2.1. The proxy premise argument

KFF and CC proponents traditionally share the intuition that cases like Fancy watch and Party Hats are genuine examples of inferential knowledge. However, while supporters of KFF accept these cases as genuine instances of inferential knowledge, they do so at the cost of the CC principle. In contrast, CC advocates maintain that the principle’s plausibility remains intact. To defend CC, these authors typically argue that the false premise does not play any epistemic role in such cases. Instead, only a known premise truly serves this function. This line of defense is known as the proxy premise argument.

Now, let’s examine the proxy premise argument put forward by defenders of CC. As we will soon explore, this strategy may ultimately offer stronger support for KFF cases than for CC itself. Before diving into that, however, it’s important first to address the objection raised against KFF cases.

Traditionally, proponents of CC address the problem posed by KFF cases by asserting that a dispositional true belief – called the “proxy premise” – closely related to the false belief provides the sole epistemic basis for KFF inferences. According to this view, the proxy premise must logically follow from, or be evidentially supported by, the false premise and must also be dispositionally known by S (cf. Fitelson Reference Fitelson2010; Ball and Blome-Tillmann Reference Ball and Blome-Tillmann2014; Montminy Reference Montminy2015, Reference Montminy, Borges and Schnee2023). For instance, in the Fancy watch case, a plausible proxy premise might be, “It is approximately 2:58 p.m.”; in the Party hats case, it could be, “There are fewer than 100 children.”

Now, to prevent the risk of epistemic overdetermination between the false belief and the proxy premise in KFF cases, CC advocates need to argue that the proxy premises are not just useful but also indispensable for establishing inferential knowledge. To do so, they claim that certain beliefs must be essential for S to know a conclusion inferentially. To ratify this requirement, CC advocates introduce the following condition:

Within this framework, CC advocates avoid epistemic overdetermination by positing that the proxy premise – being both dispositionally known and epistemically robust – constitutes the sole essential ground for S’s inferential knowledge of q.

For instance, consider a KFF scenario. According to CC advocates, if S genuinely knows the conclusion q, S is dispositionally warranted in believing the proxy premise, such as “It is approximately 2:58 p.m.” This belief, they argue, is evidentially supported by both the watch’s reading and is entailed by the false premise that S explicitly believes, namely, “It is exactly 2:58 p.m.” However, according to CC advocates, the false premise does not satisfy the essentiality condition. After all, if removed from S’s belief set, S would still know the conclusion because it remains grounded in the dispositionally warranted proxy premise. Indeed, the proxy premise remains essential to the inference: without it, S would lose the inferential knowledge of q, which CC advocates seeking to preserve. Moreover, the proxy premise is also dispositionally known by S, as accurate watches are typically slightly off rather than drastically inaccurate (cf. Montminy Reference Montminy, Borges and Schnee2023, 66).

In light of this, CC advocates argue that the false premise becomes epistemically irrelevant in the inferential process. Focusing on the dispositionally known proxy premise, they argue that it alone supports S’s inferential knowledge, defending CC by arguing that the inference relies on true, known premises.

Without delving into the finer details of this argument, we wish to emphasize what we consider its central issue. According to advocates of CC, for S to know the conclusion of the alleged KFF inference, S must possess dispositional knowledge of a proxy premise (e.g., “It is approximately 2:58 p.m.”). Crucially, however, CC proponents maintain that this proxy premise must not merely co-occur with the false belief but must either logically follow from it or be evidentially supported by it. In other words, the proxy premise derives its epistemic credibility precisely from its relationship to the false belief, “It is exactly 2:58 p.m.” This dependency is not incidental but essential – without the false belief, the proxy premise loses its epistemic support. Thus, by their own epistemic standards, the false belief remains epistemically indispensable. As a result, their purported solution reintroduces the very problem it aimed to resolve, since knowledge of the proxy premise itself becomes entangled in another instance of KFF.Footnote ³

Moreover, as Luzzi (Reference Luzzi2019, 23) rightly observes, even if one were to argue that S’s original reasons provide independent dispositional warrant for the proxy premise (e.g., “My watch reads 2:58 p.m.” supporting “It is approximately 2:58 p.m.”), a pressing question remains: in an indistinguishable scenario where it is actually 2:58 p.m., why would S’s inferential knowledge be based on the correct (precise) premise, while when the premise is false, knowledge shifts to rely on an (approximate) proxy premise? This inconsistency further undermines the proxy premise argument. After all, it seems highly counterintuitive to suggest that the accuracy of the clock – the only changing variable in this new envisaged scenario – could somehow dictate the reasoning process of S.

In conclusion, the challenges posed by CC proponents highlight a critical issue in their reliance on the proxy premise argument. While they assert that this premise serves as the essential basis for inferential knowledge in KFF cases, they inadvertently open themselves to the risk of KFF at a meta-level if they lean too heavily on it. Alternatively, if they argue the proxy premise stands independently, they must confront a troubling thesis about the connection between real-world facts and inferential reasoning. Thus, these considerations reveal deeper complexities in understanding how we derive knowledge from premises, whether true or false.

2.2. Toward a (preliminary) solution

Despite objections from CC supporters, KFF advocates have found a valuable ally in the proxy premise idea. As a matter of fact, KFF proponents have skillfully turned some of the criticisms of the proxy premise argument into a strong defense of their position. Notably, both Klein (Reference Klein and Smith2008) and Feit & Cullison (Reference Feit and Cullison2011) present a case for KFF that aligns with some ideas of the proxy premise argument. Specifically, they argue that, unlike in Gettier cases, knowledge can still be derived from a false premise in KFF cases because a warrant for the conclusion remains intact, even if the subject later discovers their belief in p is false.

To illustrate, consider the Fancy watch case. Suppose S learns the time is not exactly 2:58 p.m. Despite this, KFF proponents argue S can still know, “I am not late for the 7:00 p.m. meeting,” based on a dispositionally warranted belief in a related but approximate premise, like “It is approximately 2:58 p.m.” This suggests that even with contradictory information, the dispositional warrant for the approximate belief is strong enough to support the conclusion.

Now, contrast this with the Ford car case. Here, S believes “Someone in our class owns a Ford,” based on the incorrect premise that “Nogot owns a Ford.” However, in this case, S’s belief does not qualify as knowledge because the foundational grounds are false. If S learns “Nogot does not own a Ford,” this removes any warrant for the belief that “Someone in the class owns a Ford.” Crucially, S’s initial reasons cannot transfer to an alternative premise that would preserve their inferential knowledge after the false premise is compromised. Therefore, S’s inferential belief does not count as knowledge in this case.

To clarify these ideas further, we will now introduce some well-known technical terms that will be useful in our subsequent discussion. The next section will examine these terms more in-depth.

The main argument posited by proponents of KFF appears to be the following: the same initial reasons that provide a doxastic warrant for a false belief p also provide a propositional warrant for a related, approximate true proposition a. Now, since a logically follows from p – because any precise proposition p entails its approximation a – the two remain closely connected from an epistemological standpoint. As a result, even if S encounters a defeater for p (for example, learning that “The time is not exactly 2:58 p.m., but 2:57 p.m.”), their inferential knowledge of q remains intact. This is because p, despite being false, serves as a premise in the inference to q and warrants q propositionally in much the same way a does. Thus, in a counterfactual scenario where S acquires a defeater for p, the knowledge of q is preserved. In this way, a false premise can still generate knowledge, provided its negation does not serve as a defeater. Therefore, even when the doxastic warrant for p is defeated, the propositional warrant for a continues to support the inference.Footnote ⁴ Below is a graphical representation of these ideas (see Figure 1).

Figure 1. These graphs illustrate the epistemic support relations for CC (left) and KFF (right) supporters. Dotted horizontal arcs indicate entailment between propositions p and a, while vertical and diagonal arcs represent evidential or inferential dependencies. DW and PW refer to doxastic and propositional warrant, respectively.

Hence, KFF advocates argue that while secondary reasons (a) may serve as a fallback in counterfactual situations where primary reasons (p) fail, the primary reasons still support S’s inferential knowledge in the actual world. The distinction is crucial: counterfactual reasons do not carry the same epistemic weight as real-world reasoning. Therefore, KFF supporters stress the need to differentiate between counterfactual support (a) and the actual basis of S’s inferential knowledge (p).

The above considerations explain why KFF cases constitute knowledge, unlike Gettier cases. The solution relies on the role of defeaters and the distinction between doxastic and propositional warrant. These concepts distinguish cases where knowledge arises from false but stable premises (cf. Warfield Reference Warfield2005) from those where true belief is merely a product of luck (cf. Pritchard Reference Pritchard2005). This raises an important question: how general is this solution, epistemologically speaking? Specifically, is it externalistically kosher? In the next section, we will argue that this is not the case.

3.1. The reason-first approach

Proponents of KFF address the proxy premise argument by distinguishing between doxastic and propositional warrant. This distinction is traditionally introduced as follows:

• • S has propositional warrant to believe proposition p iff they have sufficient epistemic reasons to do so, regardless of whether they actually hold that belief.

• • S is doxastically warranted in believing p iff they have propositional warrant for p, believe p, and base their belief on these sufficient epistemic reasons (cf. Firth Reference Firth, Goldman and Kim1978).

Put simply, doxastic warrant is propositional warrant combined with its connection to S’s belief, specifically through the basing relation. Under this account, propositional warrant is conceptually fundamental: understanding the concept of doxastic warrant requires first grasping the concept of propositional warrant (cf. Melis Reference Melis2018). This framework is often referred to as the reason-first approach (cf. Silva and Oliveira Reference Silva, Oliveira, Lasonen-Aarnio and Littlejohn2023).

The reason-first approach to epistemic warrant stems from an internalist view of non-factive positive epistemic support, which has long been dominant among epistemologists. Philosophers such as Ayer, Russell, Carnap, and C. I. Lewis, building on the work of modern thinkers, supported the sense-data theory of reasons. In nuce, this theory posited that empirical knowledge and its warrant depend entirely on internal mental states and procedural norms – meaning rules – that allegedly align an individual’s sensory experiences with the world’s facts. However, this theory declined due to the challenge of identifying these procedural norms.Footnote ⁵

In response, epistemologists shifted towards a new dominant view: epistemic warrant is grounded exclusively in reasons. According to this view, S has propositional warrant for p just in case S’s psychological states support p. However, under this account, having adequate reasons for p does not automatically mean S is warranted in believing p. In other words, the rationality or credibility of p fundamentally depends solely on the objective relations between S’s reasons and p. The following quotations further illustrate this idea:

These quotations emphasize that propositional warrant arises from the objective, or supervenient, relations rooted in an agent’s internal psychological states. Notably, Conee and Feldman’s quotation, drawing from Firth’s (Reference Firth, Goldman and Kim1978) work, underscores the alignment between propositional warrant and the evidential support relations among propositions. Firth aptly captures this idea, stating:

So, the reason-first approach focuses on determining when S has a warrant for a proposition rather than on when they are warranted in believing it. To be warranted in holding a belief, one must not only possess good reasons but also believe based on those reasons. Thus, propositional warrant remains independent of S’s cognitive limitations.

As a matter of fact, Firth adopts a broad interpretation of the term “logical,” encompassing both deductive and non-deductive objective relationships. This suggests that, in the reason-first framework, propositional warrant depends on the objective connections between propositions and the reasons available to S, even if S is unaware of or unable to fully grasp them. Thus, on this view, the ultimate epistemic support for a belief lies in its logical relation to the agent’s reasons, regardless of the agent’s understanding (cf. Smithies Reference Smithies2015; Melis Reference Melis2018; De Toffoli Reference De Toffoli, Silva and Oliveira2022). Kornblith further develops this idea, noting:

Now, this focus on the objective structure of propositional warrant is particularly illuminating in the context of KFF cases, where S’s initial reasons may support both a false proposition and a true proxy premise.Footnote ⁷ For instance, consider the Fancy watch case: S’s initial reason – “My watch reads 2:58 p.m.” – provides evidential support for both the false proposition “It is exactly 2:58 p.m.” and the true proposition “It is approximately 2:58 p.m.”Footnote ⁸ However, as responses to the proxy premise argument emphasize, S actually believes p, not a. This distinction is crucial: in the actual world, S has a doxastic warrant for p but not for a because S does not believe a. In the Fancy watch case, for instance, S’s doxastic warrant applies only to “It is exactly 2:58 p.m.” because they base their belief on the specific reading of the watch, not the approximate time.

Therefore, KFF advocates address objections to the proxy premise argument by distinguishing between propositional and doxastic levels of warrant, a distinction that aligns with the reason-first approach. Specifically, they argue that while new reasons may defeat S’s propositional (and doxastic) warrant for “It is exactly 2:58 p.m.,” the propositional warrant for “It is approximately 2:58 p.m.” remains intact. Importantly, this propositional warrant, supported by S’s reasons evidentially and deductively, does not lead to a belief but provides a solid ground for a stable inference. As a result, a continues to inertly support the true conclusion, “I am not late for the 7 p.m. meeting.” Thus, S can know this latter fact, even if the inference stems from a false belief about the exact time.

In contrast, while the reason-first approach can also be applied to explain Gettier cases, KFF proponents argue that their strategy does not fit these cases, further highlighting the plausibility of their rejoinder. In Gettier cases, the reasons available to S do not provide a propositional warrant for a proxy premise, making it difficult to infer conclusions like “Someone in my class owns a Ford,” even if the conclusion is true and based on a false belief.

Therefore, if our considerations so far are accurate, it would be reasonable to conclude that the KFF proponents’ solution to the proxy premise argument is at least internalistically adequate. Specifically, if one accepts a conception of warrant where doxastic warrant entails propositional warrant – with the latter being fundamental – they would agree with the KFF rejoinder. However, this raises the question: is this solution acceptable to externalists? The next subsection will explore the key differences between internalist and externalist approaches to KFF.

3.2. But what about externalism?

Traditionally, epistemic externalists, including Goldman (Reference Goldman and Pappas1979), Kornblith (Reference Kornblith1980), and more recent authors such as Graham and Lyons (Reference Graham, Lyons, Brown and Simion2021), reject the reason-first approach. Consider, for example, the views of Goldman and Kornblith:

Hence, for epistemic externalists, the relationship between propositional and doxastic warrant is understood in reverse compared to the reason-first view: doxastic warrant is conceptually fundamental, and propositional warrant is defined in terms of it. Here is Goldman:

Within this framework, S has doxastic warrant for p if and only if the belief in p results from a reliable belief-forming process type.Footnote ⁹

Now, the shift in focus from propositional to doxastic warrant carries significant implications for the externalist position. First, it shows that a warrant for a belief that p does not ensure a backup warrant for another proposition a. This is because the reliable processes or cognitive states needed to warrant an alternative belief may not be available to S at a given moment, thus failing to meet reliabilist standards. Second, it underscores that the ultimate grounds of rationality – aiming to believe the truth and avoid error – are now determined not by the logical relations between internal states and propositions (cf. Pollock and Cruz Reference Pollock and Cruz1999), but by the agent’s contingent cognitive features (cf. Graham Reference Graham and Fernández Vargas2016). Consequently, and more importantly, it appears that externalists may not benefit from the rejoinder proposed by KFF proponents against CC advocates.Footnote ¹⁰

Consider Fancy watch: S checks their watch, feels confident in its accuracy, and, after careful observation, forms the belief, “It is exactly 2:58 p.m.” Although this belief turns out to be false, it is still doxastically warranted. According to the reliabilist framework, epistemic warrant accommodates occasional errors (cf. Burge Reference Burge, Graham and Pedersen2020). What matters is that the belief was formed through a generally reliable process, even if it led to a mistake in this particular instance.

However, a key question arises: at t ₀ , when S checks the watch and before forming the belief that p at t ₁ , does S have available a reliable process that could lead to an alternative epistemic path toward concluding q – specifically, one that concludes q by way of a propositional warrant for a proxy premise? According to Goldman’s account, having a propositional warrant requires that, at time t, S must have a reliable process available that could provide a doxastic warrant for the relevant belief. Thus, the central issue is whether S has such a reliable process available at t ₀ to form an approximate belief.

This issue introduces additional complexity. In general, belief-forming processes for approximate beliefs tend to be significantly less reliable than those that lead to exact beliefs (cf. Kornblinth Reference Kornblinth2017; Grundmann Reference Grundmann2022). Several factors contribute to this difference.

First, belief-forming processes for exact beliefs typically rely on well-defined mechanisms and feedback loops that can be measured objectively, thereby ensuring the accuracy and truth of the belief (cf. Burge Reference Burge, Graham and Pedersen2020). For instance, we can assess whether a person’s belief about the time is correct by comparing their watch to an atomic clock. This kind of verification process provides clear and reliable feedback. By contrast, the formation of approximate beliefs often lacks such transparency, making their evaluation more difficult.Footnote ¹¹

Now, some might argue that approximate beliefs, being logically weaker, are more likely to be true in a broad or general sense. However, this supposed advantage is offset by the difficulty of assessing their accuracy. As a matter of fact, without clear methods of evaluation, their reliability becomes questionable (cf. Goldman Reference Goldman and Pappas1979). For instance, consider someone who looks at an accurate watch and concludes, “The time is about 2:58 p.m.” While this belief may be roughly correct for everyday purposes, its vagueness introduces a greater risk of error due to the inherent ambiguity of approximation.

This ambiguity becomes even more apparent when we try to assign a clear truth value to approximate beliefs. Unlike exact beliefs, which can be directly tested against objective standards, approximate beliefs often resist straightforward evaluation. For example, if an atomic clock shows 3:02 p.m., and someone believes “The time is approximately 2:58 p.m.,” what truth value can we reasonably assign to that belief? The lack of precision makes it difficult to provide a definitive answer, suggesting that approximation introduces a degree of interpretive openness that objective criteria alone may not resolve. In such cases, our judgments about truth may depend more on social conventions than on measurable facts.Footnote ¹²

Second, belief-forming processes for approximate beliefs may be prone to systematic biases. Approximations can often underestimate or overestimate values, further reducing the reliability of these processes and weakening the epistemic support for our beliefs (cf. Tversky and Kahneman Reference Tversky and Kahneman1974). For example, if S tends to round up the time (such as rounding 2:57 to 2:58 p.m.), this rounding bias introduces minor inaccuracies that, over time, could lead to consistently flawed beliefs. In contrast, exact beliefs, like “It is exactly 2:58 p.m.,” are less affected by such biases; S’s watch provides a specific time, and, unless it is consistently misread, there is no intermediary where rounding or similar biases could interfere.

Third, the cognitive complexity in forming approximate beliefs introduces a margin of error that is absent when forming exact beliefs. Epistemic externalism favors processes that minimize error and maximize truth. Thus, if an approximate process is more error-prone, it may not meet the reliabilist criteria for epistemic warrant (cf. Williamson Reference Williamson2000). For instance, if S tries to approximate the time while multitasking – glancing at the time while also focusing on the meeting – split attention could lead to errors. Estimations in such scenarios rely on less precise perceptual cues, resulting in less reliable beliefs. In contrast, forming the belief “It is exactly 2:58” by simply reading the watch enables an immediate and cognitively effortless belief formation with minimal mental interference. This simplicity reduces error and supports the reliabilist criteria for doxastic warrant.

Therefore, although approximate beliefs may be logically weaker, reliabilism does not grant them higher reliability simply because they are easier to align with the truth. The potential for error in approximation undermines their reliability by externalist standards, which require a high ratio of true to false beliefs, not just close enough beliefs. Given these challenges, it becomes clear that typical a posteriori processes for forming approximate beliefs are inherently unreliable. Their lack of transparency, susceptibility to biases, and cognitive complexity compromise their capacity to produce reliable beliefs consistently.

Furthermore, it is also noteworthy to highlight that even employing a priori processes would not resolve this issue. For instance, consider the claim that the proxy premise “It is approximately 2:58 p.m.” could be warranted simply by understanding the statement “My watch reads 2:58 p.m.” For this to be the case, there would need to be an inherent semantic link between the two, grounded in the meanings of terms like “approximately” and “reads” or in the sentences themselves. Yet such a link is not guaranteed. An epistemic agent might reliably understand “approximately” without understanding “reads” and vice versa. Moreover, a competent agent might reliably grasp the meanings of these sentences without recognizing any necessary semantic link between them (cf. Williamson Reference Williamson2007). Thus, since understanding one term or sentence does not inherently depend on the other, the warrant for this approximate belief cannot be derived solely from the meanings of the words or sentences.

Finally, before moving on, it is also important to clarify a potential misunderstanding regarding S’s ability to infer the proxy premise a from the false belief p. At time t ₀ , belief p has not yet formed in S’s cognition. This is a crucial point. Unlike internalists, who rely on perceptual or doxastic reasons that are logically connected to other propositions to explain S’s warranted states, S here operates within an externalist framework, where S’s only cognitive input is a perceptual (non-doxastic) state showing the exact time on their watch at t ₀ .

This clarification carries two key implications. First, unless this perceptual state engages in a reliable cognitive process available to S at t ₀ , it cannot independently provide epistemic support on its own. However, as discussed above, S lacks a reliable method to form a at t ₀ . Second, this latter point is reinforced by the fact that S, in the actual world, interprets the time displayed on the clock only in an exact way at t ₀ . Thus, from a strictly externalist standpoint, S also lacks any approximate cognitive states at t ₀ that could lead to the formation of the proxy premise.Footnote ¹³

In conclusion, any attempt to explain how S could derive a warrant for a from p (or their initial cognitive states) overlooks the limitations of S’s cognition at t ₀ . These limitations preclude the formation of the proxy premise and prevent S from acquiring propositional warrant for a at that time.Footnote ¹⁴

Given these considerations, if externalists supporting KFF’s thesis cannot identify a reliable process available to S at t ₀ to form an approximate belief based on initial cognitive states, we should conclude that S lacks propositional warrant for a. This suggests that externalists may not only miss the benefits of the proxy premise strategy proposed by reason-first advocates of KFF to counter CC but also struggle to distinguish KFF cases from Gettier cases. Notably, the epistemic triangularization proposed by the reason-first approach in KFF cases – linking initial reasons, doxastic warrant, and propositional warrant – appears incomplete within the externalist framework. Specifically, an epistemic link between the initial grounds and the approximate belief appears to be missing. Consequently, the lack of a stable path available to proponents of the reason-first approach further weakens the externalist position. Below is a graphical representation of these ideas (see Figure 2).

Figure 2. The graph illustrates the externalist account of KFF at t ₀ , where S has a PW for p but lacks one for a due to the absence of reliable processes available to S at t ₀ .

Despite these challenges to the externalist position, concluding that these considerations fully support the reason-first approach is premature. To understand this view’s limitations in KFF cases, it is crucial to remember that the reason-first approach emphasizes the role of objective evidential relations between propositions and S’s available reasons in structuring (propositional) warrant. In the next subsection, we will present an argument against the reason-first view, highlighting the latent risks of this assumption.

3.3. Countering the standard explanation of KFF

To illustrate the limitations of the reason-first approach, let us consider the Fancy watch case, assuming the same initial conditions remain in place. To recap, in this scenario, S has a doxastic and propositional warrant for a false belief p and a propositional warrant for a true proxy premise a. Notably, both warrants are supported by the same reason, “My watch reads 2:58 p.m.”

Now, according to the reason-first approach, if S has a propositional warrant for p, S possesses a reason that, though false, can generate infinite propositional warrant tokens through logical links independently of S’s cognitive capacities. As a matter of fact, since propositional warrant is closed under logical consequence (assuming classical logic to adopt KFF proponents’ assumptions), an infinite set of warranted propositions can be derived from an arbitrary belief $\psi $ using Disjunction Introduction, i.e., $\psi \vdash \psi \vee \omega $ .Footnote ¹⁵ For example, if S has a propositional warrant for $\psi $ , S also has a propositional warrant for $\psi \vee \omega $ , where $\omega $ represents any atomic or non-atomic proposition like “the wall is blue” or “if the wall is blue, then the wall is blue.”

Interestingly, both friends and foes of KFF cases support this perspective. For instance, Klein (Reference Klein and Smith2008, 50–1), building on Hilpinnen’s work, argues that for KFF cases to be genuine cases of inferential knowledge, the proxy premise a must be “sufficiently close” to the false premise p. While Klein suggests something less strict than an entailment relation might work to explain this epistemic closeness, his brief remarks on this point indicate he favors entailment. Thus, he seems to argue that S has propositional warrant for a because a is entailed by p (and because S’s initial reasons support it).

A similar view is expressed by Montminy, who defends the role of entailment in supporting CC via the proxy premise argument (cf. §2.1). In a recent work, however, Montminy (Reference Montminy, Borges and Schnee2023, 65) adds an important detail: while some true propositions may not be explicitly believed, an infinite set of “obvious” propositions remains dispositionally warranted and epistemically available to S. This highlights the wide scope of propositional warrant within the reason-first approach, showing that even a false premise can enable S to access a vast network of warranted propositions through entailment, such as $\psi \vee \omega $ .

However, the reason-first approach faces challenges due to the arbitrary nature of the propositions that can be substituted into the disjunct $\omega $ . For example, consider the following substitution: “All my approximate belief-forming processes are unreliable.” If this instance of $\omega $ were derivable from the false premise p, it would present serious issues for the proxy premise.

Now, to see how this instance of $\omega $ could easily be derived, consider the relevant counterfactual situation where S, after forming the doxastically warranted belief “I am not late for my 7 p.m. meeting” based on the false belief p, learns that “It is not exactly 2:58 p.m.” In this case, S gains a reason to reject the false belief, “It is exactly 2:58 p.m.” However, since propositional warrant is closed under logical consequence, S would also acquire propositional warrant for any claim substituted into $\omega $ via Disjunctive Syllogism, i.e., $\;\psi \vee \omega, \;\neg \psi \vdash \omega $ . Consequently, if ω represents the problematic proposition mentioned earlier, S would now have a reason that defeats the previously warranted proxy premise (cf. Pollock Reference Pollock1986).Footnote ¹⁶

To illustrate this more clearly, consider $\omega $ as “All my approximate belief-forming processes are unreliable.” In this case, $\psi \vee \omega $ can be read as “It is exactly 2:58 p.m., or all my approximate belief-forming processes are unreliable.” Now, when we introduce the true information $\neg \psi $ (“It is not exactly 2:58 p.m.”), we can apply Disjunctive Syllogism to derive $\omega $ . This leads to the result that S has propositional warrant for ω. However, this instance of $\omega $ challenges the credibility of the proxy premise’s propositional warrant. Using Pollock’s terminology, the derivation of $\omega $ introduces an undercutting defeater for S. Specifically, if R is a defeasible reason for S to believe p, R* undercuts R just in case R* shows S that their reasons for believing p are inadequate.Footnote ¹⁷ Thus, the derivation of $\omega $ undercuts the proxy premise’s warrant, preventing S from confidently knowing, “I am not late for my 7 p.m. appointment” (cf. Klein Reference Klein and Smith2008, 32).

A similar dynamic applies when considering rebutting defeaters. Again, following Pollock’s framework, a rebutting defeater arises when if R is a defeasible reason for S to believe p, R* is a rebutting defeater for R just in case R* is a reason for S to believe ¬p. Let’s break down this case to understand the issue. Previously, we saw how Disjunctive Syllogism worked in our case: given $\psi \vee \omega $ and $\neg \psi $ , we were able to derive $\omega $ . Now, consider $\omega $ as a conditional statement where the antecedent is ¬p and the consequent is any proposition that contradicts a. By applying Modus Ponens – i.e., $\psi, \;\psi \to \omega \vdash \omega $ – to ¬p and the conditional statement that instantiates ω, it follows that S is propositionally warranted in believing a rebutting defeater for a upon learning ¬p. Therefore, deriving one’s preferred $\omega $ undermines the propositional warrant of the proxy premise, preventing S again from confidently knowing, “I am not late for my 7 p.m. appointment.”

But there is more! The problems with the reason-first approach become clearer when we consider Gettier cases. Suppose S has a propositional warrant for “Nogot owns a Ford.” By applying Disjunction Introduction, S would also have a propositional warrant for “Nogot owns a Ford, or Havit owns a Ford.” Yet, if we assume S later discovers that “Nogot does not own a Ford,” we can use Disjunctive Syllogism again to derive “Havit owns a Ford” and thus argue that S acquires propositional warrant for this proxy premise.

However, this result contrasts with the conclusions typically drawn by the reason-first proponents in KFF cases. As a matter of fact, in the relevant scenarios where S discovers that “Nogot does not own a Ford,” S might still retain a propositional warrant for a proxy premise supporting the knowledge that “Someone in the class owns a Ford.”Footnote ¹⁸

Therefore, these cases highlight a significant challenge for the reason-first framework: by imposing objective constraints on propositional warrant rather than cognitively fitting ones, this approach can become too permissive and yield counterintuitive results.

With this in mind, we will now turn to four potential objections that reason-first advocates might raise regarding KFF cases, and we will address each in turn.

4.1. Graham & Lyons’s account

Graham and Lyons have recently put forward a general externalist account of propositional warrant and defeaters. To engage meaningfully with their view, it is important to first clarify a key aspect of their terminology. Specifically, they use the term “warrant” in a way that encompasses both propositional warrant and defeaters, effectively treating the possession of a defeater as equivalent to having propositional warrant. In contrast, throughout this paper, we have employed the term “warrant” in a more precise sense, meaning non-factive, positive epistemic support for a proposition or belief. Hence, to accurately reflect Graham and Lyons’s usage while avoiding confusion, we will use the term “warrant/defeater” when referring to their conception of warrant as inclusive of defeaters. This terminological distinction is essential for a clear and accurate presentation of their account. Thus, with these clarifications in mind, we can now turn to Graham and Lyons’s account:

Graham and Lyons’s account rests on two central ideas: (i) expanding Goldman’s account of defeaters and propositional warrant and (ii) preserving Pollock’s insights on defeaters in line with externalist tenets (cf. Graham and Lyons Reference Graham, Lyons, Brown and Simion2021, 57). In the remainder of this subsection, we will explore these two ideas in some detail, laying the groundwork for our subsequent analysis of KFF cases.

Regarding (i), Graham and Lyons’s account closely parallels Goldman’s account of propositional warrant. To recap, Goldman asserts that S has a propositional warrant for p at t just in case S has available a reliable process that would yield a doxastic warrant for p if applied to S’s cognitive states at t. Now, when Goldman’s notion of defeaters is considered alongside his account of propositional warrant, Graham and Lyons’s connection to it becomes clear. In Goldman’s (Reference Goldman and Pappas1979) account, S’s warranted belief that p at t is defeated just in case S has another reliable process available at t that, if used, would have prevented S from believing p. Thus, for Goldman, Graham, and Lyons alike, a defeater for p is closely tied to having propositional warrant: if S were to apply an alternative reliable belief-forming process to their current cognitive states, S would arrive at a different judgment regarding p – such as disbelief, suspension of judgment, or a reduction in the degree of belief in the proposition.Footnote ²⁴

Despite having clear similarities to Goldman’s view, Graham and Lyons introduce important adjustments in their account of warrant/defeaters. First, they argue that warrants/defeaters function as reasons in an internalist sense, but without requiring the agent to re-evaluate their beliefs consciously. That is, warrant/defeaters do not necessitate agent-level epistemic basing (cf. Graham and Lyons Reference Graham, Lyons, Brown and Simion2021, 58). Notably, their choice to characterize warrant/defeaters as “outputs” rather than beliefs reinforces this idea, suggesting that even non-doxastic states can act as warrant/defeaters for doxastically warranted beliefs.

For example, if S believes in p due to a reliable process but then perceives ¬p through another reliable process, that non-doxastic state acts as a warrant/defeater for S’s belief in p. Importantly, the existence of ¬p does not compel S to reassess their beliefs consciously; instead, S may immediately adopt the belief in ¬p or suspend judgment regarding p. Conversely, if S learns p but is misled by an illusion, such as the Müller-Lyer illusion, they may still maintain their belief in p despite having a non-doxastic defeater (cf. Graham and Lyons Reference Graham, Lyons, Brown and Simion2021, 62).

Second, Graham and Lyons’s account deliberately leaves open the question of whether the outputs of a belief-forming process are always available to S (cf. Graham and Lyons Reference Graham, Lyons, Brown and Simion2021, 62). As a matter of fact, their account suggests that the available process “does or would likely produce p as an output,” which implies that propositional warrant and defeaters may not always be immediately available to S. This indicates that Graham and Lyons view the availability of propositional warrants or defeaters as dependent on both the cognitive complexity involved in acquiring these epistemic resources and the individual’s cognitive capacities.

For example, if forming a warrant/defeater for p (¬p) requires minimal cognitive effort – such as a few basic steps of reasoning – then ¬p could serve as a warrant/defeater for p, provided that S has the cognitive ability to form ¬p in their current state. However, if S has another available cognitive process that blocks ¬p, they may retain their belief in p. This aligns with Graham and Lyons’s view, where S could use a defeater-eater to neutralize the original warrant/defeater. Conversely, if the process behind ¬p is reliable but beyond S’s cognitive capacities, ¬p would not function as a warrant/defeater for p (cf. Graham and Lyons Reference Graham, Lyons, Brown and Simion2021, 63).

Thus, while there is indeed some ambiguity in this account regarding how reliable cognitive abilities are identified or how modally stable they must be to ensure the availability of their outputs to S, this vagueness ultimately supports Graham and Lyons’s account. As they note, their framework offers a flexible guide for externalists rather than a rigid blueprint for understanding propositional warrants and defeaters. Consequently, while the precise details of how reliable competencies are acquired can be filled in by different externalist accounts to address concerns about the scope and applicability of these reliable epistemic processes in various contexts, we can set aside such specifics for now and focus on the broader contours of this general approach.

Building on these considerations, Graham and Lyons propose the following analysis of warrant/defeaters:

• • S has a rebutting defeater for her belief that p iff she has a warrant to believe ¬p.

• • S has an undercutting defeater for her belief that p iff she has a warrant to believe that her warrants for believing p are inadequate.

• • S has a defeater for her belief that p iff she has a rebutting or an undercutting defeater for her belief that p (Graham and Lyons Reference Graham, Lyons, Brown and Simion2021, 59).

So, regarding (ii) above, Graham and Lyons’s analysis also aligns with Pollock’s reason-first view of defeaters (cf. §3.3). By integrating the notion of warrant/defeater into Pollock’s framework, Graham and Lyons offer a theory that bridges internalist and externalist perspectives, with defeaters becoming more dependent on one’s cognitive capacities.

Given this overall framework, the critical question we must now address is whether this account enables proponents of KFF to counter CC effectively. In the next subsection, we will apply Graham and Lyons’s account to resolve the tension encountered in the standard explanation of KFF cases.

4.2. Toward an(other) explanation of KFF cases

If we recall correctly, the main issue with the reason-first interpretation of KFF cases is that the requirement for propositional warrant to be closed under logical consequence needs to be revised. Specifically, reasons derived from the false belief p can, through simple logical inferences, generate a defeater that blocks the credibility of the proxy premise a. In contrast, the externalist faces a more troubling issue: the lack of propositional warrant for a at t ₀ . Thus, in order to coherently reconsider this latter perspective, the first question we must address is whether S has a reliable backup warrant for the proxy premise at subsequent times t ₁ through t _n that enables them to sustain their inferential knowledge of q even when confronted with defeaters.

According to accounts from Graham, Lyons, and Goldman, it appears that S can possess a propositional warrant for a proxy premise at t ₁ . To illustrate, consider Fancy watch and assume that S has doxastic warrant for p at t ₁ . In this case, if we can reasonably assume that S can make inferences, S could form the belief that a at t ₁ through a belief-dependent, conditionally reliable process (cf. Goldman Reference Goldman and Pappas1979; see also Grundmann Reference Grundmann2022). This process involves two key elements: (i) the slightly inaccurate but doxastically warranted premise p, which, although false, remains competently produced, and (ii) the conditional true belief a: “If it is not exactly 2:58 p.m., then it is approximately 2:58 p.m.” available to S.

Now, to see how S could arrive at this latter belief, we can consider two approaches. The first, based on a Kantian-inspired concept of containment, suggests that S could infer that the exact time is contained within the approximate time. Alternatively, a more sophisticated reasoner might use the formal concept of a set, concluding that the set labeled “Approximately 2:58 p.m.” includes the subset of “Exactly 2:58 p.m.”Footnote ²⁵

Consequently, since S is confident that their watch is accurate, they can reliably rely on the false belief that p to derive a. Notably, given S’s normal reasoning abilities and reliable methods, S has a propositional warrant for a at time t ₁ . Even if p is slightly inaccurate, a remains warranted because S can engage in a conditionally reliable process. The negligible inaccuracy in p does not compromise this reasoning, as p is sufficiently dependable to assure the reliability of a. Thus, if the conditional statement holds reliably and the reasoning process is executed competently, a can be propositionally warranted for S at t ₁ .

This conclusion is reinforced by recognizing that when S forms the false belief p, they already have propositional warrant for the conclusion q. For example, the inference from “It is exactly 2:58 p.m.” to “I am not late for my 7 p.m. appointment” is straightforward and reliable. The inferential process (likely simple arithmetic or time reasoning) is highly reliable, as there is minimal room for error in calculating that over four hours remain before the appointment. Although the input belief “It is exactly 2:58 p.m.” is off by one minute, this inaccuracy is negligible because the difference between 2:57 p.m. and 7 p.m. is so large. Thus, the reliability of the conclusion, “I am not late for my 7 p.m. appointment,” remains unaffected by the small error in the input belief p.

So, the key difference between the externalist and reason-first approaches lies in how they structure epistemic support in KFF cases. According to the externalist perspective, at t ₀ , S begins with a perceptual state. At t ₁ , this state is processed by a reliable belief-forming process that provides doxastic warrant for p. Simultaneously, another reliable process secures propositional warrant for a.

Conversely, the reason-first framework holds that after S acquires initial reasons at t ₀ , they obtain doxastic warrant for p but lack it for a at t ₁ . However, S retains propositional warrant for both p and a at t ₀ . Below is a graphical representation to illustrate these ideas (see Figure 3).

Figure 3. These graphs illustrate the epistemic dependency relations between the key epistemic elements of the internalist approach (on the left) and the externalist approach (on the right).

The graph on the right highlights two key aspects of the externalist approach to countering CC. First, it shows that the epistemic status of the inferential belief that q depends inherently on the false belief that p. In contrast, the left graph shows that, under the reason-first approach, the belief that q is not inherently dependent on the belief that p. Thus, while the reason-first approach can be criticized for relying on an alternative path that may be epistemically available but cognitively inaccessible to S in supporting knowledge of q, this criticism does not apply to the externalist approach.

Second, and relatedly, the graph on the right highlights the essential role of doxastic warrant in acquiring knowledge from falsehoods within the externalist framework. Recall that, for externalists, propositional warrant requires satisfying conditions for doxastic warrant, which, in turn, requires holding a belief. This then implies that even when inferential knowledge is obtained through an alternative stable path, the propositional warrant for elements along that path depends on S’s doxastic abilities. Thus, externalism consistently argues that the epistemic availability of any alternative path is linked to S’s cognitive capacities, underscoring the importance of doxastic warrant in acquiring and maintaining knowledge.Footnote ²⁶

Now, a potential objection to KFF’s core intuition – that these cases involve genuine inferential knowledge – is the absence of epistemic triangulation within the externalist framework. Some might argue that this absence weakens the idea that such cases count as knowledge from falsehoods. However, we contend that this lack of triangulation actually strengthens the view of knowledge as an achievement rooted in S’s cognitive abilities.

For example, we previously assumed that S could infer an approximate time from an exact one. However, this ability is not universally shared. A developing epistemic agent, like an infant, may lack the competence to make such inferences at certain stages of their development. For example, the child might not yet grasp concepts such as set membership or containment. In such cases, it seems intuitive to deny that the agent knows whether they are late, since they cannot reliably interpret the information provided by their belief about the watch. Without the necessary conceptual understanding, these agents clearly lack the epistemic competence required for such inferences. Therefore, by adopting a broad conception of relevant epistemic agents at play, the lack of triangulation in externalist accounts does not weaken the plausibility of KFF cases but allows for a more nuanced, context-sensitive understanding of inferential knowledge.

Having established the epistemic grounds of the inferential knowledge of q, we can now analyze defeaters. For an externalist, two key aspects of defeaters are crucial: the cognitive complexity involved in obtaining them and the nature of their semantic content. These factors are central to understanding how defeaters affect KFF cases within the externalist framework. Hence, we will begin by discussing the cognitive complexity component of defeaters.

In examining Graham and Lyons’s account, we observed that the mere fact that a warrant/defeater logically follows from a proposition believed by S does not automatically make it epistemically available to S. For a warrant/defeater to be available, S must be able to infer it through a reliable cognitive process using the cognitive states at their disposal. Therefore, when considering the cognitive complexity of defeaters, the key issue is whether S can cognitively engage with the reasoning that leads to the undercutting and rebutting defeaters discussed in §3.3.

Unfortunately, the KFF cases we’ve analyzed in this paper provide little information about S’s cognitive background. Nevertheless, we can make three relevant observations about the processes involved in generating the defeaters discussed in §3.3.

First, both undercutting and rebutting defeaters in §3.3 depend on the effective application of deductive principles. Although these principles are generally straightforward and reliable, integrating them into a coherent argument presents a significant challenge for many epistemic agents. For instance, chaining Disjunction Introduction and Disjunctive Syllogism may stretch the cognitive capacities of some unskilled agents, hindering their ability to form the relevant defeaters.

Second, forming these defeaters requires the epistemic agent to engage with unfamiliar propositions that may not be immediately available to S. This suggests that the availability of these defeaters depends on S’s capacity to work with abstract or unfamiliar premises, a requirement that may not always be met.

Finally, the interplay of deductive reasoning with arbitrary premises complicates the process further. Such cognitive tasks may be beyond S’s reach and risk leading to unreliable outcomes. Hence, this complexity highlights the challenges S faces in generating and utilizing defeaters, illustrating the tension between reliable cognitive capacities and available epistemic states.

Based on these considerations, we can draw two general conclusions about the cognitive complexity involved in generating the defeaters discussed in §3.3. First, producing these defeaters requires greater cognitive effort than other inferences in KFF cases, as it involves combining at least two deductive principles. In contrast, forming the proxy premise a or inferring q typically involves only one inference.

Second, although S may be able to apply deductive processes reliably, the presence of arbitrary premises complicates the process of defeater formation within an externalist framework. This highlights the importance of S having available cognitive states that relate to their epistemic abilities and connects back to our earlier discussion regarding the semantic content of defeaters.Footnote ²⁷ Therefore, let us now turn to the issues surrounding the semantic content of defeaters.

In an externalist framework, S would likely struggle to acquire the defeaters discussed in §3.3., as doing so would demand a level of conceptual sophistication that is typically beyond the cognitive reach of a non-expert agent. Specifically, S would need to trace the inferential chain to falsify a and understand the conceptual content of each step in that chain. This level of understanding is crucial. For example, grasping undercutting or rebutting defeaters requires advanced meta-epistemological skills, including a good understanding of abstract terms such as “reliability,” “defeater,” and “approximate beliefs.”

Given this, while the KFF scenarios offer limited information and do not rule out the possibility that S may possess these skills, it seems safe to expect that an unskilled agent would lack the cognitive resources necessary to defeat the warrant for a. Additionally, even without considering these details, we have already observed that combining arbitrary premises with deductive reasoning typically leads to unreliable results. Therefore, unless the KFF cases are further clarified, it seems reasonable to conclude that these defeaters would not be epistemically available to S.

In conclusion, the abovementioned externalist analysis suggests that S can follow an undefeated, stable path to q, aligned with their doxastic abilities. Hence, this externalist account better predicts that S has the inferential skills to support the KFF advocates’ thesis within a reasonable epistemic framework. Specifically, if S can derive the proxy premise a from p, we can clarify how S might know, “I am not late for my 7 p.m. appointment.” By applying similar reasoning to that used for a, S can combine their belief in a with relevant propositions, such as “if it is approximately 2:58 p.m., then I am not late for my 7 p.m. appointment.” This allows S to establish an undefeated propositional warrant for q, even if ¬p is later discovered.

Crucially, this alternative inferential process remains reliable in counterfactual and real-world scenarios. As shown in the comparison between the externalist and internalist graphs, if S discovers that p is false, they can still infer both a and q. This consistency satisfies the externalist’s criteria for propositional warrant, preserving S’s epistemic position regarding q and meeting the conditions for doxastic warrant and knowledge in the actual world.

Lastly, the externalist framework also effectively addresses Gettier cases. In these circumstances, S lacks evidence regarding Havitt’s car make, and their only method of forming a warranted belief about a is to guess which peer owns a Ford. Therefore, since S lacks reliable means to establish a stable alternative path, they cannot recover their knowledge of q.