2.1. Inexact belief

Williamson’s argument in the previous section is directed specifically at the luminosity of knowledge. While it is not intended to apply more broadly, it is natural to ask whether the argument extends to belief, given the close connection between the two states.

Mr. Magoo is a reasonable average observer and cautious believer. For no natural number i, he does believe that the tree is i inches tall; and for many natural numbers i, he does not believe that the tree is not i inches tall. Moreover, for no natural number i, the tree is i + 1 inches tall, while he believes that it is not i inches tall.Footnote ⁵ Reflecting on his beliefs, he believes this latter general fact:

1. 1. Mr. Magoo believes that (if the tree is i + 1 inches tall, he does not believe that it is not i inches tall).

About Magoo, we make three assumptions. First, the uncontentious:

1. 2. Mr. Magoo believes that the tree is not 1 inch tall.

We further assume that belief is bright to Mr. Magoo, stated in the following weakened form:

1. 3. BB. For any pertinent p, if Mr. Magoo believes that p, then he believes that he believes that p.

His courses on logic made him sharp enough to satisfy:

1. 4. C*. If p and all members of the set X are pertinent, p is a logical consequence of X, and Mr. Magoo believes each member of X, then he believes that p.Footnote ⁶

Again, we assume:

1. 5. The tree is 666 inches high.

The argument is analogous, except for the last step. We end up with the conclusion:

1. 6. Mr. Magoo believes that the tree is not 666 inches tall.

However, since belief is not factive, this does not contradict the fact about the tree.

We may even grant:

1. 7. ∼∃i (Mr. Magoo believes that the tree is i inches high)

At the same time, it remains highly plausible that:

1. 8. Mr. Magoo believes that ∃i (60 < i < 6000 inches and the tree is i inches high).

These are perfectly consistent. One might surely believe that some fair lottery ticket is the winner, while for no particular ticket, one does believe that it is the winner.Footnote ^{7 ,} Footnote ⁸

Similar considerations show that the argument does not bear on the luminosity of belief, provided that we do not assume KK. To show this, we make the same assumptions as in the former argument, with the exception of KK , which we substitute for:

KB. For any pertinent p, if Mr. Magoo believes that p, then he knows that he believes that p.

1. 1. Mr. Magoo knows that (if the tree is i + 1 inches tall, he does not know that it is not i inches tall).

2. 2. Mr. Magoo knows that the tree is not 1 inch tall.

3. 3. C. If p and all members of the set X are pertinent, p is a logical consequence of X, and Mr. Magoo knows each member of X, then he knows that p.

4. 4. The tree is 666 inches tall.

We also assume:

1. 5. If Mr. Magoo knows that p, then he believes that p.

Unsurprisingly, the argument flounders where KK was employed. Applying 5. to 2., we conclude that

1. 6. Mr. Magoo believes that the tree is not 1 inch tall.

Now applying KB to 6., we arrive at:

1. 7. Mr. Magoo knows that he believes that the tree is not 1 inch tall.

Since 6. is logically compatible with Mr. Magoo not knowing that the tree is not 1 inch tall, we cannot apply C to conclude that Mr. Magoo knows that the tree is not 2 inches tall. Of course, adding the plausible assumptions on knowledge will suffice for this and reasonably close steps, but for some sufficiently high i, the assumptions cease to be plausible enough to restore the argument.

3.1. Defending knowledge-safety: First pass

How could one argue for K-SAFE – and the more general scheme of which it is an instance, Knowledge-safety?

A first natural step is to consider:

Belief-Safety . ∀α (at α: Kp → ∀β(β is similar to α → at β: (Bp → p)))

According to Belief-Safety, one knows that p in a situation only if in every possible similar situation in which one believes that p, it is true that p. The principle arguably captures an intuitive application of the idea that knowledge requires a modal notion of safety: one’s belief qualifies as knowledge only if it is not hostage to falsity in every similar scenario (Pritchard Reference Pritchard2005: 146f.).

Belief-Safety does not suffice for Knowledge-Safety, since it requires only that in every scenario in which one believes that p, it is true that p. To derive K-SAFE, one might resort to Srinavasan (Reference Srinivasan2013: 302):

Doxastic-Disposition for feeling cold

∀α_i(at α_i: B(one feels cold) → ∃β_{i + 1} (β_{i + 1} is a phenomenal duplicate of α_{i + 1} & at β_{i + 1}: B(one feels cold))).

Accordingly, if one believes that one feels cold at a situation α_i, there exists a possible situation β_{i + 1} in which one’s cold feelings are exactly the same as in the next actual situation α_i+1 and in which one believes that one feels cold. That one’s cold feelings are exactly the same in those scenarios, that α_{i + 1} and β_{i + 1} are ‘phenomenal duplicates’, means: at α_{i + 1} one feels cold if and only if at β_{i + 1} one feels cold.

Doxastic-Disposition and Belief-Safety (where ‘one feels cold’ instantiates ‘p’) entail K-Safe. Suppose K(one feels cold) at α_i. Since knowledge implies belief, B(one feels cold) at α_i. Modus ponens applied to Doxastic-Disposition implies that there is a possible scenario β_i+1 similar to α_i in which B(one feels cold). Thus by Belief-Safety, at β_{i + 1}: one feels cold. Since β_{i + 1} and αi + 1 are phenomenal duplicates, at αi + 1: one feels cold. Conditionalization discharges the initial supposition, and we obtain K-SAFE.

Doxastic-Disposition is an instance of a more general principle:

Doxastic-Disposition (general) . ∀α(at α: Bp → ∀β(β is similar to α → at β: one is disposed to Bp))

That is, if one believes that p at a scenario, then in every sufficiently similar scenario, one is disposed to believe that p. Although the similarity relation allows for a familiar degree of flexibility, the principle draws its plausibility from a natural requirement: that similar situations preserve the bases on which the relevant beliefs are formed – or would be formed. In the example above, this means, for instance, that one’s feelings of cold are preserved across the relevantly similar cases.

However, this derivation of K-SAFE falls prey to a reply raised by many in response to Williamson’s argument (Leitgeb Reference Leitgeb2002; Weatherson Reference Weatherson2004; Berker Reference Berker2008; Ramachandran Reference Ramachandran2009). Namely, that it presupposes the falsity of:

Phenomenal-Belief Constitution (CON). ∀α (one attends to one’s cold feelings at α→ (at α: one feels cold ↔ B(one feels cold)))

That is, in every situation where one attends to whether one feels cold, one’s beliefs about feeling cold track one’s cold feelings. Although the important point is the left-to-right implication across possible situations, this is usually assumed to be supported by a more intimate essential or constitutive relation between one’s feelings and the beliefs about them.

If CON holds, then Doxastic-Disposition is false with respect to the case at hand. For, as the case is described, there is an α_i which is the last where one feels cold; thus by CON the last where one believes that one feels cold. But then there cannot be a possible β_{i + 1} which is a phenomenal duplicate of α_{i + 1} – thus, where one does not feel cold – and yet one believes that one feels cold, since that contradicts CON.

3.2. Defending knowledge-safety: Second pass

Despite numerous replies that rely on CON, brace yourself: Williamson does not employ that strategy in defending K-SAFE, or Knowledge Safety more broadly. Rather than invoking Belief-Safety, Williamson formulates a safety condition for knowledge in terms of degrees of confidence. To a first approximation, one’s degree of confidence in a proposition reflects the extent to which one is disposed to rely on it in practical reasoning – that is, to treat it as a premise in deliberation.

Degrees of confidence are not credences, or degrees of subjective probability reflecting betting behavior. Williamson describes the following case to illustrate the difference (see also Reference Williamson2000: 98-99):

Lottie is not at fault for refraining from believing that the ticket will lose – even if she assigns it a high probability. The case is consistent with her having no inclination whatsoever to rely on that proposition in practical reasoning, that is, with her having confidence 0 in it.

Though one might be tempted to respond by equating outright belief with credence 1, the following case runs against the temptation:

In cases involving an infinite number of cases, assignments of probability 1 do not correspond to outright belief, since it is consistent with such an assignment that the claim does not hold (cf. Williamson Reference Williamson2007; see Haverkamp and Schulz Reference Haverkamp and Schulz2012 for discussion).

The underlying idea of the principle from which K-SAFE might be derived is that knowledge requires absence of nearby misplaced confidence:

Confidence-Safety. If in case α one knows with confidence c that p, then in any sufficiently similar case α* in which one has an at-most-slightly-lower degree of confidence c* that p, it is true that p.

To derive K-SAFE from Confidence-Safety, we need an additional premise. Srinavasan (Reference Srinivasan2013) suggests (she labels it ‘Conf*’):

Bridge Principle (‘Bridge’). If at α_i one has degree of confidence c that one feels cold, there exists a sufficiently similar possible case β_{i + 1} in which one’s cold-feelings are a phenomenal duplicate of one’s cold-feelings at α_{i + 1} and in which one has an at-most-slightly-lower degree of confidence c* that one feels cold.

Bridge has two main elements: it asserts on the one hand that if one has degree of confidence c that one feels cold at α_i, there is a possible scenario β_{i + 1} which is such that one feels cold at α_{i + 1} if and only if one feels cold at β_{i + 1} (this is intended as a consequence of one’s cold feelings being a phenomenal duplicate of one another in these scenarios); and on the other hand, at that same β_{i + 1}, one has an at-most-slightly-lower degree of confidence c* that one feels cold.

Suppose at α_i one knows that one feels cold, and thus has a sufficiently high confidence c that one feels cold. By Bridge, there is a possible β_{i + 1} which is such that at β_{i + 1}: one feels cold if and only if at α_{i + 1}: one feels cold; and at which one has an at-most-slightly-lower confidence c* that one feels cold. Since at α_i one knows that one feels cold, by Confidence Safety at β_{i + 1} one feels cold. By Bridge, at α_{i + 1} one feels cold. Hence, K-SAFE.

Crucially, this defense of K-SAFE is independent of the falsity of CON above. If a constitutive relation between feelings of cold and beliefs about them holds, then at exactly the point where one ceases to feel cold, one stops believing that one feels cold. As we saw, if the safety requirement for knowledge would amount simply to absence of nearby untrue belief, CON blocks the argument. But Confidence-safety amounts to more than this, namely absence of nearby misplaced confidence (even if it falls short of outright belief). As Srinavasan (Reference Srinivasan2013) writes:

Thus, Confidence-Safety offers a robust defense of the broader argument against luminosity.Footnote ¹⁰ However, as we shall see, even if this response succeeds with respect to first-order attitudes, it leaves room for a luminist reply in the case of higher-order beliefs – particularly if a constitutive link can be established between belief and the underlying degrees of confidence.

5.1. Translucent minds

If the argument is to be resisted, we should turn to specific accounts of self-knowledge – those that seek to explain how we form beliefs about our own beliefs. The views considered in this section need not rely on luminist preconceptions. They are typically aimed at making sense, on the one hand, of the peculiar access one seems to have to one’s mental states: that we seem to know what we believe in a way that differs markedly from how we know other things, including the mental states of others; and of the privileged stance we appear to occupy with respect to our doxastic lives, on the other. The peculiarity as such is neutral on the epistemic status of higher-order beliefs. The privilege might fall short of luminosity, and in at least some degree should be recognized even by anti-luminists. Though we may not be luminous believers, we are far from fumbling in the dark: we are, by and large, impressively good at knowing what we believe. The principles we’ll examine shortly rely only on how we form higher-order beliefs and are therefore independent of whether Luminosity* holds.

A recurring theme across otherwise divergent positions is the notion that first-order beliefs are, in some sense, necessarily linked to higher-order beliefs about them. Stated in such general terms, the idea lends itself to several distinct formulations.

Transparency theorists hold that, when deciding whether they believe that p, the typical way to proceed is by considering whether p is true. As Evans famously puts it – in a passage by now cited almost to exhaustion – ‘in making a self-ascription of belief, one’s eyes are, so to speak, or occasionally literally, directed outward – upon the world. (…) I get myself in a position to answer the question whether I believe that p by putting into operation whatever procedure I have for answering the question whether p’ (Evans Reference Evans and McDowell.1982: 225). The core idea is that we come to believe that we believe p by way of considering p itself and taking a positive stance toward it. In self-ascribing belief, attention need not be directed toward the believer or their internal states, but only toward what the first-order belief is about.

The transparency idea can be implemented in different ways. A higher-order belief might be formed inferentially – for example, by following a rule such as: if p, then believe that you believe that p (Byrne Reference Byrne2018). Alternatively, it might be formed non-inferentially – for instance, if the higher-order belief arises from the same (not necessarily inferential) basis as the first-order belief that p (Fernández Reference Fernández2013).

Another related family of views connects higher-order beliefs to first-order beliefs in a different way. Peacocke (1994, Reference Peacocke1999, ch. 5) develops an account of self-knowledge in which phenomenal conscious states and events serve as non-inferential justifiers of higher-order states and events that share part of their content – assuming the subject possesses the relevant concepts. On this view, for example, having the apparent memory that Williamson was born in Sweden can itself justify the judgment that Williamson was born in Sweden. One need not observe the memory, infer the content of the judgment from it, or let the justification reside within the judgment itself.

For higher-order beliefs, the typical justifiers are judgments, which may themselves be non-inferentially grounded in further conscious states or events. For instance, one’s judgment that Williamson was born in Sweden can justify the self-ascription of the belief that Williamson was born in Sweden, itself a judgment which in turn justifies the belief that one believes that Williamson was born in Sweden. The justification of a higher-belief is described by a process that typically has four parts: first, one undergoes a conscious episode or state with the content that p; taking the episode at face value, one then judges that p on that basis; then that judgment non-inferentially justifies the self-ascription of the belief that p; finally, that self-ascription justifies the belief that one believes that p.

Yet a last family of views holds an even more intimate, constitutive, correlation between first-order and second-order beliefs. Wright in earlier work, for instance, suggests that one’s (first-order) beliefs are constituted by what one takes themselves to believe, that is, their higher-order beliefs, under appropriate, optimal conditions (Wright Reference Wright1998; Khani Reference Khani2020). These include having the concepts in question, not being subject to self-deception, and so on. Facts about belief, and mental contents more broadly, are accordingly response-dependent, in an analogous way as being red might be taken to be dependent on how observers respond to objects in appropriate, optimal conditions. As their response-dependence attaches to the nature of belief, that constitution is meant to hold necessarily.

A similar result is pursued by Shoemaker (Reference Shoemaker1996), who, however, grounds the necessary correlation between first-order and higher-order beliefs in a necessary connection between the states themselves – more precisely, in the partial identity of their neural realizations and the overlap in their causal profiles.

It hardly needs saying that this brief overview does not do justice to the intricacies of the individual accounts; its aim is merely to illustrate a common tendency – namely, to posit a necessary connection, sometimes conditional on some conditions we may realistically presume the subject in the argument satisfies, between second-order beliefs and the first-order states they are about.

Applied to degrees of confidence, the idea shared by those views might be spelled out in two main ways (we assume that degrees of confidence assume values in the real interval [0,1]):

Higher-Order Constitutivism (HCON). For some c, necessarily,

1. i) one has confidence 1 in Bp if one has at least confidence c in p;

2. ii) one has confidence 0 in Bp otherwise. (cf. ‘Vindication’ in Goldstein and Waxman Reference Goldstein and Waxman2020: 9)

‘c’ might vary with the range of substitutions of ‘p’. HCON draws on those accounts being accounts of our knowledge of outright belief, marked by a threshold of confidence which one’s higher-order beliefs reliably track.Footnote ¹¹

Alternatively, one might let degrees of confidence in Bp mirror the degrees of confidence in the embedded propositions, thus obtaining:

Higher-Order Constitutivism * (HCON*). Necessarily, for all c: one has confidence c in Bp if and only if one has confidence c in p. (Cf. ‘Calibration’ in Goldstein and Waxman Reference Goldstein and Waxman2020.)

According to HCON*, one’s confidence in one’s belief exactly matches one’s confidence in the proposition believed.

HCON and HCON* are consistent with means of arriving at second-order beliefs that do not turn on first-order ones – by testimony of a trusted therapist, for instance. One might relativize them to a specific way of knowing to render this explicit. Importantly, the methods they describe plausibly align with how the person in the argument forms second-order beliefs.

What evidence is there that we conform to either HCON or HCON*? Any defense of the specific views on higher-order belief discussed earlier already lends support to this contention. Yet typical armchair philosophy – or psychology confined to the conceptual, foundational level – can hardly be expected to settle the issue. Still, taken together, these diverse views may provide cumulative evidence for either of the confidence profiles: the fact that multiple, well-developed accounts have been advanced by reasonable theorists, each underwriting either HCON or HCON*, constitutes, however inconclusive, considerable evidence in their favor.

Can HCON and HCON* be tested empirically? Recent research on metacognition has established a strong connection between confidence and metacognitive sensitivity (De Gardelle and Summerfield Reference De Gardelle and Summerfield2011; Spence et al. Reference Spence, Dux and Arnold2016; Boldt et al. Reference Boldt, de Gardelle and Yeung2017; Fleming Reference Fleming2024). In this context, metacognitive sensitivity denotes the ability to reliably monitor one’s own cognitive functions – for instance, perceptual discrimination – while confidence refers to the degree to which one believes oneself to be correct, typically concerning a particular decision or action (often termed ‘propositional confidence’ in this literature; Pouget et al. Reference Pouget, Drugowitsch and Kepecs2016; Fleming Reference Fleming2024).

There are, of course, notorious methodological challenges: metacognitive bias, task evaluation, and unconscious processing may all distort sensitivity in ways difficult to capture quantitatively (Fleming and Lau Reference Fleming and Lau2014). Still, under certain idealizing assumptions, an observable positive correlation between perceptual discrimination and metacognitive sensitivity – hence with confidence in the above sense, especially in visual discrimination tasks – may be regarded as indirect evidence for either HCON or HCON*. Specifically:

1. i) if the rate of success in perceptual discrimination reflects one’s belief or degree of confidence in sensory input, and

2. ii) if the assessment of the correctness of the ensuing action or decision reflects one’s higher-order attitude toward that belief or those degrees, then the observable correlation parallels, however imperfectly, the sort of relation predicted by HCON and HCON*.

Assumption (i) is plausible in light of Williamson’s idea that one’s degree of confidence tracks the extent to which one relies on the object of that confidence in action or reasoning. If agents tend to form beliefs and adjust their first-order confidence levels reliably – and if visual discrimination, being among the least contaminated and most practiced first-order perceptual functions, exemplifies this – then (i) is well motivated.

Assumption (ii) may be justified as follows. In such experiments, participants are asked to classify a given stimulus – for example, to press or refrain from pressing a button depending on how they categorize it. Their decision is thus tied to their belief (and degree of confidence) that the stimulus falls under a certain category. Consequently, their assessment of whether their action was correct can reasonably be taken as an assessment of the correctness of their belief.

To be clear, even if these assumptions hold, the observed correlation cannot conclusively establish either hypothesis – partly because of the methodological complications just noted. Nonetheless, current research on metacognition may be taken to provide defeasible empirical support for the idealized correlation predicted by HCON and HCON*.

Both principles offer responses to the anti-luminosity argument as applied to belief: HCON is inconsistent with Bridge*, while HCON* undermines the central motivation for Confidence-Safety*.

To see the first point, note that for some α_i in the case, α_i is the last where one has confidence c that one feels cold, that is, at α_{i + 1}, one has a lower confidence that one feels cold (for c a particular value of ‘c’ in HCON). Bridge* would have us conclude that there is a possible situation β_{i + 1} which is a duplicate of one’s first-order doxastic states about feeling cold at α_{i + 1}, where one has an at-most-slightly-lower degree of confidence c* that one believes one feels cold. But by HCON, one’s confidence drops at β_{i + 1} to from 1 to 0, that is, from one extreme to another, thus c* cannot be at-most-slightly-lower than c.

HCON describes a confidence profile of higher-order beliefs about beliefs that is utterly implausible when applied to first-order beliefs in the original case. It seems far more plausible that one’s confidence in the proposition that one feels cold does not abruptly drop from 1 to 0 once the sensation of cold falls below a certain threshold. However, if our higher-order beliefs are constitutively based on our first-order ones, as the account behind HCON hold, this kind of profile is not always reserved for highly idealized reasoners. Interestingly, this comes close to invoking the constitutive connection strategy in the context of higher-order belief. Even if Srinivasan’s (Reference Srinivasan2013) ingenious reply succeeds for first-order mental states, it does so only by permitting the necessary connections to resurface in the specific case of higher-order belief.

To better probe HCON, it is instructive to consider a case where the first-order belief targets a non-gradable, or at least not so explicitly gradable, phenomenon.

Imagine looking out of a window in one of Oxford’s buildings and seeing, in the distance, an average tall, slender figure in a well-tailored suit. His silvery hair carries the scholarly disarray of someone caught between a paradox and a comb. As he draws nearer, at a slow but steady pace, the bright British accents of his colorful socks catch your eye – followed by his glasses: serious enough for logic, stylish enough for metaphysics. Then, as he steps fully into view, it suddenly strikes you: that’s Timothy Williamson.

The case employs a gradient from not recognizing Williamson to recognizing him – contrasting with the ‘feeling cold’ case. Our focus lies on the formation of the belief – formed through visual experience – that that person (pointing demonstratively) is Timothy Williamson and of the accompanying higher-order belief that one believes that is Timothy Williamson.

To visualize the confidence profile predicted by HCON, imagine a graph where the X-axis represents time as Williamson approaches, and the Y-axis represents one’s confidence in the proposition that is Williamson. According to HCON, the graph remains flat at 0 for a stretch, followed by a sharp jump to 1 at the moment when the visual evidence becomes sufficient to form the belief. (Strictly speaking, this need not be a strict discontinuity, but steep enough so that there is no situation ‘in between’.) The jump marks the point at which one sees Williamson and thus comes to believe it is him. It contrasts with the original ‘feeling cold’ case, where confidence drops gradually in step with a slowly fading sensation; the corresponding graph would resemble that of a constant linear function.

The conflict between HCON* and Confidence-Safety* is more nuanced than a straightforward logical inconsistency. On the latter view, if an agent knows that p with confidence c in a given situation, then p must be true in all sufficiently similar situations where the agent retains a slightly lower level of confidence, c*. The standard rationale for safety requirements appeals to our limited discriminatory capacities: we are often unable to reliably distinguish between very similar situations, and our beliefs do not systematically vary in a way that tracks these fine-grained differences. As a result, for a belief to amount to knowledge, it must be appropriately insulated from error across nearby cases. The application of safety constraints thus reflects our status as cognitively limited agents who are prone to misplaced confidence. That such requirements apply to us is a consequence of our epistemic fallibility and is far from an idealization of our capacities (Williamson Reference Williamson2009; Srinivasan Reference Srinivasan2013: 314–315).

However, if our confidence profile regarding propositions about our own beliefs conforms to HCON*, then our discriminatory capacities appear far from limited. To appreciate the implausibility of imposing a safety condition on confidence in such cases – and analogous ones – consider the following example. Suppose we have a thermometer (T) inserted into a liquid that is being continuously warmed, and a secondary device (D) that registers what T measures. D reliably tracks T’s readings: at each moment α_i, D displays that T measures the liquid at i degrees Celsius. Although D knows that T registers i degrees at α_i, at the subsequent moment α_i+1, when the temperature has increased slightly, T no longer measures i degrees. Yet D’s knowledge is not compromised by this, because it reliably tracks T’s evolving measurements – were T to register i, for any i, D would indicate as much. A safety condition on confidence, when applied here, rules out too much: it fails to accommodate cases in which the knower is reliably tracking change across a range of closely related situations. In such contexts, the knower’s discriminatory success undermines the rationale for imposing safety constraints.Footnote ¹²

If HCON* holds, we are like the device tracking the thermometer’s measurements. Confidence Safety* does not apply to us. We are undoubtedly imperfect – but that does not mean certain domains of knowledge cannot offer glimpses of perfection.