Reductive Evidentialism and the Normativity of Logic

Abstract

‘Reductive Evidentialism’ seeks to explain away all ‘structural’ requirements of rationality—including norms of logical coherence—in terms of ‘substantive’ norms of rationality, i.e., responsiveness to evidence. While this view constitutes a novel take on the source of the normativity of logic, I argue that it faces serious difficulties. My argument, in a nutshell, is that on the assumption that individuals with the same evidence can have different rational responses (interpersonal permissivism), the view lacks the resources to maintain its central tenet that an individual’s body of evidence cannot make it rationally permissible for the individual to believe logical inconsistencies (intrapersonal nonpermissivism).

1. Introduction

Does logic play a normative role in guiding our deliberation about what to believe? Many have found it plausible to answer this question in the affirmative. After all, logical incoherence seems to be a distinctively bad kind of cognitive failure, so much so that “being logical” is virtually always assumed to be a necessary condition of being rational.

But exactly how could logic play such a role? ¹ What, if anything, is it that could account for the normative authority of logic? Let us call this “‘the source question’” of the normativity of logic. ² My aim in this paper is to formulate an increasingly popular answer to this question and raise some challenges against it.

The dominant strategy to answer the source question these days seeks to outsource the normativity—that is, it attempts to explain the normative role of logic in terms of other norms that govern our doxastic attitudes. The idea is to analogize the role of logic in our deliberation to that of any other science, such as physics or biology: Roughly, one might think that if one is interested in forming a true belief about what the net force (F) exerted on an object, then one should believe that (F) equals the product of the mass and acceleration of the object in accordance to Newton’s second law. Just as physics itself is not in any interesting sense normative and yet can have normative implications, according to the standard answers, logic can also play a “normative” role insofar as there are independent norms that govern belief. ³ Let us call the category of views that try to outsource the normativity of logic in this way the ‘extrinsic’ views of the source of the normativity of logic, since it sees the normativity of logic as something extrinsic to it. ⁴

One version of the extrinsic view is Reductive Evidentialism (RE for short), which says that all epistemic norms of rationality can be entirely explained in terms of our possessed evidence. RE should be situated within the more general reasons-first program, according to which all norms of rationality (theoretical or practical) can be explained in terms of our responsiveness to possessed reasons. ⁵ As an epistemic thesis, RE focuses specifically on epistemic norms of rationality and thus assumes evidentialism, i.e., that only evidence can constitute epistemic reasons.

RE is a particularly interesting version of the extrinsic view because it denies a distinction that is often made in the literature on epistemic rationality: that of ‘structural’ versus ‘substantive’ norms of rationality. ⁶ While the former is taken to refer to norms that govern how our attitudes are related to each other, the latter is supposed to govern how our attitudes should respond to our evidence. According to RE, all appearance that there are such things as norms of structural rationality can be explained in terms of substantive norms of rationality, and thus there are no such norms.

It is worth emphasizing why RE is an example of an extrinsic view. As we shall see in more detail in section 2, RE tries to explain away the appearance that logic plays a normative role in favor of norms that govern how we should respond to evidence. The basic idea is that logic’s contribution is in determining (purely descriptively) certain restrictions on evidence, but we can still get the appearance of it playing a normative role because there are evidential norms that govern belief.

My goal in this paper is to examine whether RE offers a viable strategy for explaining the source of the normativity of logic. I argue that it does not—not at least without significant costs. My argument, in a nutshell, is that the view lacks the resources to maintain one of its central commitments, namely, that no body of evidence can permit the same individual to both believe $ P $ and $ \neg P $ , without thereby denying another plausible thesis that in general different individuals might respond differently to the same body of evidence.

Before I begin, I should say a few words about what our target notion of “normativity” will be throughout the paper. As my characterization of the topic thus far should make clear, my focus is on whether we can make sense of logic as playing a normative role in our deliberation about what we should or shouldn’t believe from a first-personal perspective. One kind of example that seems to manifest this role particularly helpfully is, what we might call cases of ‘constructive advice.’ What I have in mind are the cases in which one provides second-personal advice by attempting to occupy the other person’s perspective. In such cases, one seems to make explicit one’s deliberation about what one should or shouldn’t believe.

The deliberative notion of normativity is to be contrasted with what has been labeled “merely classificatory.” ⁷ In this sense, a normative role functions by allowing the classification of certain objects—in the case of norms of thinking, doxastic attitudes—into those that are correct and those that are not. To the extent that this sense allows for the “evaluation” of its proper objects, it might be viewed as normative. For our discussion, this will not be the relevant notion of normativity. ⁸

The plan is as follows. In section 2, I offer a fuller sketch of RE by considering two toy examples: (1) simple contradiction, and (2) modus ponens incoherence. The aim is to bring out some important presuppositions of RE—in particular, what we can label ‘nonpermissivism’: that one’s available evidence restricts what attitudes one is rationally permitted to take toward a given proposition. In section 3, following the literature, I distinguish between interpersonal vs. intrapersonal permissivism, and argue that if RE accepts the interpersonal permissivism, then it must also accept the intrapersonal permissivism. In other words, I argue that the view doesn’t have the resource to deny intrapersonal permissivism, which it needs to explain the normativity of logic, without thereby denying interpersonal permissivism.

2. Reductive evidentialism

To see how RE tries to explain the normative role of logic, let’s look at two toy examples:

2.a Simple contradiction

Why should one avoid believing a simple contradiction, $ P $ and $ \neg P $ ? Above, I said that RE tries to explain this in terms of our responsiveness to our evidence. How can the logical relation between $ P $ and $ \neg P $ enter into one’s handling of the evidence? RE’s basic idea is that logic enters into the scene by, in the words of Kolodny (n.d.), “structuring the evidence” in the sense of explicating restrictions on what evidence could be. The idea is roughly that just as the true laws of physics, for instance, rule out the possibility of physical worlds that violate those laws, logic too lays out principles that one’s evidence respects independently of the particular situation in which one finds oneself. For instance, no evidence, one might think, can be sufficient for both $ P $ and $ \neg P $ . Given this basic idea, RE claims that the standard norms requiring that one should proportion one’s beliefs to the evidence allow logic (and physics, too), which in and of itself is normatively inert, to play a normative role. This is an example of the ‘extrinsic view’: What generates rational requirements is evidence; logic plays a role only in a roundabout way through structuring evidence.

Let us spell this out more concretely. RE proposes that we explain the irrationality of believing a simple contradiction by appealing first to a descriptive principle about the role of logic—again, in the sense that it rules out certain combinations of what total evidence is possible:

Evidence Claim for Simple Contradiction (EC_sc): If one has sufficient evidence that $ P $ , then one lacks sufficient evidence that $ \neg P $ .

The second principle that RE’s mode of explanation requires is a normative one about how one should respond to one’s evidence:

Evidence-Rationality Permission Relation (ER_p): One is rationally permitted to believe $ P $ only if one has sufficient evidence that $ P $ .

(ER_p) is a fairly standard norm of rationality that having sufficient evidence is a necessary condition of having permission to believe. Together, these two principles imply:

Nonpermissivism for Simple Contradiction (NP_sc): If one has sufficient evidence that $ P $ , then one is not permitted to believe $ \neg P $ .

This gives us an explanation of why someone who believes both $ P $ and $ \neg P $ is guaranteed to not respond to their evidence appropriately. For, either they have sufficient evidence for $ P $ or they don’t. If they don’t, then by (ER_p) they are not permitted to believe $ P $ . If they do, then by (NP_sc) they are not permitted to believe $ \neg P $ because they don’t have sufficient evidence for it. Either way, they believe something that they are not permitted to believe.

2.b Modus ponens incoherence

Consider now a second kind of case: you present someone with an argument for $ Q $ . You get them to believe $ P $ and $ P\to Q $ . But they just continue to believe $ \neg Q $ .

RE offers the same pattern of explanation of this situation as before: your interlocutor has failed one way or another to respond appropriately to their evidence. To secure this result, however, RE needs an additional assumption about how evidence is structured by logic, this time bringing the notion of implication explicitly into the mix. Here’s an initial suggestion:

Evidence Claim Generalized (EC_g): If $ {P}_1,{P}_2,\dots, {P}_n\models Q $ , then if one has sufficient evidence for $ {P}_1,{P}_2,\dots, {P}_n $ , then one has sufficient evidence for $ Q $ .

Unlike (EC_sc), (EC_g) is a highly contested principle. On the one hand, there are the Preface and Lottery considerations, ⁹ which seem to suggest that (EC_g) does not hold for many-premise arguments, and, on the other hand, there are the more traditional arguments from skepticism ¹⁰ that speak against even the single-premise version of (EC_g).

Let us, however, bracket these worries and assume that a weaker version of this principle that is sufficiently strong for our purposes is true. For instance,

Evidence Claim Restricted (EC_r): If $ {P}_1,{P}_2\models Q $ , then if one has sufficient evidence for $ {P}_1,{P}_2 $ , then one has sufficient evidence for $ Q $ .

My target lies elsewhere in RE’s account. If it turns out that even (EC_r) cannot be maintained in the face of some challenges, then so much the worse for RE. ¹¹

Now, (EC_r) together with assumptions (ER_p) and (NP_sc) from the previous section, provide an explanation of what has gone wrong with the interlocutor. Either they have sufficient evidence for $ P $ and $ P\to Q $ or they do not. If they don’t, then by (ER_p) they believe something that they are not rationally permitted to believe. If they do, then by (EC_r) they have sufficient evidence for $ Q $ . But then, by (NP_sc) they are not rationally permitted to believe what they happen to believe—that is, $ \neg Q $ . So, either way, the failure to respond correctly to the evidence that they have.

One initial worry about this account is that it is not only the case in which one believes the premises of a valid argument and disbelieves the conclusion that needs explanation; we also need an explanation of what has been labeled ‘logical obtuseness’ (MacFarlane, 2004). The idea is that in certain circumstances if your interlocutor believes the premises, and instead of disbelieving the conclusion, simply refuses to form a belief (or suspends), they are still exhibiting a distinctive kind of rational mistake. ¹²

Kiesewetter (2017) has proposed to respond to this demand by adopting a more radical Evidence-Rationality Relation than (ER_p). Kiesewetter argues that since the only reasons for belief are given by the evidence supporting their content, in cases where one has sufficient evidence for $ P $ , one’s balance of available evidence counts decisively in favor of $ P $ . And since if one has decisive evidence for $ P $ , one is required to believe $ P $ , it follows that (181):

Evidence-Rationality Requirement Relation (ER_r): If one has sufficient evidence for $ P $ , and one attends to $ P $ , then one is required to believe that $ P $ . ¹³

The “attend” clause is necessary to avoid a similar “clutter avoidance” objection: without it, the principle would be too strong because it would imply that one is rationally required to believe all kinds of irrelevancies as long as the evidence is available to them. ¹⁴

If (ER_r) is accepted, RE has a ready explanation of the obtuseness challenge: the interlocutor either does or does not have sufficient evidence for $ P $ and $ P\to Q $ . If they don’t, then that’s their failure. If they do, then they have sufficient evidence for $ Q $ . But then, by (ER_r) they would be failing to respond appropriately to their evidence if they adopted any attitude, but a belief toward $ Q $ .

Many, however, will find (ER_r) to be too strong. For instance, borrowing the following example from the literature on epistemic contextualism, Way (2018) argues that there are “merely permissible” cases—that is, cases in which while one’s evidence is sufficient to rule out disbelief in a proposition, it leaves both belief and suspension as permissible options:

Low-stakes bank: You would like to deposit a check, but there is a long line at the bank. If the bank is open on Saturday, it will be much easier to deposit it then. If the bank is not open on Saturday, you will only be mildly inconvenienced. You recall that the bank was open on Saturday a couple of weeks ago, but also know that banks sometimes change their hours. (11) ¹⁵

Way contends that in this case, it is plausible to think that one could justifiably either believe that the bank is open or suspend judgment on the matter (11). He writes: “Since it is not a trivial matter whether the bank will be open tomorrow, this is a fair case of merely permissible but non-trivial belief” (11). The idea is that sometimes sufficient evidence is not strong enough to decide the rational response between suspension and belief. One can be merely permitted to believe, in the sense that it is rationally open to one to either believe or suspend. If this is right, then (ER_r) has to be rejected and RE cannot use this strategy to meet the obtuseness challenge.

Although I agree with Way that (ER_r) is too strong, I am not convinced that this poses a fatal challenge for RE. For, even if Way’s argument succeeds, it only shows that the strategy of using (ER_r) to meet the obtuseness challenge fails. On its own, that only shows that an alternative strategy is needed, not that RE somehow fails. ¹⁶

I believe that the real problem for RE runs much deeper. In the next section, I argue that RE faces a serious challenge in maintaining a principle that is crucial for its explanation of both simple contradiction and modus ponens incoherence, namely, (NP_sc).

3. Nonpermissivism

In the growing literature on epistemic permissivism, it has been argued that a single body of evidence can sometimes be sufficiently strong to justify both belief and disbelief in the same proposition. Consider, for instance, the following example from Schoenfield (2014):

Community: You have grown up in a religious community and believe in the existence of God. You have been given all sorts of arguments and reasons for this belief, which you have thought about at great length. You then learn that you only have the religious beliefs that you do, and only find the reasoning that you are engaged in convincing, because of the influence of this community. If you had grown up elsewhere, you would have, on the basis of the same body of evidence, rejected those arguments and become an atheist. (205)

For this discussion, I will simply assume that Schoenfield is right to maintain that there are cases such as Community in which the evidence is sufficiently strong to justify both belief and disbelief in the proposition that God exists.

On its own, however, this does not show that (NP_sc) is false. It is now common to distinguish between two interpretations of permissivism: ¹⁷ first, that for any agent there is more than one doxastic attitude that they can rationally take toward any proposition based on the evidence, and, second, that different agents who have the same evidence can rationally take different doxastic attitudes toward a given proposition. These interpretations are labeled intrapersonal and interpersonal, respectively. ¹⁸ In light of this distinction, as Schoenfield would agree, the most cases like Community can show is the interpersonal interpretation of permissivism. ¹⁹ Notably, they do not show that one and the same individual is permitted to both believe and disbelieve that God exists. Schoenfield explains that what allows for different individuals to have different attitudes toward the same proposition is their different “epistemic standards.” But she thinks that “crucially, no one set of epistemic standards will ever warrant belief in $ P $ and $ \neg P $ ” (200). ²⁰

So, the existence of cases like Community seems to fall short of posing a threat to (NP_sc) even though it does undermine the interpersonal version of permissivism, i.e., that the same body of evidence can permit one person to believe $ P $ and another to believe $ \neg P $ . This seems to suggest a straightforward position that RE can occupy in reaction to cases like Community: accept interpersonal permissivism while rejecting intrapersonal permissivism, the thesis that, as we have seen, is at the core of RE’s explanation of the normativity of logic.

I believe, however, that due to the reductive ambitions of RE, cases like Community do ultimately pose a serious challenge to the view. The problem, in a nutshell, is that RE lacks the resources to make the crucial distinction between intrapersonal versus interpersonal cases. To see this, consider what it would be like for RE to accept the interpersonal while rejecting the intrapersonal interpretation of permissivism. According to this view, while in so far as one’s evidence is concerned, one could be permitted to believe or disbelieve a given proposition, once one’s epistemic standards (or perhaps the different ways that agents might weigh the epistemic goals of pursuing truth and avoiding error ²¹) kick in at the individual level, there is a unique attitude which one is rationally required to have. ²² So, for instance, if it is true that one who rationally believes $ P $ and $ P\to Q $ and yet disbelieves $ Q $ is not responding as they should to their evidence that $ Q $ , this is not because of “the structure of the evidence” alone. After all, the acceptance of the interpersonal interpretation of permissivism means that one and the same evidence can permit different attitudes toward the same proposition for different individuals. Thus, the agent’s failure in such a case would have to be explained by the contribution that is introduced at the individual level about how one is to handle the evidence.

We can begin to make this more precise by relativizing the evidence claim for simple contradiction to epistemic standards:

Evidence Claim for Simple Contradiction-Relativized (EC_sc-r): If one has sufficient evidence that $ P $ relative to epistemic standard $ E $ , then one lacks sufficient evidence that $ \neg P $ relative to $ E $ .

The injection of the idea of epistemic standards can be viewed as a way of effectively introducing the contribution of logic at the individual level: while in abstraction from epistemic standards the evidence is unrestricted by logic, given a permissible epistemic standard at the individual level, inconsistent evidence is ruled out.

Now, we saw earlier (section 2.a) that RE’s central assumption (NP_sc) is derived from (EC_sc) and (ER_p). Having relativized (EC_sc) to individual epistemic standards, can RE maintain (NP_sc), and with it, its explanation of the irrationality involved in believing a simple contradiction? It seems to me that the answer turns on RE’s ability to do two things: (1) provide a defense of (EC_sc-r)—that is, of why epistemic standards rule out inconsistent evidence—and (2) stipulate that one can only have a single epistemic standard at a time.

Let’s consider each of these in the reverse order. RE needs (2) because otherwise, even if each epistemic standard rules out inconsistent evidence, one may have different epistemic standards $ {E}_1 $ and $ {E}_2 $ such that according to $ {E}_1 $ the evidence is sufficient for $ P $ , while according to $ {E}_2 $ it is sufficient for $ \neg P $ . I take it that RE can establish (2) if it can show that there is a rational requirement against switching epistemic standards when one is deliberating about what to believe. This would follow from a general requirement against switching one’s epistemic standards. However, this strategy to establish (2) fails because a general requirement against switching is not plausible. Consider, for instance, the case of Derek Black, who was born and raised in a white supremacist environment and groomed by his father to become a neo-Nazi leader. Thankfully, Black managed to redefine himself after having been confronted in college by his peers and invited to Shabbat dinners at a Jewish families home. ²³ Arguably, there is no change in the available evidence in cases likes this. People like Black are not deprived of the relevant evidence. What allowed Black to undergo his radical transformation is rather a change in his epistemic standards about whom to trust. Although it may be true that the process through which he realigned his trust and was able to exit his echo chambered existence as a neo-Nazi was not a particularly rational one, there is surely nothing irrational about it. ²⁴

Thus, if RE has to stipulate a requirement against switching standards, it would have to be a synchronic one—that is, a requirement not to switch in one breath. It is worth emphasizing that it would be ad hoc for RE to argue that this requirement comes from logic. Recall that as an instance of the extrinsic view, the basic explanatory strategy of RE is to outsource the normativity of logic: that logic only plays a purely descriptive role in “structuring” the evidence, spelled out by (EC_sc). If RE tries to explain the synchronic requirement by appeal to logic, it would have to say that logic “structures evidence” not simply in the sense that it implies restrictions on which combinations of evidence are available, but more substantively on how individual believers are to adopt epistemic standards. But at best, this seems too close to a kind of psychologism: What is it about logic that makes it fit to regulate how individuals adopt epistemic standards? And, at worst, it seems to commit RE to a robust normative role for logic that flies in the face of the reductive ambitions of RE.

Fortunately for RE, however, it doesn’t seem too farfetched to try to explain the synchronic requirement independently of logic. After all, the idea that double standards are problematic has no obvious connection to logic. We don’t need logic, for instance, to explain why I would be treating my children unfairly if I let one have the candy and the other not in similar circumstances. While going into the details of this explanation in the case of epistemic standards is not trivial, I think the idea is plausible enough that, for our purposes, we can assume that RE is capable of explaining the synchronic requirement without appeal to logic, and therefore establishing that one can rationally only have a single epistemic standard at a time. In other words, let us assume that RE has a way of establishing (2).

But what about (1)? Can RE provide a satisfactory explanation of (EC_sc-r)—that is, why any given epistemic standard must rule out inconsistent evidence? This brings us to the heart of the problem for RE. For, unlike the case of the synchronic requirement not to switch standards in one breath, it seems that an explanation of (EC_sc-r) would indeed have to draw on logic. That’s because (EC_sc-r) is explicitly formulated in terms of the logical relationship between $ P $ and $ \neg P $ . And, having to draw on logic in explaining (EC_sc-r) seems to be detrimental for RE because, on the one hand, as a psychological thesis, it is ad hoc and therefore implausible to think that our psychologies are constituted such that they rule out inconsistent evidence. On the other hand, as a suggestion about accepting a normative role for logic in determining the permissible epistemic standards, the explanation would be inconsistent with the basic explanatory strategy of RE. If logic does indeed determine which epistemic standards are permissible (i.e., those which rule out inconsistent evidence), then logic doesn’t play a purely extrinsic role and is not normatively inert as RE wants to say.

We can further appreciate the tension between introducing the contribution of logic at the individual level and RE’s mode of explanation by considering a plausible account of what it is for an individual to possess evidence. Take, for instance, Lord’s (2018) account. According to Lord, there are two conditions that must be met for one to count as “possessing” a piece of evidence: first, an epistemic accessibility condition, and second, what he calls a “practicality condition.” The former is a more or less standard condition, which for our purposes we can leave aside. ²⁵ The latter, Lord argues, is necessary to rule out cases in which one rationally does not take a piece of objective evidence which one has epistemic access to (in the relevant sense) as a reason—for instance, a case in which one is told by a reliable source that something that is, in fact, a sufficient evidence for $ P $ is not. Lord proposes to understand this condition in terms of an ability or know-how of the agent to respond to the evidence that they have as the evidence that they are (26).

Now, it might be suggested that part of the ability to respond to the evidence is a disposition not to take one’s evidence as sufficient for both $ P $ and $ \neg P $ . However, notice that, in general, an ability to $ \phi $ is consistent with very different ways of doing $ \phi $ . When I approach a turn on my bike, for instance, I might exercise my ability to ride the bike in radically different ways. I might lean into the curve and pickup some speed; or, I might put on my brakes and turn my handlebars to navigate the turn. These are both perfectly fine manifestations of the same ability.

I see no reason why the same is not true of the general ability which Lord builds in the possession conditions of evidence. Just as people can manifest their ability to ride a bike in radically different ways, we don’t seem to have any reason to resist the idea that the general ability to respond to the evidence can also be manifested in radically different ways—including potentially ways that are jointly inconsistent with each other.

It might be argued that, for at least some of the ways of exercising a given ability, attempting more than one of them is ruled out by the possession of the ability. For instance, I won’t be riding a bike if I try to both slowdown and lean into the turn. Similarly, this line of thought continues, while the ability to respond to the evidence can be exercised in different ways, it rules out taking one’s evidence to be sufficient for both $ P $ and $ \neg P $ .

However, notice that in the case of riding the bike, there is a ready explanation of these kinds of limitations: those who try to both lean and slowdown fall. What would be the corresponding explanation in the case of the requirements against contradictions? What is it about logical relations that, for any given epistemic standard, require not taking one’s evidence as sufficient for both $ P $ and $ \neg P $ ? The answer for RE can’t be that one is guaranteed to have a false belief because, of course, the latter is what RE is trying to explain in the first place. The trouble is explaining the relevance of the purely logical relation between $ P $ and $ \neg P $ to one’s ability to respond to the evidence. On the one hand, if the logical relation is taken to be part of the evidence, and to thereby play a normative role, then this flies in the face of the explanatory ambitions of RE to outsource the normative role of logic to the norms that govern how we should respond to the evidence. On the other hand, if logic is not itself part of the evidence, then it is unclear how it could play the role RE wants it to play in structuring the individual epistemic standards. If RE is to maintain the stipulation that there is a rational requirement on permissible epistemic standards not to treat the same evidence as both sufficient for $ P $ and $ \neg P $ , it owes us an explanation of how the logical inconsistency between $ P $ and $ \neg P $ generates such a requirement. And, as far as I can see, RE is not in a position to do that.

I thus conclude that accepting interpersonal permissivism while denying intrapersonal permissivism is not a stable position for RE. If that’s true, given the plausibility of interpersonal permissivism in light of cases such as Community, it turns out that RE is incapable of maintaining one of the most central assumptions that its mode of explanation depends on—namely, (EC_sc): if one has sufficient evidence for $ P $ , then one lacks sufficient evidence for $ \neg P $ . To say the least, then, in order to explain the normativity of logic, RE can’t rest content with the relatively conservative position of rejecting merely the intrapersonal permissivism; it must rather reject both forms of permissivism.

4. Conclusion

Reductive Evidentialism promises an ambitious version of the standard view of the source of the normativity of logic. It tries to account for a deliberative normative role for logic purely in terms of responsiveness to evidence. The account is attractive because it promises simplicity and parsimony. However, I hope to have shown that it faces serious challenges in accounting for the most basic kind of target phenomenon—namely, the irrationality of believing a simple contradiction. My argument relied on the possibility of interpersonal permissive cases: i.e., cases in which one and the same evidence is sufficient to permit one person to believe $ P $ and another to believe $ \neg P $ . I showed that accepting the possibility of these kinds of cases creates trouble for RE because given the reductive ambitions of the view, it commits RE to a related but different thesis which undermines its very mode of explanation—namely, intrapersonal permissivism: that the same body of evidence can be sufficient for one and the same individual to believe both $ P $ and $ \neg P $ at the same time.