2. The modalist solution to the Gettier problem

The core idea behind modalism is that knowledge cannot be accidentally true belief; there must be a stable modal connection between belief and truth. Within this theoretical family, the safety principle has gained much popularity:

At first glance, the Safety Principle seems to offer plausible diagnoses of both environmental and intervening luck. With respect to cases of environmental luck, Barney’s belief that he sees a barn is unsafe: in many nearby possible worlds where his gaze shifts slightly, he sees a fake barn and forms a false belief by the same visual method. Such an easy possibility of error makes his belief unsafe and thus excludes it from knowledge. Turning to cases of intervening luck, Smith’s belief that the man who will get the job has ten coins in his pocket is also unsafe. We can imagine many nearby possible worlds in which Smith still gets the job but happens not to have ten coins. In these worlds, his belief, formed via the same method, would be false. By attending to these counterfactual possibilities, the Safety Principle seems to capture epistemic luck involved in Smith’s belief.

There are, however, more recalcitrant cases of intervening luck. Consider the following:

On a natural reading of the case, John’s belief appears safe. On the one hand, given how the machine has been set up, in all sufficiently close possible worlds it outputs only “Type B.” On the other hand, it is a modally stable biological fact that John’s blood type is B and, given the local culture, he could not easily have formed beliefs about other people’s blood types. Together, these facts ensure that, by consulting this machine, John could not easily have formed a false belief. But intuitively, John does not know that his blood type is B. His true belief results from a coincidence: the machine is programmed to output “Type B” regardless of the user’s actual blood type, while John independently happens to have blood type B.

A modalist might argue that the safety principle should be supplemented with a more refined account of which possible worlds are relevant to the safety assessment. One influential way of refining safety appeals to normality. On a normality-based conception, safety is assessed not simply across nearby worlds, but across normal possible worlds:

A defender of this view might offer the following reply to the blood type case. There may be normal worlds in which the machine malfunctions in a different way, producing a blood type result other than “Type B.” In those worlds, John forms a false belief. His belief then fails to be safe.

However, normality depends at least partly on the actual states of affairs (see Valaris Reference Valaris2022, Zhao Reference Zhao2025 Reference Zhao). In the present case, the actual states of affairs can be easily stipulated to block the above response. In particular, suppose that the manufacturer of the blood type machine has a strong interest in impressing the hospital director, whose blood type is B, and so deliberately programs the machine to produce only Type B results. Under this stipulation, worlds in which the machine outputs a different result appear abnormal – or at least more abnormal than the actual world in which the machine produces only Type B results. This verdict is well supported by Smith’s (Reference Smith2016) influential account of normality, according to which abnormal events are those that call for special explanation. Given the manufacturer’s interest and the machine’s actual design, a deviation from the “Type B” output would itself require some further explanation. Such worlds may therefore be excluded from the relevant normal worlds, in which case John’s belief once again satisfies the safety condition.

Blood type exposes a burden that a purely modal approach must bear.Footnote ³ Whether John’s belief counts as safe depends on which counterfactual possibilities are considered as relevantly close or normal. If worlds in which the machine outputs something other than “Type B” are included, John’s belief may be unsafe; if they are excluded, the belief may remain safe. Modalism therefore requires a principled account of relevant-world selection that delivers the appropriate verdict across such cases. Blood type illustrates how difficult this task can become when the truth of a belief results from two modally stable but explanatorily independent features of the actual situation. In this sense, Blood type reveals a genuine difficulty for a purely modal treatment of intervening epistemic luck.

3. The explanationist solution to the Gettier problem

In recent years, explanationism has gradually developed into a rival to modalism.Footnote ⁴ In contrast to modalism, explanationists reject the move of grounding knowledge in a counterfactual relation between belief and truth. Rather, they insist on an appropriate explanatory relation between belief and truth in the actual world. Explanationists capture this thought in the slogan “knowledge is believing something because it’s true” (Bogardus and Perrin Reference Bogardus and Perrin2022). More precisely:

This immediately raises a question: what does it mean for the truth of p (or the fact that makes p true) to constitute a core explanation of one’s belief that p? In response, explanationists appeal to the salience and indispensability of the explanation: the truth-making fact must be the irreplaceable, pivotal factor in the explanatory chain.

More specifically, Bogardus and Perrin (Reference Bogardus and Perrin2022) adapt the kairetic test developed by Strevens (Reference Strevens2011) in the philosophy of science to give this idea precision. The procedure works by progressive abstraction: beginning with a full set of potential explanantia for the belief, one strips away potential explanantia in the set until further abstraction would make the explanation of the belief fail. The remaining explanantia in the set are the difference-makers. Consider a subject who knows it is raining outside through an abundance of perceptual experience: she sees raindrops on the window, hears rain on the roof, and feels the moisture in the air. The abstraction procedure strips away these details one by one. As such, we can abstract away any particular perceptual modality – whether she saw the rain, heard it, or felt it – and a recognizable explanation of her belief still stands. But once we remove the fact that it is actually raining outside, the explanation collapses entirely. That fact is indispensable for the explanation of why she holds the belief. It is precisely this indispensability, revealed through the abstraction process, that qualifies the fact that it is raining as the core explanatory item.

It is also worth noting that the explanatory relation invoked by explanationists is broader than causal explanation. This is important when dealing with non-empirical knowledge. Take mathematical or logical truths. The abstract fact that 2 + 2 = 4 does not “bring about” our belief through a causal process. So if the conditions for knowledge were strictly confined to causal explanations, it would be impossible to account for how subjects acquire knowledge of this kind. The explanationist response is that even in the absence of causal interaction, mathematical facts can be grasped through the subject’s rational intuition, and it is the fact itself – apprehended through that rational intuition – that plays the core explanatory role in belief formation (Faraci Reference Faraci2019, Bogardus and Perrin Reference Bogardus and Perrin2022). The same holds for future facts not yet realized. Although future facts cannot, in any temporal sense, be “causes” of present beliefs, a subject can still apprehend them through rational intuition and/or reliable inference, allowing them to figure as core explanatory items that account for our beliefs about the future (see Bogardus and Perrin Reference Bogardus and Perrin2022, Zhao Reference Zhao2025 Reference Zhao).

The explanationist framework is well suited to diagnosing Gettier cases that involve intervening luck. In the blood type case, where a modalist diagnosis depends on how the relevant counterfactual possibilities are selected, explanationism’s verdict is rather straightforward: the immediate explanation of John’s belief that his blood type is B is the machine’s output, but – given the setup of the case – this output has nothing to do with John’s actual blood type. That is, the output is not at all explained by John’s actual blood type. As such, there is an explanatory break between the belief and the fact. Consequently, John does not know. Likewise, in the classic job-seeker case, the core explanation for Smith’s belief involves his mistaken assumption that Jones will get the job and his observation that Jones has ten coins in his pocket, not the fact that the person who gets the job (Smith himself) has ten coins in his pocket. By requiring the truth-making fact itself to constitute the salient explanation for why the belief is held, explanationism effectively excludes intervening luck.

However, this focus on the actual-world explanatory chain also becomes explanationism’s Achilles’ heel when it comes to cases of purely environmental luck. In the classic barn case, Barney’s belief seems to satisfy the explanationist requirement: what explains why he believes that he sees a barn is exactly the fact that there really is a barn. The explanatory chain is wholly intact, so that the explanationist seems forced to conclude that Barney possesses knowledge.

Faced with this objection, Bogardus and Perrin (Reference Bogardus and Perrin2022) argue that the genuine core explanation in the barn facade case should encompass the subject’s overall environmental information – namely, that the subject is situated in an environment populated with fake barns. They offer a vivid analogy: if a truck scatters nails across the road and the car behind gets a flat tire, what figures crucially into the explanation is the ubiquity of nails on the road, not the presence of the particular nail that caused the puncture (Ibid.: 193). Similarly, in the barn facade case, the difference-maker is the pervasive presence of barn-like objects, not the real barn in front of Barney. On the difference-maker test, even if we abstract away the presence of the single real barn, the explanation of Barney’s belief still stands (thanks to the presence of numerous barn facades). So the real barn is not indispensable as an explanans.

We maintain, however, that the explanationists’ diagnosis above is untenable. Consider first the following variant of the barn facade case:Footnote ⁶

Intuitively, given that the epistemically unfriendly barn facades have been physically shielded, Baum knows that he sees a barn. To reach this verdict, the explanationist seems compelled to grant that the core explanation for Baum’s belief is the fact that there really is a barn there. After all, with the fake barns shielded, if the presence of the real barn were abstracted away, nothing would explain why Baum forms the belief.

Now recall the explanationist reply to the original barn facade case: it is the pervasive presence of the fake barns that constitutes the core explanation of Barney’s belief. On this basis, Bogardus and Perrin (Reference Bogardus and Perrin2022) deny that Barney knows. This diagnosis, however, sits uneasily with the Shielded Barn case. Both Barney and Baum actually perceive the one real barn in the area; both are actually surrounded by fake barns. The two cases share the same features regarding the actual perceptual situation, differing only in one respect: the presence of the screens. Yet the explanationist’s verdicts diverge sharply – Barney lacks knowledge, while Baum knows. The explanationist therefore owes us an account of why the screens make this difference.

Obviously, on an intuitive level, the screens make a significant epistemic difference. But for explanationists, they need to provide an answer to the question of why the presence of screens would make a difference to the explanation of the subject’s belief. The screens stand in no direct causal or constitutive relation to Baum’s perception of the real barn: his visual chain runs from the real barn to his retina to his belief, and the screens figure nowhere in this chain.

Why, then, does the presence of screens affect the explanatory chain? Only one answer seems available: the screens matter because they affect what Baum could have perceived. Without the screens, Baum’s gaze could easily have fallen on a fake barn, and he would have formed a false belief. With the screens in place, however, this possibility is closed off. The screens, in short, do not alter what Baum actually perceives; they alter what he could have perceived in other possibilities. And this is surely a modal property: a fact about possible worlds in which the subject’s gaze, position, or perceptual conditions vary.

The explanationist’s treatment of the two cases therefore tacitly relies on the modal profile that the screens determine. What looks like a difference in core explanation is, on inspection, a difference in modal accessibility. Strip away this modality, and the asymmetry between Barney and Baum collapses. This leaves explanationism with an uncomfortable dilemma. On the first horn, suppose the explanationist lets the actually present fake barns enter the core explanation in both cases. This would correctly diagnose Barney’s lack of knowledge, but would undesirably deny Baum’s knowledge as well. On the second horn, suppose the explanationist concedes that modal accessibility to the fake barns – as determined by the presence of the screens – does make a difference to the core explanation of the subjects’ beliefs. In that case, explanationism diagnoses these cases only by appeal to modal considerations and thus becomes a modal account in disguise, failing to constitute a genuine alternative to modalism. Either way, explanationism cannot plausibly diagnose cases of environmental luck without importing the very modal considerations it was designed to replace.Footnote ⁷

4. Modal explanationism

The respective strengths and limitations of modalism and explanationism reveal that each captures an important yet incomplete dimension of epistemic non-accidentality. This motivates the search for a framework that integrates the two dimensions rather than treating them as competing alternatives.

We propose such an alternative; consider:

Condition (1) draws on explanationism’s explanatory connection requirement between belief and truth, whereas Condition (2) is a modal reinforcement of this connection. ME incorporates both modal and explanatory aspects. However, unlike extant explanationism, ME requires the explanatory connection to hold not only in the actual world but also in counterfactual possibilities. And unlike the modal safety principle, which requires beliefs to be robustly true in counterfactual possibilities, ME instead requires the explanatory connection demanded by explanationism to continue to hold in relevant counterfactuals.Footnote ⁸

These two conditions form coherent and complementary evaluative dimensions, jointly defining knowledge as a non-accidental epistemic success. In particular, Condition (1) examines the belief’s etiology in the actual world, ensuring its source is traceable to the fact that renders the belief true. Condition (2) examines the modal reliability of this explanatory connection, requiring that it remains robust in counterfactuals where the subject still forms the belief in the same way as in the actual world. By drawing on insights from both modalism and explanationism, ME inherits the theoretical advantages of each while avoiding their major pitfalls in dealing with Gettier cases.

Starting with the barn facade case, Barney’s belief satisfies Condition (1) of ME. Unlike Bogardus and Perrin’s treatment of this case, ME can maintain the intuitive observation that the fact making Barney’s belief true – that there is a real barn before him – is the salient explanation for his belief. The crux is that the belief does not satisfy Condition (2) of ME. Consider the possible worlds in which Barney continues to form the belief that he sees a barn via perception. Given the abundance of fake barns in his cognitive environment, in the vast majority of such worlds Barney would be perceiving a fake barn and thereby forming a false belief. In those worlds, Barney’s belief is no longer explained by the presence of a real barn – his perceptual chain is linked to a fake one – and the explanatory connection between truth and belief no longer exists. Condition (2) thus captures the environmental luck involved in Barney’s true belief and correctly excludes it from knowledge.

Second, in the blood type case, ME also delivers the desired verdict. John’s belief that his blood type is B may exhibit considerable modal stability, at least in the sense that, across the relevant nearby worlds, the machine continues to output “Type B” and John’s blood type remains B. The crux, however, is that the fact that John’s actual blood type is B plays no explanatory role in his believing that his blood type is B. He forms the belief because the machine produces the Type B result, but the machine produces that result not because it correctly detects John’s actual blood type. The belief therefore fails Condition (1). Nor is Condition (2) satisfied: in nearby possible worlds in which John forms his belief in the same way, namely by using the machine, the explanatory connection between the truth-making fact and his belief still does not obtain. This clarifies a comparative advantage of ME. A purely modal diagnosis of the blood type case depends on whether worlds in which the machine produces a different output count as sufficiently close or normal. ME need not settle that question in order to reach a verdict here. Since the fact that John’s blood type is B does not explain his belief in the actual world, this actual epistemic defect already suffices, according to ME, to rule out John’s belief as knowledge.

Zagzebski (Reference Zagzebski1996) famously argues that the crux of the Gettier problem lies in the decoupling of justification and truth, with the gap between them leaving room for knowledge-undermining luck. Traditional epistemology attempts to bridge this gap by refining the justification condition, or by replacing justification with some alternative epistemic condition, but often with limited success. Our suspicion is that this is largely because most theories locate knowledge-undermining luck either in the actual-world etiology of belief or in its counterfactual profile, but not in both. ME addresses the gap at both levels. At the level of the actual world, Condition (1) anchors the source of the belief to the truth-making fact itself, establishing a direct connection between belief and truth and thereby blocking the familiar form of “believe because of A, true because of B” luck. At the counterfactual level, Condition (2) ensures that this explanatory connection is stable and non-accidental. It is precisely this dual grasp of the actual and the counterfactual that gives ME a more comprehensive framework for addressing the Gettier problem.

5. The challenge from deviant causal chains

Since ME inherits the core resources of explanationism, it also inherits a challenge that explanationist theories commonly face: deviant causal chains.Footnote ⁹ We can construct a Gettier case in which the fact that makes the belief true explains the subject’s belief via a deviant causal chain, and in which this explanatory relation remains stable across nearby possible worlds. Then both of ME’s conditions appear to be met, forcing ME to deliver a knowledge verdict that intuitively seems incorrect. Here is such a case:

Intuitively, Harold does not know there is a sheep in the field. Yet his belief seems to satisfy both conditions of ME. Take Condition (1) first: the fact that there is a real sheep in the field explains why Harold believes that there is a sheep in the field, because it explains why the fake sheep – the disguised sheepdog – is there. And the presence of the fake sheep in turn explains why Harold forms his belief. Now consider Condition (2). In nearby worlds where Harold forms the belief in the same general way, he either sees the fake sheep or the real sheep. Given the case’s stipulation, even when he sees the fake sheep, the fake sheep is present only because there is a real sheep nearby. Thus, the explanatory connection between truth and belief remains intact across nearby worlds. If Harold sees the fake sheep, the real sheep explains the fake sheep’s presence, and the fake sheep’s presence explains Harold’s belief. If he sees the real sheep, the real sheep’s presence explains his belief directly. The explanatory relation between truth and belief therefore appears to be modally stable. Thus, one may argue that ME still struggles to handle the Gettier problem.

In reply, the issue lies in how we interpret the spatial constraint implicit in the “keeping company” stipulation of the case. There are two plausible readings of this constraint, and ME can plausibly accommodate both.

On the weak reading, the fake sheep “keeping company” with a real sheep requires only that the real sheep be spatially close to the fake one. It does not require that the two occupy the same strictly demarcated region. To be sure, Harold believes there is a sheep in the field because he sees a very convincing fake sheep, and the fake sheep is there to accompany a real sheep. But notice that, for this explanation, the precise location of the real sheep – whether in the field, on the hillside, or down by the stream – does not really matter. As Strevens (Reference Strevens2011) might put it, the real sheep’s precise location is not a difference-maker. We can easily imagine, for example, that the fake sheep is at the edge of the field while the real sheep grazes on the adjoining hillside: the two are close enough to satisfy the “company” stipulation, yet strictly speaking the real sheep is not in the field. This means that the real sheep’s being in the field (rather than on the neighboring hillside) is a purely accidental spatial fact – one that plays no crucial role in the explanation of Harold’s belief. In that case, Condition (1) is not satisfied: Harold’s belief that there is a sheep in the field is not saliently explained by the fact that there is a sheep in the field. ME correctly rules that Harold does not know there is a sheep in the field.Footnote ¹¹

Alternatively, one may read “keeping company” more strictly, as requiring genuine spatial co-location: wherever the fake sheep is placed – in a field, on a hill, or anywhere else – the real sheep must be within that very same strictly demarcated region. On this strong reading, the co-presence of the real and fake sheep in the field is no longer a mere coincidence but is guaranteed by the standing arrangement described in the case. Under this setup, ME does appear to count Harold’s belief as knowledge. But this, we think, is not an implausible result. At the very least, it is much less clear that Harold lacks knowledge in this scenario. The driving force behind the Gettier intuition is knowledge-undermining luck – that the subject’s belief is true merely by luck or accident. In the traditional, coincidental version of the sheep case, the real sheep’s presence behind the decoy is a sheer fluke, which drives our intuitive verdict that Harold lacks knowledge. By contrast, under the strict co-location setup, the real sheep’s presence in the same area is no longer a coincidence: the fake sheep is placed there because a real sheep is there, and this relation is maintained across the relevant nearby circumstances. Once this feature is built into the case, the classic Gettier intuition is considerably weakened.

Indeed, under this stipulation, the relation between the fake sheep and the real one resembles the relation in the following case. A person looks at what he takes to be a window and has a visual experience as of a tree outside, thereby coming to believe that there is a tree outside. Unbeknownst to him, however, he is looking at an ultra-high-definition screen that displays, perfectly and in real time, the actual tree standing directly behind it. In such a case, it does not seem unreasonable to say that he knows there is a tree outside.Footnote ¹² Harold appears to be in a relevantly similar position. On the strict reading, the fake sheep is present in the relevant region because a real sheep is there, and this dependence is not a one-off coincidence but a stable feature of the setup. Thus, on the strict reading, the intuition that Harold lacks knowledge seems substantially weakened. The case is no longer a straightforward counterexample to ME.

6. Comparison with virtue epistemology: testimonial knowledge and Gettier cases

In contemporary epistemology, another attractive approach to knowledge is virtue epistemology. According to it, knowledge is a kind of cognitive achievement – a true belief whose correctness is credited to the agent’s intellectual virtues or competences. However, virtue epistemology faces a dilemma when it comes to the social dimension of knowledge – a dilemma that highlights the distinctive advantage of ME.

The dilemma faced by virtue epistemology, as Lackey (Reference Lackey2007) has pointed out, stems from the inherent tension in its “credit” condition. On the one hand, to effectively eliminate Gettier-style epistemic luck, virtue epistemologists need a strong credit condition, requiring that the acquisition of a true belief be primarily credited to the agent’s own intellectual virtues. In paradigmatic Gettier cases such as the blood type case, the correctness of the belief is primarily a product of luck rather than the agent’s own intellectual virtues. So a strong credit condition correctly excludes it from knowledge.

On the other hand, when we consider testimonial knowledge, the situation is reversed. Consider the following case:

Intuitively, Morris acquires knowledge regarding the Sears Tower’s location through this exchange. Yet under a strong credit condition, the correctness of Morris’s belief is primarily credited to the local resident’s geographical knowledge and honest character, not to Morris’s own intellectual virtues – he merely happens to ask a passing local. If the correctness of a belief must be primarily credited to the agent’s own intellectual virtues, Morris’s knowledge is in jeopardy. To accommodate such testimonial knowledge, virtue epistemologists are typically forced to weaken their core condition, adopting a weak credit condition according to which it suffices for knowledge that the correctness of the belief is only partially credited to the agent’s own intellectual virtues.

But an overly lenient credit condition would wrongly classify Gettier cases as knowledge. In the blood type case, John does exhibit some basic intellectual virtues (following instructions for drawing blood, reading the report, reasonably trusting hospital equipment). The weak condition thus wrongly attributes knowledge to John.

Virtue epistemology thus faces an uncomfortable trade-off: strengthening the credit condition to handle Gettier cases undermines its explanation of testimonial knowledge, and weakening the condition to accommodate testimonial knowledge reopens the door to Gettier cases.

Although we do not think this difficulty is insurmountable for the virtue epistemologist,Footnote ¹³ the fact that ME does not face it at all speaks in its favor. ME’s theoretical framework naturally circumvents this dilemma. Virtue epistemology focuses on the question of agential attribution: to what extent can the correctness of the belief be credited to the agent’s own virtues? ME, however, focuses on a question of objective connection: is the belief explained by the relevant worldly fact? This objective explanatory chain can – and often does – span across multiple agents.

In the Chicago case, ME delivers the following diagnosis. Condition (1) is satisfied because a complete explanatory chain runs from the fact regarding the actual location of the Sears Tower, through the local resident’s genuine knowledge and sincere testimony, to Morris’s belief. This is a chain that originates in a worldly fact and passes through a reliable intermediary before reaching its endpoint in Morris’s belief. Condition (2) is also satisfied: in the vast majority of nearby worlds in which Morris forms his belief in the same way – say, by sizing up the situation and asking someone who appears to be a local adult – the person he asks is reliable, and the explanatory connection between his belief and the actual location of the Sears Tower is preserved.Footnote ¹⁴ ME therefore correctly classifies Morris as knowing where the Sears Tower is.

At the same time, ME remains immune to knowledge-undermining luck involved in Gettier cases. As mentioned earlier, in the blood type case, although John exercises his ordinary cognitive abilities in reading and trusting the result, the fact that makes his belief true (i.e., that his blood type is indeed B) and the source of his belief (i.e., the report provided by the malfunctioning machine) are explanatorily disconnected. Condition (1) therefore fails, and John does not know. In sum, ME is flexible enough to accommodate the social dimension of knowledge, yet rigorous enough to rule out luck involved in Gettier cases.