chapter 5.).Footnote ³ In what follows, I shall demonstrate why the epistemic externalist should, in principle, find the latter argument convincing while suspending their support for the former.

Let us first understand why the margin-for-error principle (MAR) may be at all compelling for the broad externalist audience, not necessarily aligned with Williamson’s “knowledge-first” approach to epistemology. The principle in question is usually formulated as follows:

where α _i+1 is a case sufficiently similar to α _i . Such cases may be understood as subsequent points of time (e.g., consecutive seconds or milliseconds), situations differing in intensities of a given quality (e.g., minimal-degree increases in temperature), or the size of a perceived object (e.g., 1-inch differences in height of a perceived tree); crucially, the difference between any case α_i and α_i+1 may be as small as one pleases, up to the point of absolute imperceptibility. The intuitive sense of MAR is that for a specific belief to count as knowledge, it need not only be true, but safely true, that is true in all cases which unrecognizably differ from ours. In this sense, MAR can be seen as a generalization of the factivity condition for knowledge, which covers similar cases, and expresses (with important caveats) the idea that knowledge is supposed to “track truth.”

Williamson’s first formulations of the principle are used to motivate the epistemicist semantics for vague predicates; later on, he came to hold that much (or all) of our knowledge is inexact, and to all such knowledge states, according to Williamson, MAR applies. Here is how he introduces the concept of inexact knowledge in Vagueness:

To see how MAR applies in those cases given most plausible epistemic externalist views, consider the library case mentioned in this quote. If I am trying to tell how many books there are in a library room by looking, and my eyesight is not perfectly accurate, my belief that there are fewer than 1000 of them, when there are in fact exactly 999, would constitute merely “misplaced confidence,” not knowledge. In Williamson’s analysis, this is because my belief-forming mechanism (perception) is not sufficiently well correlated with the actual number of books for me to tell the difference between what is actually the case (α ₉₉₉ ) and the case in which there is one more book present (α ₁₀₀₀ ). If, on the other hand, I form a belief that there are fewer than 2000 books in the room, this belief counts as knowledge, for it is true in similar cases, while false only in those in which the number of books is 2000 and more, which I would plausibly perceptively distinguish from the actual one and would not form such a belief. These sufficiently similar cases in which p remains true form, in Williamson’s words, a safety “buffer zone” (Reference Williamson2000, p. 19) between the epistemic agent’s cognitive success and failure. In this sense, MAR explicates a condition of “safety” of knowledge and ensures its “reliability,” as present in the traditional externalist analyses of the concept and may be taken to be their formal expression. Hence, even if one believes that knowledge can be given an externalist analysis along the lines of being “safe,” “reliably formed” or “truth-tracking” belief, one may be tempted to see MAR as approximating their preferred view.

Let us now look at the first, more general argument for the thesis that we may label General Anti-Luminosity. According to this thesis, no non-trivialFootnote ⁴ condition C is luminous (L), in the following sense:

Accordingly, if condition C is “knowing that p,” L is equivalent to KK, and if C is “believing that p,” L is equivalent to KB – General Anti-Luminosity entails the rejection of both principles. To set up the argument, we begin by constructing a sorites sequence of ever-so-slightly changing worldly conditions, modeled as a set of cases α₀, …, α_n. If we assume that in α₀ one is in a luminous condition C, we then proceed by utilizing L and MAR to take us to the conclusion that one is in C in any case in the series. To construct the argument against KK along the lines of this general strategy, we need the following schema derived from MAR for knowledge (K-MAR):

and against KB, following schema for belief (B-MAR):

If K-MAR and B-MAR are true, then knowledge and belief are non-luminous, and hence, KK and KB are false, by the following argument. Following a classic example, let us imagine Mr. Magoo looking at the modernist clock with just one hand and no numbers and dashes, and no way of knowing the time besides observing the face of the clock from a distance. By looking at the position of the hand, Mr. Magoo, despite his poor eyesight, nevertheless comes to believe something, e.g., that the time is between 3 and 5, or that it is not 6 o’clock; those beliefs are accurate and intuitively amount to knowledge (suppose it is 4 o’clock). Let the sequence of cases α₀, …, α₁₂₀ be the sequence of imperceptibly different positions of a hand from 4 to 6; for simplicity, take the difference between any position to be 1 minute. Given that Magoo knows and believes that it is not 6 o’clock given its current position (α₀), and, by KK and KB – he knows that he does know and believe that. By K-MAR and B-MAR, he would then also believe and know that it is not 6 o’clock if the hand of the clock was 1 minute closer to 6 (α₁); and here, as well, he would be in a position to know that he does. By applying the same reasoning 120 times, we arrive at the soritical conclusion: Magoo comes to know and believe that it is not 6 o’clock in α_120, that is when the position of the hand shows precisely 6 – which is, of course, a false conclusion given that Magoo’s perceptual abilities are roughly accurate. Since this result is unacceptable and the fact that Magoo knows in α₀ that it is not 6 o’clock is indisputable, MAR excludes the possibility of KK or KB being true. Neither knowledge nor belief is luminous.

The result Williamson arrives at is very strong, given that it generalizes not only over beliefs but also over other mental states that are more commonly thought to be intimately connected with one’s awareness – most prominently phenomenal mental states sometimes labeled “self-presenting” (Gertler Reference Gertler, Smithies and Stoljar2012), such as pains or ticklesFootnote ⁵ . What may seem initially suspicious about Williamson’s argument is that it derives a highly general and strong conclusion concerning our introspective abilities without even mentioning what sort of process governs the acquisition of introspective knowledge. While intuitive examples used to justify MAR usually appeal (as I did two paragraphs above) to our limited abilities of perceptual discrimination, blurry memory, or impreciseness of someone’s testimony, making a similar case for introspection proves at least tricky. However we gain it, introspective knowledge is accompanied by a strength of authority unseen among other species of knowledge. In other words, if any knowledge is not inexact, the knowledge that one believes, perceives, or remembers something to be the case seems to be a good candidate, unlike knowledge of the believed, perceived, or remembered fact. Shouldn’t introspection be exempt from MAR until proven guilty?

Even those generally sympathetic to externalist intuitions have taken issue with this lack of distinction between different sources of knowledge and proposed that MAR should be either content-constrained – and not applied to self-presenting states of mind (Weatherson Reference Weatherson2004, Conee Reference Conee2005) – or applied only to certain source-indexed knowledge operators, like knowledge-by-perception, knowledge-by-memory, and so on (e.g., Dokic, Égré Reference Dokic and Égré2009). These proposals however would plausibly not save any of the introspection principles in question. While the former proposal’s justification would not typically cover knowledge or belief among self-presenting states, the latter proposal is problematic since it severely limits our ability to model the interplay between knowledge coming from different sources (e.g., gaining knowledge-by-inference from applying an inferential rule to premises known by perception and memory). But provisionally exempting self-knowledge from MAR does not require such radical moves. To make introspective knowledge immune to MAR without rejecting the principle altogether, we may simply assume that the syntactic form of p in MAR is limited to that of a sentence free from epistemic or doxastic agent-specific operators (Restricted MAR)Footnote ⁶ :

If we adopt the restricted reading, then K-MAR nor B-MAR no more straightforwardly follow as special cases of the general principle. What then happens to KK and KB? To answer this question, let us turn our attention to the second version of Williamson’s anti-KK argument (as featured in Reference Williamson2000, chapter 5.). Unlike in the first argument, let us now stipulate that although Mr. Magoo’s higher-order knowledge is not sensitive to MAR, he also knows that such an amended version of MAR is true of his first-order knowledge. If Mr. Magoo knows that he knows that it is not 6 o’clock in α_i, by his knowledge of MAR, he may still come to know that it is not 6 o’clock in α_i+1, for he knows that whenever he knows that it is not 6 o’clock in α_k for some k, it remains true in α_k+1. This, in turn, validates K-MAR as applied to knowledge of one’s first-order knowledge (assuming additionally one-premise closure) without assuming anything of substance about one’s introspective abilities – and we may again derive a paradoxical conclusion. In effect, to reject KK via the second route, we trade the unrestricted-order application of MAR for an additional assumption concerning one’s knowledge of MAR.

How far does this argument extend? Quite intuitively, one may still apply it to BK, essentially replicating the above reasoning under the scope of belief operator. If Mr. Magoo believes that it is not 6 o’clock in α_i, by BK he also believes that he knows that; and if he believes even in our restricted, first-order MAR, he still may come to believe that it is not 6 o’clock in α_i+1 by modus ponens reasoning; and so, the argument further replicates. The same does not, however, easily extend to KB. If one rejects the idea expressed by BK – that we are committed to treating our beliefs as knowledge states – then one has no good reason to believe that the restricted MAR principle holds of one’s belief; they may deny that they would still believe that p is true in α_i+1 even if they claim to know that they believe p in α_i. Therefore, accepting the latter, but not the former Williamsonian argument allows for accepting KB (and BB as a result), but not BK and KK. Let me label this combination of views ∼KK+KB thesis.

As shown above, upholding ∼KK+KB without completely abandoning MAR, but just restricting it, is at least consistent. Whether it is a good position to occupy comes down to answering whether self-knowledge of beliefs is sufficiently exact to justify the mentioned restriction. In line with the idea that the truth of these principles ought to be derived from a characterization of our introspective abilities, answering the above questions requires rooting in a sufficiently strong epistemological account of self-knowledge. To fulfill our purpose, such a theory should both (a) vindicate the idea that higher-order knowledge of one’s beliefs is not subject to MAR and at the same time (b) stay faithful to the externalist intuitions about knowledge motivating restricted MAR’s application. The following sections will discuss the prospects, limitations, and challenges of exactly such a project.

3. Transparency of belief and KB

In The Principles of Psychology, William James pronounces that “introspective observation is what we have to rely on first and foremost and always. The word introspection need hardly be defined; it means, of course, the looking into our own minds and reporting what we there discover.” (James Reference James1890, p. 185). While his epistemic optimism towards the infallible reliability of introspection is (as witnessed above) not as popular today, his vision of the source of self-knowledge is more prevalent and widely represented. Starting from Locke and Hume, many philosophers characterized introspection as an “inner sense” or “inward perception,” and knowledge of one’s mental states as essentially perceptual – most notable contemporary examples include Armstrong (Reference Armstrong1968) or Nichols and Stich (Reference Nichols and Stich2003). This vision of introspection also seems to be the one endorsed by Williamson, e.g., when he justifies the intuitive rejection of KB by noting that it is “at least as hard to introspect the time according to [one’s] phenomenal clock as to see the time according to a real clock” (Reference Williamson2014, p. 989), presupposing that we come to know what we believe by observing how things appear to us – presumably, with a “mind’s eye.” It is easy to see why this account of introspection makes the application of MAR to self-knowledge compelling. For if knowing that one is in mental state M comes from one perceiving M, this perception plausibly must admit at least some possibility of error. As such concession would validate MAR when applied to introspective knowledge, I believe this approach to introspection should be abandoned, if we are set on justifying KB.

Among competitors to “inner sense” theories, we might crudely distinguish at least two of their bundles: acquaintance theories, building upon Bertrand Russell’s remarks on acquaintance with the mental, and transparency theories, building on Gareth Evans’ work (Reference Evans1982).Footnote ⁷ As the former’s stronghold field of application is knowledge of one’s phenomenal states (as, e.g., Gertler Reference Gertler, Smithies and Stoljar2012, pp. 102–104 directly concedesFootnote ⁸ ) and the latter’s – knowledge of one’s epistemic states, it is natural to focus on transparency theories when arguing for the luminosity of belief expressed by KB. According to transparency accounts of self-knowledge, the agent’s knowledge of their beliefsFootnote ⁹ is not acquired by anything close to perception of our “inside world,” but rather by attending directly to the content of such beliefs. In the often-quoted words of Gareth Evans, “in making a self-ascription of belief, one’s eyes are, so to speak, (…) directed outward – upon the world”: to answer the question of whether one believes that p, one needs to attend to “precisely the same outward phenomena” as one would when answering the question whether p is true (Reference Evans1982, p. 225). Call this particular process of figuring out what we believe the transparency procedure.

Can utilizing the transparency procedure for one’s beliefs yield knowledge? And, if it can, what kind of knowledge is generated by following this procedure? There are many responses to these two questions present in the literature. Since the purpose of this paper is not to adjudicate between themFootnote ¹⁰ , I shall follow one of the most detailed and popular transparency accounts of introspection proposed by Alex Byrne (Reference Byrne2005, Reference Byrne2018). As we shall see, Byrne’s account is especially promising for our project of finding the theory fulfilling the starting desiderata due to its simplicity, wide scope, and straightforward compatibility with externalist epistemology.

In short, Byrne’s approach to the problem could be summed up as describing the transparency procedure straight-up as a form of inference, thereby grounding our capacity for self-knowledge in a naturalistic competence non-controversially possessed and mastered by human agents. More specifically, such transparent inference is described as utilizing a specific set of epistemic rules, i.e., rules explicitly guiding one’s acquisition of belief expressed by a conditional of the form R (Byrne Reference Byrne2018, p. 101):

Inspired by an earlier proposal of Gallois (Reference Gallois1996), Byrne proposes to describe the agent acquiring transparent self-knowledge of beliefs as an agent trying to follow a BEL rule of the form:

Obtaining higher-order beliefs with the use of BEL is described as follows. On Byrne’s account (Reference Byrne2018, pp. 101–102), one follows a rule if one knows the premises of the rule and tries to follow a rule if one merely (and perhaps erroneously) believes them. To be in a position to apply BEL, i.e., follow or try to follow BEL, one then has to either know or believe that p. Hence one’s higher-order belief resulting from applying BEL – that one believes that p – will come out true whenever one is in a position to apply BEL. By this process, one therefore is guaranteed to have accurate higher-order beliefs about their mental states. Moreover, it grounds our introspective process in inferential abilities which are uncontroversially possessed by ordinary agents and the success of applying the rule is properly limited to the first-person perspective. This, in turn, allows Byrne to account for both “peculiar” and “privileged” access to one’s beliefs (see Byrne Reference Byrne2018, pp. 108–112), i.e., the intuition that one knows that they have certain beliefs differently than others know it of them, and enjoys epistemic authority concerning this fact.

Can higher-order beliefs acquired by applying BEL be, however, adequately described as knowledge states? Many of the criticisms pointed against Byrne’s account (e.g., Boyle Reference Boyle2011) mention the fact that the inference supposedly performed when one applies BEL is very much unlike any other type of inference that is supposed to yield knowledge. There is neither a causal nor evidential connection between the truth of p supposed in the premise and the fact that one believes it, stated in the conditional’s conclusion; p does not make the fact that one believes that p anyhow more likely. Moreover, the explanation of BEL’s success explicitly allows one to acquire knowledge by reasoning through a false step, for p might very well be false while the true conclusion is to be granted the status of knowledge. In short, to use Boyle’s phrase, “the inference is mad” (Reference Boyle2011, p. 230) and does not fit any of the ways standardly associated with justifying an inferential rule.

While Byrne developed detailed responses to such counterarguments (Reference Byrne2018, pp. 121–127), their crucial point, which closely aligns his position with the externalist intuitions we seek, is emphasizing the unmatched reliability of BEL-based inference as a belief-producing mechanism.Footnote ¹¹ The mere honest application of the BEL rule, unlike other rules of inference, secures the truth of the resulting belief irrespective of whether one actually follows or merely tries to follow BEL – in Byrne’s terms, this makes BEL strongly self-verifying. One may usefully compare BEL with another epistemic rule:

While both rules are knowledge-generating, neither of them produces only true beliefs if applied. For example, unlike BEL, CLOSURE is not strongly self-verifying, for one’s false belief in the conditional or its premise may lead one to a false conclusion. In Byrne’s terminology, CLOSURE counts as merely self-verifying, as it yields knowledge only if one follows, not just tries to follow it. The rule like DOORBELL is not even that – one’s knowledge that the doorbell rings does not thereby guarantee the presence of someone at the door. Though arguably both CLOSURE and DOORBELL reliably produce true beliefs, it would be superficial at best to ground their validity in this fact only, as their reliability as epistemic rules is itself grounded in the validity of a logical rule of inference or a pattern of “best explanation” reasoning. A brain in a vat (or a counterpart of ours deceived by an evil demon) would be blameless in trying to follow CLOSURE and DOORBELL even though beliefs produced by applying them would turn out to be at large false, but they would be blameless only because these rules are fashioned after otherwise justified patterns of inference. On the other hand, a brain in a vat trying to follow BEL would still end up with true higher-order beliefs about their beliefs and could be advised to apply BEL on these grounds alone. One may require the above traits – “no false lemmas,” the logical or evidential connection between the premise and the conclusion – from an epistemic rule insofar as the rule is not strongly self-verifying and the risk of error arises. If a rule is strongly self-verifying, this feature alone seems to justify its use on brute, reliabilist grounds.

Does our reliable BEL in some way grounds the validity of the KB principle and allows for exempting knowledge gained through it from the scope of MAR? To answer this, let me first propose an adjusted version of KB (Adjusted KB) rooted in the assumption that BEL is a proper epistemic rule:

The adjustment (modeled after Das and Salow’s [Reference Das and Salow2018, p. 8] modification of KK that shall be discussed in the next section) counters the standard objections to introspection principles, such as a purported lack of the concept of belief necessary to form the higher-order belief at all or the lack of computational power necessary to perform relevant deductions. On the other hand, however, this modification makes more vivid the similarity between KB and other principles of doxastic and epistemic logic that may be grounded in epistemic rules as described by Byrne. If the above-mentioned standard objections are sound, very similar ones could be raised, e.g., against epistemic or doxastic closure principles. Surely, an infant might not be able to apply CLOSURE to the premises p and p implies q because of their conceptual or computational limitations; one might also be too tired or distracted to exercise their usually possessed ability. What matters in these considerations is the agent’s competence, not (the limits of) their performance. Byrne’s theory, by positing that knowledge of one’s beliefs is rooted in one’s inferential, not perceptual, abilities, then allows one to analogize between KB and other less controversial principles of epistemic-doxastic logic, such as epistemic or doxastic closure. Its crucial prediction is that once one believes that p, then one knows all there is to know to know that they believe it, just as once one knows p and that p implies q, then one knows all there is to know to know that q, given they have relevant inferential capacities.Footnote ¹²

To decisively answer whether such justification is enough to secure the needed MAR exemption, one may still consider the following modification of Williamson’s argument sometimes directed specifically against KB. Let us imagine the case, in which one’s confidence in p very gradually declines over time t ₀ – t _n , so that in the moment t ₀ one believes that p, but does not at t _n. It is independently plausible that (however we gain knowledge of our credence) our ability to determine our level of confidence is not precise. Hence, on most occasions, one would be unable to discriminate between having credence r at t _i and a slowly lower credence r’ at t _i+1 . If belief is, as many take it to be, intimately connected with sufficiently high confidence, this seems to lend independent plausibility to B-MAR, as knowing that one has high confidence in p at t _i seems to require that one maintains high confidence in p at t _i+1. To adjust this reasoning to our context, let us now stipulate that the agent in question is able to apply BEL at any of t ₀ – t _n moments. Isn’t it then possible that at some threshold point, one mistakenly applies BEL because of their misplaced confidence in p – which, even though still high, just barely falls short of full belief – and come to falsely believe that they believe that p? To put it differently: is it possible that the agent not even tries, but merely tries to try to follow BEL? If such cases are possible, then we should reject the Adjusted KB.

To see how one could reject this argument on Byrne’s account, let’s note that this reasoning rests on the premise that it is the agent’s having high confidence in p, not just their belief that p, triggering their disposition to apply the BEL rule. In line with the logic of the argument, there must be both the largest i such that at t _i a believes that p, while at t _i+1 a no longer believes that p, as well as the largest j such that at t _j a is disposed to apply BEL to the premise that p but is not disposed to do so at t _j+1, and finally the largest k such that at t _k a has (sufficiently) high confidence in p, but does not at t _k+1 . For the argument to be sound, it needs to be established that i > j ≥ k, that is, that one is disposed to still apply BEL once they lose the belief, but retain sufficiently high confidence in p. This ordering is however not at all obvious. Note that on Byrne’s characterization, to apply the BEL rule to the premise that p, one needs first to judge that p simpliciter, not only that p is likely or even that p is highly plausible (unless one wishes to find out whether they believe that p is likely). Therefore, to acquire second-order beliefs through BEL, the agent needs to be disposed to judge that p and use p as a premise in theoretical reasoning (namely – BEL). These dispositions are frequently taken to be important properties of belief states (e.g., Audi Reference Audi1994, Silins Reference Silins, Smithies and Stoljar2012, Smithies Reference Smithies2019, also: Williamson Reference Williamson2000, pp. 46–47); and even if we are not full-on dispositionalists about belief, one may plausibly claim that exhibiting such inferential dispositions is essential, and so i = j. Adopting this premise does not entail rejecting the claim that belief is intimately connected with confidence. A fully committed Lockean might hold that belief is equivalent to some subjective credence above some threshold r, and grant that on many occasions one is not in a position to know what precise confidence one has in a certain proposition p. Yet until they do not also maintain that at the same time, one is in a position to know what the appropriate threshold isFootnote ¹³ , it is consistent to maintain that one may figure out by applying BEL whether one believes p, without knowing what precise credence underlies this belief.

If such judgment-generating and inferential dispositions are essential for belief and Byrne’s characterization of the transparency procedure is right, it is then quite easy to see how the B-MAR principle may fail, and KB be true. On Byrne’s inferential picture, coming to know that you believe that p requires just exercising the relevant dispositions, not knowing their exact boundaries. Hence, even though one cannot know exactly where one’s confidence threshold for belief lies on a particular occasion, this fact does not justify the B-MAR principle. If a believes that p at α _i but not at α _i+1 , then a is disposed to judge that p and apply BEL to the premise that p – and come to know that they believe that p – only at α _i . As one is in a position to use the BEL rule only in success cases, the threat of error does not arise.Footnote ¹⁴ Byrne’s theory allows us then to account for both of our starting desiderata. It secures one’s strong introspective access to one’s beliefs as a source of knowledge by appealing to the reliability of the transparency process (thereby validating the KB rule) and on the other hand can avoid the margin-for-error considerations concerning the process’ reliability, which allows to describe our second-order knowledge generated by BEL as peculiar and distinct from first-order knowledge gained from experience or inference. Let us now see how the above discussion translates to the debate surrounding KK.

4. Transparency of knowledge and KK

One thing I demonstrated in the previous section was that Byrne’s transparency account provides an externalist case for KB; what needs to be shown is that this case extends also to ∼KK+KB. I am not the first one to point out the connection between Byrne’s theory and the justification for introspection principles – the first such case was made by Das and Salow (Reference Das and Salow2018).Footnote ¹⁵ Yet, the focus of their approach was to use the transparency account to defend KK. As I wish to do the exact opposite, I need now to rebut their argument. In the course of this rebuttal, I will also propose some independently plausible augmentations to Byrne’s transparency account and demonstrate how it may ground an externalist argument against KK different from that of Williamson.

In making their case for KK, Das and Salow do not make use of BEL but the epistemic rule KNOW, proposed by Byrne (Reference Byrne, Smithies and Stoljar2012):

On their account, if we assume that the agent is in a position to apply the KNOW rule and follows it (in Byrne’s sense of rule-following), one’s resulting belief will be guaranteed to be true in the same way as the belief produced by BEL. Because unlike BEL, KNOW is not strongly self-verifying, i.e., the resulting belief comes out true only if the agent actually follows KNOW and not merely tries to follow it, Das and Salow claim that the belief resulting from the application of a self-verifying epistemic rule qualifies as knowledge only if the agent qualifies as following the rule in Byrne’s sense. Hence, whenever one comes to believe that one knows that p, although one does not know that p, their belief is a result of merely trying to follow KNOW which, according to Das and Salow, is insufficient to generate knowledge (Reference Das and Salow2018, p. 10). If one, on the other hand, follows KNOW, one is epistemically guaranteed to succeed – and KK (in a qualified form analogous to Adjusted KB) gets validated. Das and Salow support their case for this restriction by likening the knowledge obtained by following Byrne’s transparent rules to the knowledge obtained by following rules like CLOSURE (Reference Byrne2018, p. 5).Footnote ¹⁶ Their argument is fairly simple: given that CLOSURE, like KNOW, does not generate knowledge if one does not know the premises p and p entails q, one’s epistemic base for gaining knowledge through epistemic rules needs to be following, not trying to follow a rule (Reference Byrne2018, pp. 11–12). Because we should count all sufficiently similar rules governing the acquisition of inferential knowledge as having the same type of epistemic base (by what Das and Salow call Generality Constraint [Reference Byrne2018, p. 11]), it follows that all self-verifying epistemic rules should count as knowledge-generating only if the agent follows them, not merely tries to; this, in turn, gives them the right to claim that only beliefs resulting from following KNOW may count as knowledge, and validate KK.

As we may already see, the case for KK Das and Salow proposes differs significantly from the simple reliabilist justification provided above for Adjusted KB based on BEL. Since KNOW produces true beliefs only if followed, the question of why it could be said to be knowledge-producing no longer can be answered on the grounds of the exceptional ability to produce only true beliefs. A brain in a vat will no longer easily gain higher-order knowledge by applying KNOW; it is also quite plausible that a purely statistical assessment of reliability will qualify only some – more fortunate or epistemically cautious – actual world agents as forming reliably true beliefs by applying KNOW. Their justification for why KNOW may count as a knowledge-generating rule needs then to be substantially different. BEL’s status as a good epistemic rule is made by appealing to the fact that it is strongly self-verifying, which implies its infallible reliability in producing true beliefs. A similar case cannot be made for KNOW because of its self-verifying status, for it is simply not the case that all self-verifying epistemic rules deserve a place in our cognitive repertoire. If we agree with Williamson even on Restricted MAR, then the following becomes an example of a self-verifying epistemic rule:

since knowing that p is true at α _i (i.e., following SORITES) guarantees that p is true at α _i+1. Yet, as we may readily see, thoroughly applying SORITES to one’s beliefs yields detrimental results in the form of a variety of fallacious soritical conclusions. The crucial difference between SORITES and CLOSURE in this respect is that while CLOSURE is knowledge-conducive, SORITES is not. Though if one knows that p is true at α _i , then their SORITES-based belief that p is true at α _i+1 would be true, there is a point at which it would fail to be knowledge. While KNOW guarantees that, if followed, it produces true beliefs, we have no justification as to why such beliefs should constitute knowledge.

Should we be skeptical of KNOW then? Given the fact that the rule neither shares its reliability with BEL, nor is independently justified (by logical principles, like CLOSURE, or abductive inference, like DOORBELL), I think the answer is yes. A more straightforward reason for such skepticism is that, if accepted, KNOW lends support to the (appropriately adjusted) BK principle – which, as we saw above, is inconsistent with Restricted MAR. In fact, much of the intuitions supporting KNOW, such as that we are “rationally committed to treating our beliefs as knowledge” or that “belief feels like knowledge,” seem to parallel those offered in support of BK. Footnote ¹⁷ If we, however, are on the right track in justifying introspection principles by appealing to Byrne’s epistemic rules while acknowledging the plausibility of MAR for first-order knowledge, these intuitions need to be treated with caution. For while we might concede that in ordinary cases one cannot internally distinguish between merely believing and knowing that p, it does not follow that one always takes one’s beliefs to be knowledge states, just that one usually cannot determine whether a certain belief of theirs constitutes knowledge. From an externalist point of view, it is perfectly coherent to hold that we can be agnostic concerning our beliefs constituting knowledge on a particular occasion. Think again of Mr. Magoo observing the clock. When he comes to believe at 5.50 that it is not 6 o’clock, he may still plausibly suspend judgment as to whether his eyesight is good enough to grant him knowledge that it is not 6 o’clock, and yet fully believe that the world is the way it presents itself to him, even if only for the fact that there is no better data he could rely on in his reasoning. If his skepticism towards his perceptual abilities is unwarranted, his belief constitutes knowledge, if it is warranted – then it does not. But while he has no means to assess which of the scenarios he is in, and hence cannot truly distinguish “from within” whether he merely believes or knows that it is not 6 o’clock, it does not seem like a form of irrationality on his part to stick to believing what seems to him true.

What should we then do with KNOW if we reject the intuition of the first-person “knowledge-belief collapse”? I suggest that staying true to externalist intuitions, we may propose the following Byrne-style epistemic rule to guide our self-ascriptions of knowledge:

KNOW* is a transformed CONFIDENCE rule, proposed by Byrne (Reference Byrne2018, pp. 119–121) to guide our self-ascriptions of high (and low) confidence in belief. By “high epistemic credentials” ascribed to the belief I mean, like Byrne, relevant evidence about the source of the belief and reliability of the belief-forming mechanism. Importantly, unlike in KNOW, self-ascriptions of knowledge are here “broadly Rylean,” i.e., based on the assessment of the reliability of our cognitive faculties, perception, memory, etc. that is in principle available also to others. Consider, for example, my beliefs that the Warsaw Ghetto Uprising took place in 1943 and that it will rain in 3 days. Even though both are my beliefs, I am more inclined in giving the first, but not the second one the status of knowledge. Why? In case of the first belief, I may consider a variety of factors that make such status plausible: (a) that someone trustworthy have told me that, (b) that I am generally well-acquainted with World War II history, (c) that I’m an inhabitant of Warsaw, (d) that I am good at remembering dates. All these factors speak to my reliability in forming and maintaining the belief in question, so they are directly relevant evidence for my belief constituting knowledge on the externalist picture – they objectively make it more likely that my belief constitutes knowledge.Footnote ¹⁸ On the other hand, I can think of much fewer relevant factors giving high epistemic credentials to my belief about the future weather; even if I remember checking a weather app on my phone, my assessment of the long-term reliability of meteorology is rather poor. Hence, even though this belief may constitute knowledge (the testimony of the app may be reliable), it is not subjectively plausible, and hence I would be well advised to suspend judgment on the matter. Similarly, a person who knows all these relevant factors may be in principle just as justified in granting or denying my belief a status of knowledge – and hence I have no peculiar or privileged access to the epistemic status of my belief.

Given the interpretation of MAR advanced here, it is natural to include safety from error within a reasonable margin as one of the factors that belong to the assessment of the epistemic credentials of our belief. This allows to accommodate the Williamsonian intuition that the imprecision built into our belief-forming mechanisms may prevent us from knowing that we know something even if we do know it. Mr. Magoo may very well come to believe that it is not 6 o’clock by looking at the clock showing half past 5 from a distance, but still, after reflecting on his poor sight and discrimination abilities, refrain from believing that he knows that, as his epistemic credentials do not meet the desired standard. Yet, if this reflection leads him to believe that his perceptual faculties are appropriately fine-tuned and the risk of error is minimized, he may come to believe (and, if he is right – know) that he knows that it is not 6 o’clock.

All this seems quite natural for an epistemic externalist. If an externalist takes themselves to know that knowledge that p requires sufficient margin-for-error, they may be expected to be more conservative in their self-ascriptions of knowledge than that of belief. Knowing that one knows, unlike knowing that one believes, requires also knowing that their first-order belief meets relevant externalist criteriaFootnote ¹⁹ , which feature here under the label of “high epistemic credentials.” Only when one believes that their belief satisfies these criteria, they may conclude by KNOW* that they know that p. Unlike BEL and KNOW, KNOW* is not self-verifying – the protagonist of the Gettier scenario would arrive at a false conclusion that they know that p despite following KNOW*. However, in opposition to these rules and much more like DOORBELL, KNOW* is abductively justified: that one believes that p with high epistemic credentials is good evidence for them knowing that p. Quite easily we can then classify KNOW* as knowledge-generating, without the troubling consequence of collapsing belief and knowledge from the first-person perspective, upholding skeptical conclusion that we are never in a position to know that we know anything or denying the strong introspective access to one’s own beliefs. Hence, the epistemic externalist is offered a way out by modifying Byrne’s theory to better reflect their intuitions concerning justified self-ascriptions of knowledge, while retaining its strength with respect to justification of self-ascriptions of belief. If one follows Byrne in taking introspective knowledge that one sees (Reference Byrne, Smithies and Stoljar2012) or remembers (Reference Byrne2018, pp. 184–195) as inferential, one may readily apply the above considerations to such cases as well. Given that seeing and remembering are factive mental states, the rules SEE and MEM Byrne proposes should be also weakened in a manner similar to KNOW*, while their strongly self-verifying counterparts would give one instant knowledge of the fact that one perceives or has a recollection experience.

To finish this section, let me briefly discuss the larger import this discussion has on related issues of application of introspection principles. Why go through all this trouble, you may ask? After all, even if I succeed in defending ∼KK+KB, the exercise may be just that of coherent expression of a certain class of epistemic intuitions, without any practical or theoretical import. It may seem that giving up on KK may be too much for all the relevant practical purposes. Contemporary defenders of KK (e.g., Greco Reference Greco2015a) usually point out its two important applications: the treatment of certain epistemic Moore-like paradoxes and the natural explanation of how groups of collaborating agents might achieve common knowledge. Let me now briefly sketch how upholding ∼KK+KB might allow us to mitigate worries connected with losing these applications, and why KB, not KK, actually might be the principle needed for the job.

Even though, obviously, certain results hold only if KK is assumed, many similar results hold if we accept KB instead. In doxastic logic, the BB principle often allows us to prove theorems that are obtainable in epistemic logic under the assumption of the factivity of knowledge (expressed by axiom T), but not the weaker assumption that it is merely consistent (expressed by axiom D). Moore-paradoxical sentences, i.e., sentences of the form p & ∼B _a p or p & B _a ∼p and similarly puzzling “anti-expertise” sentences of the form p ↔ ∼B _a p can be proven to be rationally unbelievable for any KD4-doxastic agent a, while their epistemic analogs – p & ∼K _a p and p ↔ ∼K _a p – are unknowable simply for epistemic KT-agents.Footnote ²⁰ Therefore, if one wishes to maintain that there is a significant analogy between the unknowability of certain epistemic Moorean constructions and the unbelievability of their doxastic counterparts, one may be advised to accept KB (and BB as a result). As for common knowledge (spelled out in terms of an infinite hierarchy of mutual knowledge about epistemic states of others), it is quite unclear in the first place whether this assumption needs to target knowledge as a first-order mental state. We may, and often do, rationally coordinate even if we all believe something false; whether our first-order mental state is a factive state usually does not explain our success in communication, collaboration, or competition. In this vein, many philosophers write of “knowledge”-part of “common knowledge” as a concept “which can accommodate the medieval ‘common knowledge’ that the Sun circles the Earth” (Heal Reference Heal1978, p. 116) and treat the assumptions of common knowledge and common belief as essentially interchangeable for all intents and purposes (Lederman Reference Lederman2018). For the latter, but not the former, the assumption of BB is all that is required. While there has been some doubt towards such interchangeability (see e.g., Yalcin Reference Yalcin2024), this skepticism typically targets the fact that common belief allows for false or Gettiered higher-order beliefs, which disables coordinational success or makes it a matter of pure luck. Opting for stronger KB escapes this problem since in the standard way it secures common knowledge of first-order belief if agents assume that their knowledge is symmetricFootnote ²¹ , as in normal doxastic-epistemic logic KB entails the following principle:

where “knowsⁿ” means “one knows that” repeated n-times for any n. Thereby KB disallows Gettiered higher-order beliefs and responds to this objection, at the same time accommodating the intuition that a group of agents may rationally coordinate or compete given they all commonly know what all agents believe even if such beliefs are false or unjustified.