Logical Akrasia

Logical Akrasia Frederik J. Andersen Forthcoming in Episteme Abstract The aim of this paper is threefold. Firstly, §1 and §2 introduce the novel concept logical akrasia by analogy to epistemic akrasia. If successful, the initial sections will draw attention to an interesting akratic phenomenon which has not received much attention in the literature on akrasia (although it has been discussed by logicians in different terms). Secondly, §3 and §4 present a dilemma related to logical akrasia. From a case involving the consistency of Peano Arithmetic and Gödel’s Second Incompleteness Theorem it’s shown that either we must be agnostic about the consistency of Peano Arithmetic or akratic in our arithmetical theorizing. If successful, these sections will underscore the pertinence and persistence of akrasia in arithmetic (by appeal to Gödel’s seminal work). Thirdly, §5 concludes by suggesting a way of translating the dilemma of arithmetical akrasia into a case of regular epistemic akrasia; and further how one might try to escape the dilemma when it’s framed this way. Keywords Epistemic Akrasia; Logical Akrasia; Epistemic Rationality; Logical Theories; Gödel’s Incompleteness Theorem; The Dilemma of Arithmetical Akrasia 1 1 Prologue The Greek word ‘akrasia’ translates literally as ‘lack of self-control’, but has come to be used as a general term for a weakness of will, i.e., a disposition to act contrary to one’s own considered judgment. It will come as no surprise that such inability to act as one thinks right has interested ethicists since antiquity. More surprising (perhaps) is the vast amount of attention that the analogous phenomenon epistemic akrasia has received lately.1 A driving force behind this interest is the appealing thought that epistemic rationality requires coherence between: (A) an agent’s doxastic attitudes in general, and (B) their specific beliefs about what doxastic attitudes are rational.2 To illustrate, consider the medical resident Anandi who correctly figures that dosage hpi is appropriate for her patient; and thus believes that hpi. Suppose she then learns she’s been drugged herself, and further that the effects of the relevant drug very often lead to cognitive errors that are hard to detect from the inside. As a result, suppose she believes that hmy belief that p is irrationali but that she maintains her belief that hpi nonetheless. Other things being equal, Anandi’s doxastic state should strike us as irrational because she believes against her own standards of rationality, or as recent epistemological parlance will have it; because she believes akratically.3 Examples like Anandi’s drug case have led some epistemologists to argue for a general anti-akrasia constraint on epistemic rationality (Feldman, 2005; Smithies, 2012; Titelbaum, 2015; Littlejohn, 2018): 1 See (Adler, 2002; Owens, 2002; Ribeiro, 2011; Williamson, 2011; Smithies, 2012; Greco, 2014; Horowitz, 2014; Lasonen-Aarnio, 2014; Williamson, 2014; Sliwa and Horowitz, 2015; Titelbaum, 2015; Roush, 2017; Brown, 2018; Littlejohn, 2018; Worsnip, 2018; Daoust, 2019; Kappel, 2019; Skipper, 2019; Titelbaum, 2019; Kearl, 2020; Lasonen-Aarnio, 2020; Chislenko, 2021; Skipper, 2021; Christensen, 2021; Jackson and Tan, 2022; Horowitz, 2022; Kauss, 2023; Christensen, 2024). 2 Examples of doxasitic attitudes: belief-tokens, credences, opinions, judgements etc. 3 According to a popular evidentialist formulation a subject is epistemically akratic when they are highly confident that proposition hpi is true while also believing that the higher-order proposition expressed by hmy current evidence doesn’t support pi is the case. So, if Anna believes that it’s going to rain tomorrow while also believing that her evidence at the time doesn’t support this, then Anna is in a state of epistemic akrasia. Prima facie—at least—Anna’s overall doxastic state should strike us as irrational. Since believing against what one takes one’s evidence to support just seems epistemically bad; if not outright paradoxical. To this end, the card-carrying evidentialist Richard Feldman wonders “...what circumstances could make [epistemic akrasia] reasonable...” (Feldman, 2005, p. 109). 2 The Akratic Principle. No [epistemic] situation rationally permits any overall [doxastic] state containing both an attitude A and the belief that A is rationally forbidden in one’s situation. (Titelbaum, 2019, p. 227)4 5 This principle is taken to imply that you should either have the attitudes you believe you ought to have, or stop believing that you ought to have those attitudes. Hence, in the name of rationality, the Akratic Principle forbids you to have certain combinations of attitudes such as not believing that hpi while believing that hbelieving p is rationally required in one’s situationi; or having credence(p) = 0.9 while believing that hhaving credence(p) = 0.9 is rationally forbidden in one’s situationi. We’ll return to the Akratic Principle in due course (cf. §3), but for now let’s consider a widely discussed case from the literature on epistemic akrasia to further guide our intuitions. The case concerns a sleep deprived detective, Sam, who possesses misleading higher-order evidence (i.e., misleading evidence about what his first-order evidence supports): Sleepy Detective. Sam is a police detective, working to identify a jewel thief. He knows he has good evidence—out of the many suspects, it will strongly support one of them. Late one night, after hours of cracking codes and scrutinizing photographs and letters, he finally comes to the conclusion that the thief was Lucy. Sam is quite confident that his evidence points to Lucy’s guilt, and he is quite confident that Lucy committed the crime. In fact, he has accommodated his evidence correctly, and his beliefs are justified. He calls his partner, Alex. “I’ve gone through all the evidence,” Sam says, “and it all points to one person! I’ve found the thief!” But Alex is unimpressed. She replies: “I can tell you’ve been up all night working on this. Nine times out of the last ten, your late-night reasoning has been quite sloppy. You’re always very confident that you’ve found the culprit, but you’re almost always wrong about what the evidence supports. So your evidence probably doesn’t support Lucy in this case.” 4 Note that the Akratic Principle is sometimes referred to as the Enkratic Principle instead, see e.g., (Skipper, 2019; Field, 2019, 2021). 5 For further details on Titelbaum’s use of the term ‘rational’ consult (Titelbaum, 2015, 2019; Skipper, 2019). See also (Bradley, 2021; Carr, 2021) for recent discussions of ideal versus nonideal epistemic rationality. 3 Though Sam hadn’t attended to his track record before, he rationally trusts Alex and believes that she is right—that he is usually wrong about what the evidence supports on occasions similar to this one. (Horowitz, 2014, p. 719) Provided the information of Sleepy Detective—and the background assumption that respecting one’s total evidence is an important standard of epistemic rationality—what is the rational response to Sam’s predicament? In other words, what doxastic attitude should he hold with respect to the identity of the thief? And what should he believe about what his first-order evidence supports?6 The literature is divided into three main camps. According to Steadfast views, Sam should simply stick to his guns. That is to say, he should keep both his high confidence that hpi (i.e., hLucy is the thief i) and his belief that this is what his first-order evidence supports (Kelly, 2005; Titelbaum, 2015). A reason in favor of this response is that Sam actually got things right to begin with. So even though the later testimony from his partner Alex is higher-order evidence suggesting that his assessment of the first-order evidence is unreliable due to sleep deprivation, this is in fact misleading on the particular occasion. In contrast, Conciliatory views hold that Sam should reduce confidence both with respect to proposition hpi and the higher-order proposition stating that hmy first-order evidence supports pi (Feldman, 2005; Christensen, 2007).7 A reason in favor of this position is that from Sam’s first-person perspective the higherorder evidence constituted by Alex’s testimony seems undefeated. Since Sam rationally trusts Alex to be right about his unreliable track record in relevantly similar circumstances, he should reduce his confidence at both first- and higherorder level.8 Notice that although steadfast and conciliatory views disagree about the rational response to cases like Sleepy Detective, they agree that Sam’s confidence 6 Sam’s first-order evidence includes (propositions about) the letters and photographs that he was looking through as he worked late at night. 7 One should realize that while a higher-order proposition like hmy first-order evidence supports that pi has positive normative force with respect to proposition hpi, i.e., it might make it rational for you to believe that hpi; other higher-order propositions like hany epistemic situation makes it rationally forbidden to believe that pi has negative normative force with respect to proposition hpi. 8 For canonical work on defeaters in epistemology the reader should consult (Pollock, 1970, 1974, 1984, 1986, 1994). See (Kelp, 2023) for a comprehensive overview. Note also that Christensen might be said to lean towards a level-splitting rather than conciliationist view about akrasia in his more recent work on the topic. 4 in hpi shouldn’t conflict with his belief about what the first-order evidence supports. That is, both camps accept that any such level-incoherence is epistemically irrational. Level-Splitting views dispute this. According to the level-splitter it can sometimes be epistemically rational to have a high confidence that hpi while also believing the higher-order proposition hmy first-order evidence doesn’t support pi. Imagine, for instance, a long deductive proof written on a whiteboard, and suppose that Beth thinks through the proof and comes to rationally believe a series of claims from which she competently deduces their conjunction, hpi.9 Assume (quite plausibly) that Beth comes to rationally believe hpi by these means. Yet Beth knows that people like her—in similar situations involving long deductions—often make inferential errors. So, it may well be highly probable on her higher-order evidence that she has made an inferential error in the current situation, which suggests that hher first-order evidence doesn’t support pi after all (even though we can stipulate that her belief in the truth of hpi is in fact correct).10 To be sure, the intended interpretation here is that the knowledge Beth possesses about people’s shortcomings in situations relevantly similar to hers should be taken as higher-order evidence against her first-order attitude towards hpi, but according to level-splitting views, what goes on at higher-order level need not affect the rationality of Beth’s first-order attitudes.11 Thus—by level-splitting lights—this is a scenario where Beth can have a high confidence in hpi while also believing the higher-order proposition hmy first-order evidence doesn’t support pi and be rational nonetheless. As with Beth’s logic case, a level-splitting response to Sleepy Detective would have it that Sam should remain highly confident that hLucy is the thiefi and simultaneously believe that this isn’t supported by his first-order evidence. Epistemologists such as Williamson (2011; 2014), Lasonen-Aarnio (2014; 2020), Wedgewood (2012), and Weatherson (2010), have all favored level-splitting views although their reasons for doing so diverge. 9 Denoting a conjunction using the symbol ‘p’ might be thought to overload the notation, but we allow this here for the sake of simplicity. 10 Notice the structural analogy between Beth’s logic case above and the well-known Preface Paradox (Makinson, 1965; Sorensen, 2020). 11 For further clarification of the distinction between first-order and higher-order evidence, see (Christensen, 2010; Skipper, 2021; Horowitz, 2022). 5 2 Logical Akrasia While epistemic akrasia is interesting in its own right, it will not be our main concern. Our primary focus will be on an analogous phenomenon in (formal) logic. The remaining sections aim to connect the discussion of epistemic akrasia from mainstream epistemology with another existing discussion in the philosophy of logic, which concerns the use of classical logic to prove metatheoretic results (such as soundness and completeness) about a weaker, non-classical logic. It will also be suggested that some level of akrasia is unavoidable in arithmetic because of Gödel’s Second Incompleteness Theorem. As a rough starting point we’ll take logical akrasia to consist in a mismatch between the deductive strength of the background logic one uses to prove metatheoretic results and the logical theory one believes (officially), i.e., a form of incoherence in logical theorizing akin to what we saw in the case of epistemic akrasia.12 So, in other words, logical akrasia will occur when one explicitly appeals to (or at least implicitly commits to) a logical principle which is not endorsed by one’s own theory.13 14 At this point we won’t distinguish between meta-logic and metatheory, but we’ll discuss the potential importance of drawing this distinction in §2.1. Now, to provide a concrete example of logical akrasia, consider the following passage from Beall and Restall: The Intuitionist. What we have presented is a straightforward account of Tarski’s model theory for classical predicate logic, and a 12 Logic Li is deductively stronger than logic L j whenever Li can prove more, i.e., for every set of well-formed sentences, Γ, the deductive closure of Γ under L j is a proper subset of the closure of Γ under Li . 13 A concrete example of an implicit commitment to a logical principle could be committing to the excluded middle via an explicit endorsement of Peirce’s Law—i.e., (((ϕ → ψ) → ϕ) → ϕ), where lowercase letters from the Greek alphabet are metavariables. For more on the topic of implicit commitments in logical theorizing, the reader should consult (Cieśliński, 2017; Horsten and Leigh, 2017; Fischer et al., 2021). 14 It’s worth flagging that metatheoretical results (like soundness and completeness) use mathematical theories (about sets, functions, models, etc.) which are themselves the kinds of things we might explicitly endorse by our own lights, or not. Questions about the match of what we explicitly endorse and what we implicitly rely on arise at a number of levels in and around logic, both about theories that are properly logic (in the narrowest sense), and those formal theories we use in reasoning about logic. The example of arithmetic we’ll consider below is just the most simple example of a formal theory in this vicinity, and concerns around the epistemology of this theory and any akrasia involved therein, is therefore very much on topic, and not just an illustration of a similar phenomenon. 6 simple account of truth conditions in a possible worlds semantics. We have claimed that such accounts deliver classical logic. Is this indeed the case? It is commonly thought that this is the case, but in present company we may have reason to question this thought. Upon an inspection of the usual soundness and completeness proofs, we shall see that the full power of classical logic is required to complete the proof. To show, for example, that in every model A ∨ ¬A is satisfied, we need to show, for each model M, that M A or M 1 A. But this is an instance of the excluded middle! An intuitionist (for example) who rejects the law of the excluded middle will not endorse this reasoning. What can we say about this? (Beall and Restall, 2006, p. 39) One thing we could say—to answer Beall and Restall’s query—is that the intuitionist is in a state of logical akrasia. The case of the intuitionist logician who, when doing the metatheory of intuitionistic logic, finds themselves using classical (nonconstructive) principles, is an illustrative example of logical akrasia as it involves a clear incoherence of the kind we are interested in. In sum: this logician happens to presuppose logical principles, when producing metatheoretic proofs, that are not endorsed by their own (intuitionistic) standards, and as was the case with the epistemic counterpart, logical akrasia seems self-undermining and irrational. Or, to put this point more vividly: the intuitionist searching for an acceptable metatheory using a classical background logic seems akin to fixing a leaky roof by accustoming oneself to a wet floor. 2.1 Defining Logical Akrasia So far, so good! Let’s now define logical akrasia in a more regimented fashion: Logical Akrasia. Subject S, believing logical theory T, is in a state of logical akrasia if and only if S commits to a logical principle that S’s logical theory T fails to endorse as valid.15 15 Define a logical theory in the standard way. A logical theory is an ordered pair T = hLi , LiM i such that Li is an object-logic and LiM a metatheory. On the one hand, we have object-logic Li = hΦ, Σ , Pi, where Φ specifies a language of both logical and non-logical vocabulary while Σ gives a syntax for Φ (determining its well-formed formulas). P in turn provides a set of inference rules and/or axioms for syntactic manipulation of the symbolic strings that Φ gives rise to. On the other hand, we find metatheory LiM = hΦM , I, |=Φ , ⊢Φ i, the first element of which specifies a meta-language ΦM . The second element I gives a semantics expressed in that meta-language, 7 As this definition is too coarse grained to capture all relevant cases, we further distinguish between: Weak S, believing logical theory T, is weakly akratic if S commits to a logical principle that S’s logical theory T fails to endorse as valid; Strong S, believing logical theory T, is strongly akratic if S commits to a logical principle that S’s logical theory T rejects as invalid. Based on this weak/strong distinction we get two non-equivalent versions of Logical Akrasia. To appreciate this, consider a case where an intuitionist commits to an instance of the excluded middle in a restricted situation, and suppose that their theory doesn’t endorse this as valid. Then, the intuitionist can extend their theory at a later stage such that the instance of the excluded middle becomes endorsed as valid (just stipulate that the situation is decidable)—e.g., it happens to be a case concerning a quantifier-free sentence of arithmetic like 2 + 2 = 4. Before the extension they were being weakly akratic, but not afterwards.16 Yet one should acknowledge that the tenability of the strong/weak distinction depends on the possible division: meta-logic/metatheory. There is, for instance, no difference between failing to endorse as valid and rejecting as invalid if one holds a formal and complete meta-logic rather than a non-formal and incomplete metatheory.17 If one’s meta-logic is formal and complete over its domain, then the distinction between strong and weak logical akrasia collapses (since in that case everything which is not valid is simply invalid). When, say, an intuitionist holds a formal and complete meta-logic and the law of the excluded middle has counterexamples, then failing to endorse the excluded middle as valid and rejecting it as invalid amount to exactly the same. i.e., it lays down individual truth-conditions for the logical constants of Φ using ΦM . The third element |=Φ authorizes a semantic consequence relation between well-formed formulas of Φ. Finally, ⊢Φ defines a syntactic consequence relation for well-formed formulas in Φ. 16 Note that insofar as the distinction between weak/strong akrasia is relevant at all it isn’t just relevant to the case of intuitionism. In fact it seems relevant to many, perhaps even most, nonclassical logicians like Field, Kripke, Ripley, Beall etc. For they all take classical logic to be valid in non-problematic contexts. Some non-classical logicians are more “hardcore” and go nonclassical all the way down (Priest (2006), Weber et al. (2016) etc.), but this isn’t the norm. 17 For more on logic(s) and formality, see for example (MacFarlane, 2000; Beall and Restall, 2006; Dutilh Novaes, 2012; Mortensen, 2013). 8 2.2 Logical Akrasia and Incoherence As we have seen above, states of logical akrasia seem to be incoherent. A point which is also frequently underscored in the literature: If you take ‘logically valid’ to obey a logic weaker than classical, you shouldn’t ultimately be satisfied with developing your theory of that logic using inferences that are merely classically valid... (Field, 2017, p. 14) ...what a strange approach to take, if one believes logic X is the correct logic. Why use an alien logic for one’s metatheory—and if one does, why trust the result? (Read, 2006, p. 208) ...it would be untoward in a logic to appeal in proof of its adequacy to principles in which the logic in question does not believe. (Meyer, 1985, p. 13) If he rejects classical logic for the object language, how is he entitled to rely on it for the metalanguage? (Williamson, 2020, p. 6) These quotes notwithstanding logical akrasia is deeply entrenched in our contemporary logical theorizing. For it is no secret that classical logic serves as the golden standard in evaluations of non-classical logics (Schurz, 2021), i.e., it’s common practice to take classical (first-order) logic as the “neutral” backdrop against which we evaluate non-classical logics. Examples are: Łukasiewicz’ three-valued logic, Kleene’s (strong) three-valued logic, Brouwer’s intuitionistic logic, Priest’s paraconsistent logic etc.18 Where does this leave us? Is the current modus operandi of non-classical theorizing severely misguided? Well, insofar as we want reflective equilibrium (Resnik, 1985, 1996, 2004) between our logical theories and considered judgements about logicality (i.e., validity, consistency, implication, equivalence etc.), there is a sense in which the answer is affirmative. According to Michael Resnik, one’s preferred logical theory and considered judgements about logicality are in a state of reflective equilibrium when: ...the theory rejects no argument that one is determined to preserve and countenances no argument that one is determined to reject... (Resnik, 1996, p. 493) 18 Consult (Priest, 2008) for classical evaluations of each of the non-classical logics mentioned above. See (Bacon, 2013) for a discussion of non-classical metatheories for non-classical logics. 9 Thus we have at least two interpretations of reflective equilibrium, depending on how we read the term ‘reject’: (1) one’s theory judges valid every argument one is determined to preserve; and (2) one’s theory doesn’t judge invalid any argument one is determined to preserve. For weak Logical Akrasia to violate reflective equilibrium, we need interpretation (1); not interpretation (2) (cf. §2.1). Or, as one may otherwise put it, for weak Logical Akrasia to violate reflective equilibrium, we need the strong interpretation of reflective equilibrium, not the weak one. The ideal of strong reflective equilibrium is incompatible with states of weak Logical Akrasia. Hence, the golden standard of non-classical logicians—committing themselves to classical principles in their metatheoretical pursuits—appears to be a standard of fool’s gold in at least one sense.19 2.3 The Analogy with Epistemic Akrasia If we return to the analogy between epistemic and logical akrasia for a minute, we can now appreciate how both the weak and strong version of Logical Akrasia resemble the standard definitions of epistemic akrasia in various ways. Recall first the appealing thought from §1 stating that epistemic rationality requires coherence between: (A) an agent’s doxastic attitudes in general, and (B) their specific beliefs about what doxastic attitudes are rational. Epistemic akrasia occurs whenever S adopts doxastic attitudes that don’t live up to S’s own standards of rationality. Logical akrasia, similarly, occurs when S commits to logical principles that don’t live up to S’s own standards of logic. So, in both cases the problem is one of not meeting one’s own ideal (rather than pursuing a spurious ideal). Adding further to the analogy, we have no trouble imagining what Steadfast and Conciliatory responses to logical akrasia would look like. If one asserts that logical akrasia calls for a revision of one’s logical commitments or theory, then it would count as a conciliatory view. If one, on the other hand, submits that no revision is required, it would be a steadfast view. One could also cook up a special Level-Splitting kind of steadfastness without noteworthy ingenuity. Assume, say, that one is a logical pluralist (of some sort), then one may argue that there are benign cases of logical akrasia. Given the pluralist’s dictum that more than one logic can be correct, the (level-)incoherence of logical akrasia need not be problematic. Alternatively one might be able to pull off a level-splitting response by 19 See (Priest, 2006) for further discussion of this seemingly self-undermining practice of some non-classical logicians. Note also the potentially interesting distinction between sub-classical and contra-classical logics. 10 appealing to logical instrumentalism (Haack, 1974), i.e., the view that it doesn’t make sense to think of logics as being correct or incorrect rather they are simply variously useful or not. It seems plausible to suggest that appealing to instrumentalism in some way could dissolve the tension in cases of logical akrasia to the level-splitter’s satisfaction.20 21 3 Enter Gödel: Rationality and Akrasia As advertised earlier, the two sections §3 and §4 make use of Gödel’s (in)famous Second Incompleteness Theorem (Gödel, 1931) to pose a dilemma related to logical akrasia. It should be noted up front that while the puzzle of arithmetic raised in these sections is related to logical akrasia, we’ll leave it open what exactly the relation consists in, e.g., if second-order arithmetic is “logic” as per the neologicists, then the puzzle is just another case of logical akrasia; if Quine was right about bounds of logic, however, then it’s not. Gödel’s theorem establishes that: assuming Peano Arithmetic (PA) is consistent, PA doesn’t derive ConPA . In other words, if the theory PA is consistent, then PA cannot derive its own consistency.22 Consider now the following akratic puzzle: 20 The reader should consult (Russell, 2019) for an extensive overview of the many different kinds of logical pluralism on the market. It’s also worth noting the close analogy between weakness of will in logic and recent discussions of the normative significance of logic. See for example “What is Logical Monism?” (202X), where Clarke-Doane argues that logic doesn’t settle what to infer, even bracketing Harman’s complications of “Change in View” (1986), since it’s a contingency of language that we happen to have the logical concepts that we do. Thus, even assuming that, say, the disjunction of the Riemann Hypothesis and its negation is valid (out of our mouths) it remains open whether to infer this disjunction, since it remains open whether to defer to the metalogical concepts that we actually find ourselves with. Thanks to an anonymous reviewer for pointing out this connection. 21 Interestingly, Tim Williamson who is a card-carrying level-splitter with respect to epistemic akrasia, doesn’t seem to be one when it comes to logical theorizing. See for instance his review of Kit Fine’s compatibility semantics (Williamson, 2020). To be fair, however, as Williamson discusses in his review, the issue with Fine’s compatibility semantics is rather complicated and depends on the motivation for adopting the relevant non-classical logic in the first place. 22 Here we simply take the theory PA to be classical first-order logic extended with seven axioms. The language of first-order arithmetic can be specified as follows. LPA : The language of first-order arithmetic contains the usual logical vocabulary (connectives, quantifiers etc.) and auxiliary symbols such as brackets and punctuation marks. The set of primitive extralogical symbols is {+, ×, 0, S} denoting addition, multiplication, zero, and the successor function, respectively. 11 Akratic Peano Arithmetic. In using Peano Arithmetic, PA, subject S is at least implicitly committing to (and relying on) PA’s consistency. After all, if the theory were inconsistent it’s no help in sorting out truths from falsehoods. But if S’s theory is PA, then the theory doesn’t itself prove that PA is consistent (by Incompleteness). So, in that case S’s theory doesn’t prove the claim that S is committed to, and consequently S is at least weakly akratic (cf. §2.1). Of course, S can easily extend S’s theory (why not?), and then consider PA with ConPA added as an axiom. This would provide S with a proof of ConPA in a single line. But alas, now the issue of akrasia arises at the level of appeal to the consistency of that theory, i.e., PA +ConPA . And so on ad infinitum... In S’s use of PA it turns out that S is not just committed to the theory PA itself, but also the stronger theory PA +ConPA . Ergo, S is committed to a “logical” principle that their theory cannot prove, and thus S is in a state of akrasia.23 By construcTerms, formulas, and sentences, of LPA are also defined in the usual way. The formalized axioms of PA are: (Ax1) ∀x(S(x) 6= 0); (Ax2) ∀x∀y(S(x) = S(y) → x = y); (Ax3) ∀x(x + 0 = x); (Ax4) ∀x∀y(x + S(y) = S(x + y)); (Ax5) ∀x(x × 0 = 0); (Ax6) ∀x∀y(x × S(y) = (x × y) + x); (Ax7) {[Φ(0) ∧ ∀x(Φ(x) → Φ(S(x)))] → ∀x(Φ(x)) : Φ(x) ∈ LPA }. Notice that axiom 7 is really the set of arithmetical sentences falling under the axiom schema of mathematical induction, i.e., it’s an infinite set of axioms rather than just a single axiom. Obviously LPA allows us to express claims about the natural numbers in the theory PA, e.g., claims concerning addition and multiplication etc. But what is more important for our purposes below is that we’ll tacitly assume some form of coding. As Kurt Gödel (1931) showed it is possible to define a procedure, starting with assigning natural numbers to primitive expressions of LPA , and then extending the assignment to more complex syntactical objects. Eventually unique numbers become assigned to terms, formulas, and sequences of formulas; and it effect, we can then view some statements of first-order arithmetic as assertions about syntax. In other words, it becomes possible for us to use PA “introspectively”. The most famous example of this is of course the Gödel sentence (‘G’), which is at the heart of Gödel’s Second Incompleteness Theorem. The sentence G states about itself (via such-and-such substitution operations) that it isn’t a provable sentence in PA (Berto, 2011, p. 92). While it isn’t essential to us how the encoding from linguistic expressions to numbers is done—Gödel exploited the Unique Prime-Factorization Theorem to this end—it’s important to note that it can be done. For below we’ll appeal to the consistency claim—ConPA —stating that the theory PA is consistent, which in a way is just a regular claim made in LPA , and yet, this is only the case indirectly via our tacit coding procedure. 23 Some might hesitate to admit that the case results in akrasia because they don’t see Con PA as a genuine logical commitment: Why isn’t ConPA considered a further implicit, non-logical claim which S commits to? The answer is straightforward. ConPA is just a regular claim made in the language of PA (and definitely not a contingent empirical fact). There may of course be a sense in which PA doesn’t count as strictly “logical” but rather as “mathematical.” Even so the kind of 12 tion, there is no way for S to avoid committing to ConPA (or ConPA+ConPA , or...) and escaping their akratic state (on the pain of triviality). While this doesn’t necessarily show that S is irrational when using PA, it does entail that S is committed to something which goes beyond S’s own theory in such situations. Yet this is a positive kind of mismatch—i.e., S’s background logic can prove something which S’s believed theory cannot—rather than a violation of a negative rationality constraint such as the previously mentioned Akratic Principle. Recall the Akratic Principle stating that: No [epistemic] situation rationally permits any overall [doxastic] state containing both an attitude A and the belief that A is rationally forbidden in one’s situation. (cf. §1) As this is a negative principle in the sense that it involves an assertion about rationally forbidden states, the puzzle of akratic arithmetic is not in any obvious way a violation of it. Unless we are ready to grant that it’s rationally forbidden to commit oneself to something which one cannot prove, of course, but this seems overly strong. Nobody in their right mind would suggest that provability is a plausible guide to rationality simpliciter.24 Nonetheless the puzzle of akratic arithmetic does illustrate a clash with our ideals concerning epistemic rationality insofar as reflective equilibrium is among them. As logically akratic states cannot be in reflective equilibrium—given interpretation (1) from §2.2—the case above does indeed suggest that S’s doxastic state is epistemically irrational in a certain sense. How bad this sort of irrationality looks to the epistemologist does of course depend on the kind of good they take reflective equilibrium to achieve. On one interpretation reflective equilibrium merely indicates that a reasoner has done what is rationally required of them commitment S holds with respect to ConPA is not essentially different from the one S has towards the axioms of PA (though implicit). Hence—upon reflection—there is indeed a certain kind of akrasia (about the logic of the natural numbers if you like) arising in the puzzle. 24 Naturally it could be argued that provability is a plausible guide to rationality in a certain narrow sense. It’s clear enough that we can rationally believe many contingent propositions that we cannot prove to be correct, but in the case above we are not concerned with any old contingent proposition. We are concerned with the proposition that hPA is consistenti, and it’s not immediately clear how it could be rational to believe ConPA given that it cannot be proved using one’s theory. See for example (Gentzen, 1936; Chow, 2019) for further discussion of this non-trivial question. As an anonymous reviewer points out, another reason to think that provability cannot be a guide to rationality, even in mathematics, is that provability is always relative to a set of axioms, and thus an account of what makes belief in axioms rational will be needed. 13 relative to their initial data (e.g., a set of intuitions about certain logical inferences), but it would take further argument to show that a reasoner’s doxastic attitudes are also likely to be true. Under this interpretation reflective equilibrium is a rational ideal regarding the internal coherence of doxastic states rather than truth-conducive rationality.25 4 The Dilemma of Arithmetical Akrasia The upshot of §3 is what we might call the Dilemma of Arithmetical Akrasia: Insofar we take our theories to be appropriately formal and complete, then either:26 (i) we must be agnostic about the consistency of PA (on the pain of triviality), which would be extremely odd at best; (ii) or we must accept being arithmetically akratic, i.e., accept that we are trapped in an inescapable, infinite hierarchy of akratic states. Notice that while taking the second horn of the dilemma doesn’t rule out the existence of a rational fixpoint somewhere on the theoretical ladder, it does eliminate the possibility of an akrasia-free state which is accessible to us (since Gödel’s incompleteness results range over all axiomatizable theories). Further, the Dilemma of Arithmetical Akrasia is special in at least two ways. First, it involves an unsolvable case of akrasia while most cases of logical akrasia are clearly solvable, e.g., by converting to a fully classical theory. If the intuitionist we met in §2 were willing to convert to a fully classical theory, then their akratic state would dissolve. But the case of PA is different as it looks more similar to an epistemic blindspot; where proposition hpi is an epistemic blindspot for subject 25 For further discussion of positive epistemic evaluations and their connection to truthconduciveness, see, e.g., (BonJour, 1985; Alston, 1989; Littlejohn, 2012; Berker, 2013). 26 In this context the term ‘complete’ should be understood as follows: Within a given domain, every question is answerable, i.e., for any ϕ in the domain, it holds that ϕ or not-ϕ . This kind of completeness is also known as ‘Syntactic Completeness’. Note that Syntactic Completeness doesn’t entail that every particular answer has got the same epistemic status. The point is merely to suggest that we are committed to completeness in the sense that the question of whether PA is consistent has got an answer, but this is definitely not committing us to a theory which can decide every question. Different interpretations of the term ‘formal’ can be found in (MacFarlane, 2000; Dutilh Novaes, 2012). 14 S at time t if and only if hpi is consistent but unknowable by S at t (Sorensen, 1988, 2020). Similarly, the consistency presumption is fundamental to our use of PA, but it just cannot be proved (and thus known) from within the bounds of the theory itself. Second, unlike the akratic issues we focused on above (cf. §2), the Dilemma of Arithmetical Akrasia cuts across the divide between classical and non-classical logicians. In the case of PA it seems that we are all either agnostic (on the pain of triviality) or akratic! So, in the end, taking the first horn doesn’t sit well with our general intellectual outlook because we want to avoid being agnostic about the consistency of PA; on the other hand, going for the second horn is an unpleasant move as it reveals a boundary on arithmetical theorizing which seems to conflict with our rational ideals (e.g., reflective equilibrium). 5 Conclusion and Perspectives: Escaping the Dilemma of Arithmetical Akrasia While the exact details remain to be worked out, this final section provides a quick and dirty proposal of how one can translate the Dilemma of Arithmetical Akrasia into a case of regular epistemic akrasia; and further how one might escape the dilemma when it’s spelled out this way. Let’s first reformulate the dilemma such that it becomes a case of epistemic akrasia. Spelled out in terms of premises and conclusion(s), we get: 1. S believes PA [by Indispensability]. 2. S believes ConPA [by No-Miracles]. 3. S believes ConPA is a logical principle [in absence of reasons to the contrary]. 4. S believes 0PA ConPA [by Incompleteness]. 5. S believes in the strong interpretation of reflective equilibrium with respect to PA: It’s permissible for S to believe a logical principle only if PA proves it. 6. Therefore: S believes ConPA and believes that hit’s forbidden to believe ConPA i. 15 7. Ergo: S is epistemically akratic. Now, it seems fair to suggest that we don’t want to consider rejecting premises (1), (2), and (4); which leaves us with the possibility of rejecting one or both of (3) and (5) in order to escape the dilemma. That is to say: 1. S believes PA [by Indispensability]. 2. S believes ConPA [by No-Miracles]. 3. S believes ConPA is a logical principle [in absence of reasons to the contrary]. 4. S believes 0PA ConPA [by Incompleteness]. 5. S believes in the strong interpretation of reflective equilibrium with respect to PA: It’s permissible for S to believe a logical principle only if PA proves it. 6. Therefore: S believes ConPA and believes that hit’s forbidden to believe ConPA i. 7. Ergo: S is epistemically akratic. Consider first the possibility of rejecting (3), i.e., the logicality of ConPA . The principle stated by ConPA is certainly a well-formed sentence of PA; and in that specific sense it is logical. It also expresses something essential to PA (or at least our use of PA). But perhaps ConPA is still not logical in the right way for strong reflective equilibrium to apply to it. Is it really fair to expect strong reflective equilibrium to apply to paraconsistent logical theories, for example? Consider next the possibility of rejecting (5) instead. While reflective equilibrium may be initially plausible when viewed as a philosophical method or as “the prima facie epistemology of logic” (Cohnitz and Estrada-González, 2019, p. 137), it does seem like an overly demanding output to expect from applying the method in the context of logical theorizing and PA. Gödel’s theorem already suggests that epistemic principles of that kind are hopeless, because every theory with the same expressive power as PA (or more) has a Gödel sentence. Why insist on something impossible? (This is an “ought-implies-can” violation perhaps). Moreover, what reason do we have to think that strong reflective equilibrium is a norm of belief within arithmetical theorizing? It seems that you’ll need overly demanding bridge 16 principles to establish the right connections between the formal sciences and epistemology in order to get this going (cf. (MacFarlane, 2004)). Another way to put this point: the dilemma relies on it being an epistemic ideal that there is a reflective equilibrium between what one is committed to and what one’s believed theory can prove (call it ‘RE’). An alternative, and perhaps more plausible ideal is that there be a reflective equilibrium between what one’s committed to and what one’s epistemic practice can justify (call it ‘RE∗ ’). What speaks in favor of RE over RE∗ ? Supposing that at least one of the rough strategies outlined above is successful in letting us escape the dilemma—when it’s framed in terms of epistemic akrasia—we are thus left to ask whether it’s still a problem if S is arithmetically akratic after rejecting either of these premises and avoiding epistemic akrasia. Some theorists of the post-Gödel era may simply shrug their shoulders and bite the bullet here. In a way what Gödel’s second incompleteness result tells us is that we can’t both have consistency and syntactic completeness when it comes to theories with a certain amount of expressive power. So, perhaps some level of akrasia is just something that working logicians, mathematicians, computer scientists etc., have come to live with in the aftermath of Gödel. They may also want to suggest that there is an important difference between the cases of logical akrasia exemplified by the intuitionist rejecting the excluded middle (cf. §2) and the specific case of PA. In the former, the intuitionist can’t combine their official theory and the background logic they are committing to into a jointwise consistent whole, whereas this is certainly possible in the latter—it’s just that the background logic must be stronger than the theory PA itself. Acknowledgments I would like to thank the audiences at various Arché seminar meetings in 2021, University of St Andrews, for constructive feedback on earlier versions of this paper. From memory these included Matteo Nizzardo, Andrea Oliani, Thomas Radriamahazaka, Francisca Silva, and Jace Snodgrass. Further, I would like to thank the audiences at the SEECRs WiP seminar, March 2022, University of Edinburgh; 1 st Arché workshop on Logical Disagreement, April 2022, Univeristy of St Andrews; EEN conference, June 2022, University of Glasgow; 1 st BSTK conference, September 2022, University of Glasgow; 3rd Unifesp Philosophy Graduate Conference, October 2022, Federal University of São Paulo; 16th Cambridge Graduate Conference on the Philosophy of Mathematics and Logic, Jan17 uary 2023, University of Cambridge; Bergen Logic Seminar, March 2023, University of Bergen; and Joint Session of the Aristotelian Society and the Mind Association, July 2023, Birkbeck & The Institute of Philosophy, University of London, for fruitful Q&A-sessions. Finally, I want to express special gratitude to Roy Sorensen, Diego Tajer, Alessandro Rossi, Henrik Sherling, and Hugh Reilly, as well as my supervisor team, i.e., Greg Restall, Franz Berto, and Jessica Brown, for written comments on my drafts. Funding My research is funded by the Niels Bohr Foundation (Royal Danish Academy of Sciences) and the Royal Institute of Philosophy (London). References Adler, J. E. (2002). Akratic believing? Philosophical Studies, 110(1):1–27. Alston, W. P. (1989). Epistemic justification: Essays in the theory of knowledge. Cornell University Press. Bacon, A. (2013). Non-classical metatheory for non-classical logics. Journal of Philosophical Logic, 42(2):335–355. Beall, J. C. and Restall, G. (2006). Logical pluralism. Oxford University Press. Berker, S. (2013). The rejection of epistemic consequentialism. Philosophical Issues, 23:363–387. Berto, F. (2011). There’s something about Gödel: the complete guide to the incompleteness theorem. John Wiley & Sons. BonJour, L. (1985). The structure of empirical knowledge. Harvard University Press. Bradley, D. (2021). Uniqueness and modesty: How permissivists can live on the edge. Mind, 130(520):1087–1098. Brown, J. (2018). Fallibilism: Evidence and knowledge. Oxford University Press. 18 Carr, J. R. (2021). Why ideal epistemology? Mind. Chislenko, E. (2021). How can belief be akratic? Synthese. Chow, T. Y. (2019). The consistency of arithmetic. The Mathematical Intelligencer, 41(1):22–30. Christensen, D. (2007). Epistemology of disagreement: The good news. The Philosophical Review, 116(2):187–217. Christensen, D. (2010). Higher-order evidence. Philosophy and Phenomenological Research, 81(1):185–215. Christensen, D. (2021). Akratic (epistemic) modesty. Philosophical Studies, 178(7):2191–2214. Christensen, D. (2024). Epistemic akrasia: No apology required. Noûs, 58(1):54– 76. Cieśliński, C. (2017). The epistemic lightness of truth: Deflationism and its logic. Cambridge University Press. Clarke-Doane, J. (202X). What is logical monism? In Peacocke, C. and Boghossian, P., editors, Normative Realism. Oxford University Press. Cohnitz, D. and Estrada-González, L. (2019). An Introduction to the Philosophy of Logic. Cambridge University Press. Daoust, M. (2019). 16(3):282–302. Epistemic akrasia and epistemic reasons. Episteme, Dutilh Novaes, C. (2012). Formal languages in logic: A philosophical and cognitive analysis. Cambridge University Press. Feldman, R. (2005). Respecting the evidence. Philosophical Perspectives, 19:95– 119. Field, C. (2019). It’s ok to make mistakes: Against the fixed point thesis. Episteme, 16(2):175–185. Field, C. (2021). Giving up the enkratic principle. Logos & Episteme, 12(1):7–28. 19 Field, H. (2017). Disarming a paradox of validity. Notre Dame Journal of Formal Logic, 58(1):1–19. Fischer, M., Horsten, L., and Nicolai, C. (2021). Hypatia’s silence: Truth, justification, and entitlement. Noûs, 55(1):62–85. Gentzen, G. (1936). Die widerspruchsfreiheit der reinen zahlentheorie. Mathematische annalen, 112(1):493–565. Gödel, K. (1931). Über formal unentscheidbare sätze der principia mathematica und verwandter systeme i. Monatshefte für Mathematik und Physik, 38(1):173– 198. Greco, D. (2014). A puzzle about epistemic akrasia. Philosophical Studies, 167(2):201–219. Haack, S. (1974). Deviant logic: Some philosophical issues. Cambridge University Press. Harman, G. (1986). Change in View: Principles of Reasoning. MIT Press. Horowitz, S. (2014). Epistemic akrasia. Noûs, 48(4):718–744. Horowitz, S. (2022). Higher-Order Evidence. In Zalta, E. N. and Nodelman, U., editors, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Fall 2022 edition. Horsten, L. and Leigh, G. E. (2017). Truth is simple. Mind, 126(501):195–232. Jackson, E. and Tan, P. (2022). Epistemic akrasia and belief-credence dualism. Philosophy and Phenomenological Research, 104(3):717–727. Kappel, K. (2019). Escaping the akratic trilemma. In Skipper, M. and SteglichPetersen, A., editors, Higher-order evidence. New essays, pages 124–143. Oxford University Press. Kauss, D. (2023). A rational agent with our evidence. Erkenntnis, pages 1–22. Kearl, T. (2020). Epistemic akrasia and higher-order beliefs. Philosophical Studies, 177(9):2501–2515. Kelly, T. (2005). The epistemic significance of disagreement. Oxford Studies in Epistemology, 1(167-196). 20 Kelp, C. (2023). The Nature and Normativity of Defeat. Cambridge University Press. Lasonen-Aarnio, M. (2014). Higher-order evidence and the limits of defeat. Philosophy and Phenomenological Research, 88(2):314–345. Lasonen-Aarnio, M. (2020). Enkrasia or evidentialism? Learning to love mismatch. Philosophical Studies, 177(3):597–632. Littlejohn, C. (2012). Justification and the truth-connection. Cambridge University Press. Littlejohn, C. (2018). Stop making sense? On a puzzle about rationality. Philosophy and Phenomenological Research, 96(2):257–272. MacFarlane, J. (2000). What does it mean to say that logic is formal. PhD Thesis. MacFarlane, J. (2004). In what sense (if any) is logic normative for thought. Unpublished manuscript. Makinson, D. C. (1965). The paradox of the preface. Analysis, 25(6):205–207. Meyer, R. K. (1985). Proving semantical completeness ’relevantly’ for r. In Technical Report. Australian National University. Mortensen, C. (2013). Inconsistent mathematics. Kluwer Academic Publishers. Owens, D. (2002). Epistemic akrasia. The Monist, 85(3):381–397. Pollock, J. (1970). The structure of epistemic justification. American Philosophical Quarterly (monograph series 4), pages 62–78. Pollock, J. (1974). Knowledge and justification. Princeton University Press. Pollock, J. (1984). Reliability and justified belief. Canadian Journal of Philosophy 14, pages 103–114. Pollock, J. (1986). Contemporary theories of knowledge. Rowman and Littlefield. Pollock, J. L. (1994). Justification and defeat. Artificial Intelligence, 67(2):377– 407. Priest, G. (2006). In contradiction. Oxford University Press. 21 Priest, G. (2008). An introduction to non-classical logic: From if to is. Cambridge University Press. Read, S. (2006). Monism: The one true logic. In DeVidi, D. and Kenyon, T., editors, A logical approach to philosophy: Essays in honour of Graham Solomon, pages 193–209. Springer. Resnik, M. (1985). Logic: Normative or descriptive? The ethics of belief or a branch of psychology? Philosophy of Science, 52(2):221–238. Resnik, M. (1996). Ought there to be but one logic. In Copeland, B. J., editor, Logic and reality: Essays on the legacy of Arthur Prior, pages 489–517. Oxford University Press. Resnik, M. (2004). Revising logic. In Priest, G., Beall, J. C., and Armour-Garb, B., editors, The law of non-contradiction, pages 178–193. Oxford University Press. Ribeiro, B. (2011). Epistemic akrasia. International Journal for the Study of Skepticism, 1:18–25. Roush, S. (2017). Epistemic self-doubt. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Winter 2017 edition. Russell, G. (2019). Logical pluralism. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, summer 2019 edition. Schurz, G. (2021). Meaning-preserving translations of non-classical logics into classical logic: Between pluralism and monism. Journal of Philosophical Logic. Skipper, M. (2019). Reconciling enkrasia and higher-order defeat. Erkenntnis, 84(6):1369–1386. Skipper, M. (2021). Does rationality demand higher-order certainty? Synthese, 198(12):11561–11585. Sliwa, P. and Horowitz, S. (2015). Respecting all the evidence. Philosophical Studies, 172(11):2835–2858. 22 Smithies, D. (2012). Moore’s paradox and the accessibility of justification. Philosophy and Phenomenological Research, 85(2):273–300. Sorensen, R. (1988). Blindspots. Oxford University Press. Sorensen, R. (2020). Epistemic paradoxes. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Fall 2020 edition. Titelbaum, M. (2015). Rationality fixed point (or: In defense of right reason). In Gendler, T. S. and Hawthorne, J., editors, Oxford Studies in Epistemology, pages 253–294. Oxford University Press. Titelbaum, M. (2019). Return to reason. In Skipper, M. and Steglich-Petersen, A., editors, Higher-order evidence. New essays, pages 226–245. Oxford University Press. Weatherson, B. (2010). Do judgments screen evidence? Unpublished Manuscript. Weber, Z., Badia, G., and Girard, P. (2016). What is an inconsistent truth table? Australasian Journal of Philosophy, 94(3):533–548. Wedgwood, R. (2012). Justified inference. Synthese, 189(2):273–295. Williamson, T. (2011). Improbable knowing. In Dougherty, T., editor, Evidentialism and its Discontents, pages 147–164. Oxford University Press. Williamson, T. (2014). Very improbable knowing. Erkenntnis, 79(5):971–999. Williamson, T. (2020). Vagueness: A Global Approach, by Kit Fine. Mind, 131(522):675–683. Worsnip, A. (2018). The conflict of evidence and coherence. Philosophy and Phenomenological Research, 96(1):3–44. 23