Single Premise Deduction and Defeat

ORIGINAL RESEARCH Erkenntnis https://doi.org/10.1007/s10670-025-01024-5 Abstract Preface and Lottery paradoxes have shown that competently deducing a novel conclusion from multiple justied premises does not necessarily result in a justied belief. However, those paradoxes do not arise if only a single premise is used. Yet, in recent years, justication closure for single premise deduction has also been challenged. Schechter has presented a case that, according to him, thwarts any attempt to formulate a general principle that links competent single premise deduction to justication. This paper develops a justication closure principle for single premise deduction that accommodates Schechter’s case, thus showing that his pessimism regarding the possibility of such principles is mistaken. I also argue that an alternative reply to Schechter’s case fails and that safeguarding justication closure constitutes progress for establishing knowledge closure for single premise deduction. Keywords Deduction · Single premise deduction · Justication · Closure · Logic It initially seems plausible to think that if an agent believes something with justication, she is in a position to believe with justication anything she can competently deduce from it. Longstanding issues with lottery (Hawthorne, 2003) and preface paradoxes (Makinson, 1964) have shown that things are more complicated, and that competent deduction from justied belief does not always result in justied conclusions. However, these problems mostly plague deductions with multiple premises. If the agent only deduces from one premise, these paradoxes do not arise. Thus, one might think that for single premise deductions, but not multi-premise ones, competently deducing from justied beliefs results in justied conclusions. Received: 4 May 2025 / Accepted: 17 September 2025 © The Author(s) 2025 Single Premise Deduction and Defeat AukeMontessori1 Auke Montessori [email protected]; [email protected] 1 Philosophical Faculty, University of Hradec Králové, Rokitanského 62, Hradec Králové 500 03, Czech Republic 1 3 A. Montessori But even single premise deduction has been attacked. Schechter (2013) presents a case in which the risk of getting single premise deductions wrong is very high. The agent is aware of this fact and thus has a defeater for conclusions based on those deductions. Like multi-premise deductions, single premise deductions do not guarantee justication.1 According to Schechter, his challenge is strong enough to prevent us from formulating any formal principle connecting single premise deduction and justication (Schechter, 2013, pp. 450–451). This paper claims that Schechter is mistaken. A plausible principle can be given that accommodates Schechter’s case. While he succeeds in making the relation between single premise deduction and justication more complicated, things are not as dire as they seem. The rst section describes the principle I have in mind and claims it can account for Schechter’s case. The second section discusses Tang’s (2018) reply to Schechter’s case, but argues that it is mistaken. We need the principle introduced here to account for Schechter’s case. I conclude by claiming that solving problems related to justi- cation closure helps make knowledge closure more viable. 1 The No-Defeat Principle The single premise closure principle Schechter considers is the following one: (Single-Premise Closure*)2 Necessarily, if S has a justied belief that p, comes to believe that q solely on the basis of competently deducing it from p, while retaining the justied belief that p throughout the deduction, and S does not have a defeater for the claim that the deduction was competently performed, then S has a justied belief that q. (Schechter, 2013, p. 437) All the constraints of SPC seem reasonable. If the agent lacks, or loses, the justied belief that p, the ground for believing q is gone. Maintaining the justied belief in p is thus required. Competent deduction is invoked to outlaw cases in which incompetent agents deduce q from p by accident or through pure luck. The deduction should be the result of a skillful use of reason by the agent, otherwise it cannot justify. Finally, the agent must believe q on the basis of the deduction, rather than basing it on some bad reasons or not believing q at all. After all, the agent could complete the deduction and then either suspend judgement about q or endorse it for, say, emotional reasons. Schechter also includes the constraint that the agent must not have any defeaters regarding the competence of their deduction. After all, even if a deduction was 1 Lasonen-Aarnio (2008) has presented a similar case to put pressure on knowledge closure for single premise deduction. But where the agent has a defeater for Schechter, the unreliability alone is enough to cause problems for knowledge closure for Lasonen-Aarnio. For the purposes of this paper, I focus on justication closure. Note that Lasonen-Aarnio ultimately does not reject single premise closure but instead claims that it stands or falls together with multi-premise deduction. 2 I will drop the “*” going forward. 1 3 Single Premise Deduction and Defeat competent, the agent might receive misleading information that it is not. Consider the following case: Jealous Mathematician Elena is a rst-year graduate student in Mathematics. She performs a highly complex single step deduction to show that a certain theorem follows from an uncontroversial axiom in mathematics. She shows her work to a famous mathematician, whose judgement is normally highly trustworthy. The mathematician realizes that deduction is valid but is struck by a sudden and uncharacteristic t of jealousy. He falsely informs Elena that the deduction must be invalid, because it is known in the eld that the theorem does not follow from the axiom. In this case, it seems that Elena has a defeater for her deduction.3 The axiom is uncontroversial, and she has every reason to assume that the famous mathematician’s judgement is true. Especially given her relatively novice status, it seems far more likely that there is something wrong with her reasoning than that well-established ndings in the eld are wrong. Thus, the conclusion she draws is not justied, despite being the result of a competent deduction from a justied belief. SPC accounts for this by adopting the defeater constraint. SPC thus gets the right result in Jealous Mathematician. However, Schechter claims that there are other examples in which SPC does not reach the right result. Here is his example, which I will call Humility: Consider a very long sequence of competently performed simple single premise deductions, where the conclusion of one deduction is the premise of the next. Suppose that I am justied in believing the initial premise (to a very high degree), but have no other evidence about the intermediate or nal conclusions. Suppose that I come to believe the conclusion (to a very high degree) solely on the basis of going through the long deduction. I should think it likely that I’ve made a mistake somewhere in my reasoning. So it is epistemically irresponsible for me to believe the conclusion. My belief in the conclusion is unjustied. (Schechter, 2013, pp. 438–439) In Humility, the agent has a defeater. If the sequence of deductions is long enough (10,000 steps, say) it seems likely that at least one mistake is made. The agent is aware of this fact about her reasoning. There does not happen to be a mistake, but the agent does not know that nor is she justied to believe that there is no mistake. Hence, the agent should not believe the conclusion, or at least would not be justied if she did so. 3 Notice that even if the information itself is false, it is still true that the otherwise reliable mathematician claimed that the deduction is invalid. Thus, there is a defeater even if one holds that there can be no false defeaters (Williamson, 2000). 1 3 A. Montessori The problem for SPC is that it predicts that the agent is justied to believe her conclusion. This is clearly a mistake, given the defeater the agent possesses. It is worthwhile to go into detail as to why SPC gets Humility wrong, despite having a constraint about defeaters. Since all the other conditions are clearly met, it is only the defeater constraint that could save SPC in this case. In Humility, all the individual steps are simple. The rst premise is secure, by assumption. The second premise follows quite obviously from the rst, and the agent has no reason to think that that particular deduction is problematic. Hence, on SPC, the second premise is secure as well. That means that the agent now possesses a justi- ed belief that the second premise is true. Given its simplicity, the agent also has no reason to think that there is anything wrong with the deduction from the second to the third premise. She thus also obtains the third premise as a justied belief. This continues until, many premises later, the conclusion is reached. The agent has no reason to think that the conclusion does not follow from the nal premise. After all, that nal deductive step is as simple as the others. At no other point in the deduction has the agent had a reason to think that some specic step was invalid. None of the individual steps are defeated, so all the individual steps meet the requirements of SPC. Since SPC is met for every step of the deduction, the result should be that all the premises and the conclusion are justied. Yet, overall, the agent should be wary about the truth of the conclusion. A mistake somewhere, though unclear where exactly, is likely. SPC predicted a justied belief, but the result is an unjustied one. SPC is false. Instead of SPC, I propose the following alternative principle: No-Defeat Principle (NDP) Necessarily, if S has a justied belief that p, comes to believe that q solely on the basis of competently deducing it from p, while retaining the justied belief that p throughout the deduction, and S does not have a defeater for the conclusion that q, then S has a justied belief that q. The only dierence between SPC and NDP is the italicized line.4 The purpose of the change is to accommodate Humility, as it accounts for any defeater to the conclusion. In Schechter’s SPC, it reads “S does not have a defeater for the claim that the deduction was competently performed” (Schechter, 2013, p. 437). SPC thus only considers defeaters for the claim that the specic deduction in question is competent. This goes awry in Humility. After all, the problem with the nal step is not that the deduction from the nal premise to the nal conclusion might be incompetent. Nor are there any defeaters aimed at any particular deductive step. None of the specic deductions are defeated and hence all of them lead to justied beliefs on SPC. Schechter is right that the agent lacks a defeater of that kind and that this causes SPC to fail in Humility. However, the conclusion is defeated because the agent knows that there is likely to be a mistake somewhere in the long sequence. 4 Just like Schechter’s SPC, NDP is mildly internalist in that it does not assume that reliability or safety is required for justication. I take this mild form of internalism regarding justication to be plausible enough to proceed. 1 3 Single Premise Deduction and Defeat On NDP, unlike SPC, that defeater is enough to block justication by single premise deduction. After all, that defeater is a defeater to the belief that q is true. Thus, NDP reaches the right result, namely that the conclusion in Humility is not justied. NDP can also account for cases like Jealous Mathematician, as the defeater in that case defeats both the idea that the performed deduction was competent and the conclusion that followed.5 We thus have a single premise closure principle that clearly links single premise deduction to justication, without falling to counterexamples. Schechter’s pessimism about the possibility of such principles is mistaken.6 One way to read NDP is as a kind of prima facie justication closure. Considered by itself, competent single premise deduction leads to justied beliefs. However, the world might interfere, by defeating the prima facie justication provided by the deduction. Still, when all goes well, beliefs obtained through competent single premise deductions from other justied beliefs are justied. An objection to NDP might be that it is too close to being trivial. One might think there is something unsatisfying about a principle that merely states that something is justied unless defeated. It might be read as competently deduced conclusions being justied, unless some trouble occurs. However, this objection is too quick. First, it is not trivial that competent deduction grants justication by default, unless defeat is encountered. This only holds for a few choice phenomena. For example, one might think it holds true for perception, memory or inference to the best explanation. It does not hold for a plethora of other phenomena, like desire, hope or guessing. Merely hoping that some conclusion holds does not generate any justication, even if there are no defeaters. According to NDP, competent deduction is special in that it does generate default justication, which only defeat can dislodge. Secondly, defeat is a specic kind of epistemic trouble. It is not the case that any kind of epistemic trouble, like being unreliable or having a dubious etiology, will prevent competent deduction from providing justication on NDP. Only defeat can do so. Defeat only occurs under specic conditions. It is not enough if there is something that tells against the conclusion or its ground. The agent must also be aware of the defeater and appreciate its relevance. If the agent is unaware of the defeater, or fails to grasp its relevance, then the defeater cannot perform its function and thus is not a true defeater. After all, a defeater is essentially evidence agents have against a conclusion or a ground, and one can only use evidence one is aware of and appreciates (McCain, 2023; Montessori, 2025). Of course, it might also suce if the agent is merely in a good position to become aware of and appreciate a defeater, but that is still a fairly 5 In principle, NDP can also accommodate cases where only the conclusion is defeated and nothing about the performed deduction. Any kind of defeater, be it undercutting, higher-order or rebutting, can block justication. And this seems like the right result, as any defeated conclusion, no matter how exactly it is defeated, is unjustied. However, there do not seem to be cases where the conclusion, but nothing about the deduction, is defeated. After all, any reason to doubt the conclusion is a reason to doubt the deduction or the rst premise. If the rst premise is true and the deduction valid, the conclusion must be true. Any reason that suggests that the conclusion might not be true thus also throws doubt on the rst premise and/ or the deduction. 6 It also does not make use of a “thin” notion of defeat like Schechter discusses in footnote 32 (Schechter, 2013, pp. 439–440). 1 3 A. Montessori specic and constrained circumstance. On NDP, the default justication competent deduction delivers only fails under fairly specic circumstances, namely those in which defeat occurs. Thus, NDP is not as broad or trivial as it might seem at rst. To make this point clearer, take the following example. We might have an agent who is totally incapable of self-reection, perhaps due to mental handicap or some form of brain damage. She is thus incapable of having thoughts about her own fallibility when it comes to deduction. Otherwise, she is a normal epistemic agent, who is reasonably gifted when it comes to deduction. If she found herself in a situation like Humility, she would lack any defeaters to her conclusion. After all, she is incapable of realizing that she probably made a mistake somewhere along the way. Even if some outsider suggested this, she would be unable to understand the suggestion. Given her condition, she is also not in a good position to become aware of or appreciate the relevant defeater. Following NDP leads us to conclude that the unreective agent would reach a justied conclusion. After all, she reached her conclusion through competent deduction and lacks any defeaters to her conclusion. While her conclusion is unsafe in some ways, a lack of safety can be combined with being justied, as fake barn cases show (Goldman, 1976). It thus seems to me that this is the right result. What is more, it shows that NDP is not trivial. Default justication only disappears specically through defeat, which might be absent even in otherwise epistemically treacherous circumstances. Another possible objection is that the conclusion in Humility is not defeated after all. The knowledge that there probably is a mistake somewhere in the long chain of inferences is a defeater to that chain, but not the conclusion. One ought to doubt whether the conclusion follows from the chain, but that shows little about whether the conclusion is true or false. This objection, as it stands, is too quick. The only reason to believe that the conclusion is true is the long chain. If the agent is (epistemically) forced to dismiss the chain because there is likely a mistake in it, the ground for her belief in the conclusion is undermined, meaning she is no longer justied to believe the conclusion. She of course should not believe that the conclusion is false but rather suspend judgement. However, the result is still that she loses her justied belief in the conclusion. Thus, her justied belief in the conclusion is defeated by her knowledge that the long chain is unreliable, in the perfectly normal sense of defeat in which the justication for the conclusion is undermined. That said, this is not a typical form of undercutting defeat. Typically, in undercutting defeat, it is shown that the ground of a justied belief does not actually support that belief in that particular case. For example, experiencing a wall as red stops being a good ground for thinking the wall is red once you learn that there is a red light shining on the wall. In this case, the idea is rather that the agent should doubt whether the long inference ever supported the conclusion in the rst place, given her limited inferential capabilities (Christensen, 2010; Feldman, 2005). Humility is thus better interpreted as a case of higher-order defeat (Schechter, 2013, p. 442). Higher-order defeat is controversial, and several philosophers think that higherorder defeat is impossible (Smithies, 2019, Chap. 10; Tal, 2020; Titelbaum, 2015). At best, only the second-order belief that the conclusion is reliably formed is defeated 1 3 Single Premise Deduction and Defeat (Coates, 2012; Lasonen-Aarnio, 2014; Williamson, 2014). However, for the purposes of this paper I take it that higher-order defeat is possible and that Humility is an example of it. Finding a solution that can accommodate Humility is more satisfying than dismissing the case out of hand. Besides, following for example Feldman (2005), it seems obvious to me that higher-order defeaters undermine the connection between purported evidence and the conclusion that evidence is meant to support. If it has been shown that some ground is dubious, that should undermine our faith in any belief that relies on that ground. Thus, those beliefs are defeated. I will not defend higher-order defeaters any further here, as that has been done elsewhere (Horowitz, 2013; Schoeneld, 2016; Sliwa & Horowitz, 2015), including by Schechter (2013, pp. 440–450) himself. NDP runs into a problem when it comes to Humility however. On NDP, the nal conclusion is not justied. But what about the nal premise? Surely, if the conclusion is defeated due to doubts about deductive fallibility, the last premise is likewise defeated. The nal premise only took one simple step less to reach and hence is about as likely to have a mistake in its epistemic history as the conclusion. The last premise is not justied, given NDP. Once this is granted, the penultimate premise also looks suspect, for the same reasons. Then we can attack the antepenultimate premise, and so on. The worry is that this could infect the entire sequence of deductions, meaning none of them result in a justied belief. But that should be prevented, as it seems likely that the rst few single premise deductions are suciently secure. For example, the tenth premise surely is a justied belief, as the agent knows she is unlikely to make a mistake in just ten simple steps. However, premise nine thousand looks suspect. NDP needs a story to account for this phenomenon. 7 I take it that various stories are available. One explanation is that as the sequence grows longer, the doubt that there might be a mistake becomes stronger. At some specic point, the doubt is suciently strong to act as a defeater. Thus, there is a specic step t where the premise is justied, because the doubt was not yet strong enough at t-1, while the conclusion is not justied, because at point t the doubt has become strong enough. That is, there is a continuous buildup where doubt becomes stronger and the overall justication for the conclusion becomes weaker. This continues until a critical point is reached where doubt defeats the justication generated by competent deduction and the conclusion is no longer ultima facie justied. While the buildup itself is continuous, on this story, the critical point constitutes a sharp cuto point. Presumably, the agent is not in a position to know exactly when step t is reached. She presumably knows that it is not reached at step 10, but has been passed after step 10,000. This is comparable to Williamson’s claims about the anti-luminosity of knowledge (Williamson, 2000, Chap. 4). Sometimes, we know even though we cannot tell whether we know. Likewise, we sometimes have a defeater even though we cannot tell that it is a defeater. 7 SPC does not run into this problem, as SPC holds that the agent is simply justied throughout. However, should proponents of SPC wish to rescue the principle by claiming that a defeater does arise somewhere in the sequence, this problem would also arise for SPC. 1 3 A. Montessori Of course, this does not mean that the agent is unaware of the defeater or does not appreciate its strength to any degree. As we have seen earlier, in the absence of awareness or appreciation, a defeater is not a defeater. However, I take it that appreciation does not require appreciating the exact strength of a reason or a defeater. Otherwise, there would be preciously little we appreciate. For example, it seems dubious that I appreciate the exact epistemic strength of seeing a dog in the street, even if I realize that it is good enough to justify the belief that there is a dog in the street. Appreciation likely involves a rough grasp on how strong a reason or defeater is, but not an exact one. Hence, it is possible to suciently appreciate a defeater without appreciating how strong it is exactly. That also means one might have a defeater without appreciating or knowing whether it is strong enough to defeat one’s justication for a belief. I take it that this explanation would work. Other explanations are possible, like claiming that there is no sharp cuto point. Instead, the doubt eventually becomes strong enough to act as a defeater, but there is no truth to the matter as to when exactly that happens. I will not discuss those other possibilities or claim that the Williamson-inspired answer is superior.8 I merely note that there are plausible ways to account for the phenomenon. The result is that, pace Schechter, we have a general principle that accounts for the relation between single premise deduction and justication. This result has been reached without denying Schechter’s intuition that the agent lacks justication due to defeat in Humility. Of course, NDP is somewhat complicated, including four requirements, but so was SPC. Unless a serious objection can be found, NDP seems like a suitable account of justication by single premise deduction. 2 An Alternative Reply to Humility Instead of introducing a novel principle like NDP to deal with Humility, one could also try to argue that Humility does not threaten SPC. If successful, this would show that NDP is not needed. I discuss Tang’s attempt to give such an argument but claim that it fails. 9 Tang (2018) has claimed that the real problem in Humility has to do with memory.10 Due to the fallibility of memory, justication tends to erode over time. He compares this to having seen something in the past. Say I catch a brief but clear glimpse of Jesse James’s face as he robs a bank. That glimpse justies my belief that James robbed the bank. However, over time, the epistemic potency of that event erodes. If I saw James a second ago, I should be condent that he was the robber. If my experience took place ten years ago, I should be less condent. This erosion over time 8 Which option one ultimately prefers might, in part, depend on what theory of vagueness is preferred. See (Sorensen, 2023). 9 Smith (2013) has also formulated a reply, though his is aimed at Lasonen-Aarnio’s version of Humility. He claims that the kind of risk in Humility is benign. However, in Schechter’s as opposed to LasonenAarnio case, the problem is that the agent has a defeater, not the actual risk in place. Hence, Smith’s solution fails to save SPC from Schechter. 10 Like Smith, Tang also uses risk to attack Lasonen-Aarnio’s claims, but (rightly) claims that Schechter’s case requires a dierent approach (Tang, 2018, p. 1890). 1 3 Single Premise Deduction and Defeat does not mean that there was no justication in the rst place, since the experience clearly justied my belief while it was taking place. Likewise, SPC is not undermined by Humility. Single premise deduction is justication preserving. However, that justication can erode over time due to the fallibility of memory. The sequence in Humility is long enough for such erosion to occur. The intuition that the agent in Humility is unjustied is thus due to erosion, not due to a problem with closure. SPC thus remains standing. Let us call the phenomenon that Tang describes “memorial erosion”. Memorial erosion occurs when, due to the fallibility of our memory to preserve justication, the justication we have for some belief diminished to such a degree that it is no longer justied. It is unclear how exactly memorial erosion is meant to work on Tang’s account. Below is Tang on Sue, an agent who competently deduces A from B in one step and then waits ten years to competently deduce B from C in one step: Suppose that immediately upon performing the rst deduction from A to B, Sue’s belief that B is justied. But when she performs her second deduction from B to C ten years after t, it’s unlikely that her belief that B will be as justied as it was at t. First, just before Sue performs her second deduction, her memory of B being true may well be less vivid than it was immediately after the rst deduction (just as our memory of the statue in our friend’s garden is less vivid than it was before). Second, given the long passage of time that has elapsed and a less than perfect memory, all she might remember is that she deduced B from A at one point— if she even remembers that at all—without remembering the details involved in the performance of the deduction. Given that she knows that she is fallible, short of performing the deduction again, she may even doubt that she performed it correctly ten years ago (just as we may doubt whether we really had a certain visual experience on which our current belief about the statue was originally based). Such doubt may then lead to her losing some justi- cation for believing B. Similar points hold with respect to any of Sue’s subsequent deductions. (Tang, 2018, pp. 1897–1898) While Tang here gives some idea of how memory erosion works, the details remain unclear. In particular, it is unclear what exactly is forgotten. It would be implausible to insist that what is forgotten is the justifying ground of the belief. That would lead to overly demanding constraints on justication. Often, I obtain justication for some belief and later forget how I became justied. Yet, it still seems in many cases that my beliefs remain justied. For example, this is true of much of our historical knowledge. I might not exactly recall how I learned years ago that, say, William the Conqueror became King of England in 1066, but it still seems to be a justied belief of mine. If that is false, then many beliefs we considered justied are unfounded. Such a result should be avoided. But that means that, despite having forgotten how I learned that William the Conqueror became King in 1066, the justication for my belief is not so eroded that it is no longer justied. Tang’s view had thus better not be that memorial erosion occurs when the grounds for beliefs are forgotten. Similar problems would arise on a view where memorial erosion is forgetting, long after having performed a competent deduction, details like what all the exact premises or steps were or to what degree each step was checked. That again would lead us to a situation where too many of our beliefs turn out to be unjustied. 1 3