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Inferentialism and the Epistemology of Logic: Reflections on Casalegno and Williamson. By Paul Boghossian

Paolo Casalegno’s ‘Logical Concepts and Logical Inferences’ (Casalegno 2004 ) is a searching and insightful critique of my attempt to explain how someone could be entitled to infer according to a basic logical rule. I will say a bit about what I take the problem to be before considering his discussion in detail. Let us agree that we reason according to logical rules. (There are various issues about this, most forcefully pressed by Gilbert Harman (see Harman 1986 ), which I propose to set aside for present purposes.) One of the most central of the rules by which we may be said to reason is Modus Ponens, which I will take to say (again ignoring many complexities which are irrelevant for present purposes): (MP) Whenever both p and ‘if p, then q’, infer q. Those of us who have learned to formulate this rule recognize it as a rule that we operate with and, putting aside some deviant logicians, consider it valid. However, there are many perfectly rational persons – youngsters or undereducated adults – who are not aware that this is a rule that they operate with. Despite this, we think that when such people reason according to MP: (Rain) It rained last night. If it rained last night, then the streets are wet. So, The streets are wet. they are perfectly entitled to do so; and that the justification that they have for their premises transmits smoothly to their conclusion. In what does their entitlement to reason according to MP consist? Obviously it cannot consist in some argument that they have formulated for the belief that this form of inference is valid. By assumption, they do not have the belief in question so could hardly have seen the need to formulate an argument for it. But if their entitlement does not consist in an explicit justification for the validity of MP, what does it consist in? What about us sophisticated philosophers who have arrived at the knowledge that we reason according to the rule MP? In what does our entitlement to use this rule consist? Have not we formulated an explicit justification for operating according to this rule? And if we have not actually bothered doing so, is not it clear that we are at least in a position to do so, if asked and given enough time for reflection? It might seem as if the answer to the latter question is ‘Yes’. For could not we offer something like the following argument (I will not bother with the niceties of semantic ascent and descent)? (i) If ‘p’ is T and ‘p→q’ is T, then ‘q’ is T (by knowledge of the truth table)(ii) ‘p’ is T and ‘p→q’ is T (by assumption) Therefore, (iii) ‘q’ is T (by MP) Well, this particular argument obviously cannot do much for us by way of justifying our use of MP, since it relies on MP at its third step and I am assuming that we cannot explain in what our entitlement to reason with a certain rule, R, consists by showing that we have to hand an argument for R that employs R. Of course, we might be able to offer other justifications for MP that rely not on MP itself but on certain other rules, say R1 and R2. But then the question will arise what entitles us to use R1 and R2? We could now repeat the process of providing a justification for R1 and R2. Pretty soon, though, our justifications will end up appealing either to Modus Ponens, or to R1 or R2, or to some other rules for which we will owe a justification. It seems obvious, then, that even the most sophisticated and powerful philosopher will face the following dilemma: with regard to her most basic logical rules, either she has no entitlement to them, or she has an entitlement that is not grounded in her ability to provide an explicit argument for them. The skeptical alternative is dire. For if she has no entitlement to her most basic rules, then she has no entitlement to anything that is based upon them; and that means that she will have no entitlement to any of the rules of logic that she is inclined to use and therefore no entitlement to any of the beliefs that she will have based on them. This seems to me too fantastic to believe. It also seems to me to tee up an extreme form of relativism about rationality, one that I find worrisome, both philosophically and socially. For if none of us is entitled to the particular set of logical rules that we operate with, then if others among us were to find it natural to operate with a different and incompatible set of logical rules, then they would have to be deemed as rational as we are, in so far as their use of logical rules is concerned. We could not say that such people were irrational, for they are surely no worse off in their entitlements to their logical rules than we are with respect to ours. The skeptical alternative, then, is fraught with difficulty. The non-skeptical alternative, however, requires us to explain how someone might be entitled to operate according to some basic logical rule, say MP, without his being able to provide anything like a cogent argument for MP. How could someone be so entitled? In particular, how could someone be so entitled in a way that did not imply an “anything goes’ conception of ‘logical rationality?” I hope it’s obvious that this is a highly non-trivial task. Indeed, it remains unclear to me even now that there is a way of executing it that is even remotely satisfying. Large as the task may be, though, it is not quite as large as the task of providing an overall epistemology of logic. As I am thinking about it, refuting the skeptical threat involves showing that it is possible to get our entitlement to some logical rules off the ground, even if the means by which that is accomplished may not generalize to all the logical rules to which we feel intuitively entitled. What is needed, in other words, is a plausible answer to a Kantian-style “How possible?” question. We may worry about how to account for the rest of logic later. This relates to a point that shows up in an exchange that I have had with Timothy Williamson on these issues. Tim says: In previous work, Boghossian developed an epistemology of logic based on understandingassent links corresponding to fundamental rules of logic. His paradigm was modus ponens: a necessary condition for understanding ‘if’ was supposed to be willingness to assent to inferences by modus ponens involving ‘if’. The book presents a series of counterexamples, some actual, some possible, to such putative understanding-assent links, for both modus ponens and other equally fundamental rules (85–121). The counterexamples concern native speakers of a natural language who come to understand the logical words at issue in the usual way but then go in for deviant logical theorizing without losing their linguistic competence; most philosophers know such people. In response, Boghossian picks what he regards as the clearest understanding–assent link, willingness to assent to ‘and’-elimination (the inference from ‘P and Q’ to ‘P’ or to ‘Q’) as a condition for understanding ‘and’, and denies that the counterexamples I propose to it (95–6) make sense. Strategically, Boghossian’s response is not very promising. If he can rely on understandingassent links only for ‘and’-elimination and a few other equally banal rules, but not for modus ponens or other fundamental principles, then he is in no position to base either a general epistemology of logic or a general account of the understanding of logical constants on understanding–assent links. It is a little lame for him to claim in effect that not every fundamental rule of logic is a counterexample to his original account. A bolder strategy for him would be to seek a way of defending the claim that no fundamental rule of logic is a counterexample to his original account, and in particular of defending his original test case, modus ponens, as a putative understanding-assent link for ‘if’ against my counterexamples. In keeping away from the bolder strategy, Boghossian concedes so much ground that it is quite unclear what his fallback general epistemology of logic or his fallback general account of the understanding of logical constants could be. (Williamson 2011 , 500) Now, I do not, of course, deny the desirability of having a general epistemology of logic, but, as I have just been emphasizing, to my mind the fundamental difficulty in this area is to show that there might be even a single promising pathway for avoiding skepticism about our entitlement to the fundamental rules of logic. So, pace Tim, I would be happy if, in the first instance, I could come up with a plausible account just for one or two of the most ‘banal’ rules of logic. 2. Well, what are the possible anti-skeptical alternatives? We have ruled out accounts that trace our entitlement to using a basic rule of logic in terms of our ability to provide any sort of argument. Could we plausibly say that it consists in some sort of non-inferential warrant? A traditionally influential answer along these lines deploys the idea of an ‘intuition.’ A thinker is entitled to MP if he intuits its validity in some essentially non-discursive and non-inferential way. I find this answer very problematic, for reasons that I have developed elsewhere (see Boghossian 2003 ), although I am now inclined to be more sympathetic to the notion of an intuition than I used to be. If we put aside explanations in terms of intuition, then the anti-skeptical task before us becomes one of explaining how a thinker might be differentially entitled to MP blindly, without being in a position to point to any sort of justification for his use of MP, whether this be of an inferential or non-inferential variety. How could we be blindly entitled to operate according to MP? There look to be two main options. The first would consist in embracing a crudely reliabilist conception of inferential justification, according to which a thinker is entitled to the use of a rule if that rule is reliably truth-preserving, then we would have an easy answer to our problem. Any thinker would be blindly entitled to MP, since (let us assume) MP is necessarily truth-preserving. However, such a crude reliabilism is clearly false. There are lots of logically valid inferences, for example, from the Peano axioms to any instance of the inequality of Fermat’s Last Theorem, which no one would be entitled to perform merely as a result of their reliability. The second avenue for explaining blind justification involves deploying the classical notion of analyticity. For one important strand in that notion is the epistemic idea that understanding alone can sometimes suffice for entitlement: it is plausible, for example, that the mere understanding of the word ‘bachelor’ suffices for our knowing that all bachelors are male. If, on analogy with this, our understanding of ‘if’ could be shown to suffice for our being entitled to use MP, we would have the answer to our skeptic. Taking such a notion of epistemic analyticity seriously, though, required showing that it could be detached from the much more dubious doctrine of metaphysical analyticity – of truth (or validity) in virtue of meaning – with which it had always been associated, but this seemed doable (see Boghossian 1996 ). The problem then became one of explaining concretely how our understanding of ‘if’ might suffice for our entitlement to use MP. An obvious starting point was a theory of our understanding of the logical constants that had always found favor among philosophers, quite independently of epistemological issues, according to which to grasp a logical constant necessarily involves being prepared to use it according to some inference rules and not others. In its strongest form, such a theory is called an Inferential Role Semantics and says that it is in virtue of our using a constant, say ‘if’, according to some basic rule involving it, say MP, that ‘if’ means if in our idiolect.1 A weaker doctrine, which is all I will assume here, would simply have it that meaning if by ‘if’requires inferring according to MP, without necessarily being sufficient for it. As I say, many philosophers, among whom we may number Michael Dummett, Robert Brandom, Paul Horwich, Ned Block, Stephen Schiffer and Christopher Peacocke, have been partial to some version or other of an inferential role semantics. Even one of the harshest critics of this style of meaning theory, Jerry Fodor, has always maintained that when it came to the case of the logical constants, no other style of theory seemed to be in the running – certainly not causal or teleological or definitional theories. Accordingly, I got interested in the question: Suppose we assume that • (A) Inferring according to MP is necessary for someone to mean if by ‘if’. could we make it plausible that: • (B) We are blindly entitled to infer according to MP. I did not take it upon myself to argue for (A). I followed in the footsteps of the philosophers listed above and simply assumed (A). My main task was to try to show that if (A) is true (B) is true. In a series of papers, I experimented with a number of different ways of arguing for this conditional, none of which I am fully satisfied with. 3. My critics, however, Paolo and Tim included, have largely concentrated not on the arguments I provided for the conditional ‘if (A), then (B)’, but rather on the inferentialist assumption (A) itself. This has thrust me into the role of defender of an inferential role account of the meaning of the logical constants, although, as I say, my main focus was elsewhere. Nevertheless, the inferential role theory is assumed by my account and my critics have brought up many interesting points. I am therefore happy to discuss them. Now, the first two thirds of Paolo’s paper consists in some very interesting observations about why we should not take inference according to some rule to be sufficient for possessing a particular logical constant. I find some of that discussion to be very interesting, but, as Paolo realizes, it is not relevant to the sort of account that I was exploring, which depends only on the necessity claim. Paolo does eventually turn his attention to the necessity claim, and to the account of entitlement that is built upon it, and he has a number of telling objections to make against them both. 4. One objection that Paolo makes is that my story about entitlement is bound to be incomplete. Paolo claims that we are blindly entitled to many more inferences than could plausibly be said to be necessary for concept possession. When should we say that those non-concept-constituting inferences are also blameless? (When I say “concept-constituting” in this paper, I shall just mean “is necessary for possession of the concept in question”.) To provide an answer to this question, the analysis should be substantially supplemented. On the other hand, once we have found a satisfactory account of blamelessness for blind inferences which are not instances of rules belonging to the possession conditions for the logical constants, why shouldn’t we apply this account also to the blind inferences which are instances of those rules, so making the initial analysis superfluous? Unless, of course, what Boghossian intends to suggest is that only inferences which are instances of rules belonging to the possession conditions for logical constants can be at the same time blind and blameless. But this would be hard to maintain. Take Ramanujan, great Indian mathematician. We are told that he had an astonishing capacity to draw immediately very remote and complex consequences from given premisses. He was often unable to justify those conclusions by means of what most mathematicians would have regarded as an acceptable proof; in fact, we are told that he had only a very vague notion of what a proof is. Well, I think it would be wrong to deny that Ramanujan’s blind inferences were (at least in many cases) blameless, i.e. that they transferred knowledge. At the same time, it would make no sense to say that being able to perform inferences like those of Ramanujan is part of the possession conditions for logical constants, i.e. that it is necessary in order to know what they mean. (406) What I would say in response is that, in the sense I have in mind, it is plausible that Ramanujan was in a position to offer some justification for his feats of inference, even if he was not in a position to provide a rigorous proof of them. The kind of circumstance for which I coined the notion of “blind entitlement” was for the case of a logical rule so basic that no person, no matter how well informed and how good at rigorous proof, could provide any sort of justification for the use of the rule, because of the inevitable circularity that such a justification would entrain. So, I do not see that we have in Paolo’s description a clear example of a rule for which we are in principle not in a position to supply a justification, yet to which we are clearly entitled, and which could not plausibly be claimed to be concept-constituting.5. Paolo’s next point, however, is more worrisome. He maintains that there is a general recipe for generating a counterexample to any claim of the form: Reasoning according to rule R is required in order to have logical constant C. The idea that there is such a recipe strikingly anticipates an argumentative strategy developed by Timothy Williamson in his recent book (Williamson 2007 ), although the recipes that Paolo and Tim have in mind are different from one another (and were no doubt developed independently of one another). Before looking at these recipes in greater detail, let me comment on the importance of the claim that there is a recipe of this kind, a general method for generating a counterexample to any particular concept constitution claim. The point is that there is, even among the friends of inferentialism, considerable uncertainty about exactly which rules are meaning constituting for which constant. Of course, this is not meant to be a virtue and is raised by Paolo, as we shall see below, as a point of criticism of inferential theories. This uncertainty, though, can serve as a natural defense against proffered counterexamples. Faced with a counterexample to a particular concept constitution claim, the inferential theorist can always distance himself from that particular claim, while clinging to the claim that some rule or other will be constitutive. Vann McGee, for example, has developed what he takes to be a set of counterexamples to Modus Ponens (see McGee 1985 ). These examples all involve cases in which a conditional is embedded in another conditional. Does this show that MP is not necessary for possession of if? It may be hard to answer this question precisely because we may not be sure if assent to all instances of MP is necessary for possession of if. Perhaps it is enough that a thinker assents to all instances of MP that do not involve conditionals embedded in bigger conditionals? Perhaps MP is not involved in the possession of if at all. When the focus is, as mine was, on the conditional. If R is concept constituting, then we are blindly entitled to R, it can seem a matter of indifference that someone has come up with a counterexample to any particular concept constitution claim. However, if someone can show that there is a general recipe for generating a counterexample to any pair of C and R such that reasoning with R is held to be necessary for C, then clearly that goes to the heart of inferentialism and of any epistemology that might be built upon it. Here, then is Paolo’s recipe for generating such counterexamples: Apart from this difficulty, the idea that, given a logical constant C, there is a well-defined set R of rules of inference such that a subject cannot be regarded as knowing the meaning of C unless she accepts the rules in R is intrinsically problematic. No matter how a rule of inference is chosen, it seems to me that we can imagine situations in which we would be disposed to say that a subject knows the meaning of C although the subject does not accept the rule in question. Suppose Mary suffers from a cognitive disability which makes her completely incapable of performing anything which could be counted as an inference. Nevertheless, she is able to use logically complex sentences to describe visually presented scenes. (I do not know whether this kind of cognitive disability has ever been observed; but it is no doubt conceivable.) For example, we can imagine that Mary, although unable to perform conjunction-introductions and conjunction-eliminations, would be able to assert, in appropriate circumstances, “The box is red and the book is blue”, or things like that. It seems to me that it would then be possible to say that, in spite of her disability, Mary knows the meaning of the word “and”. At least: my intuition is that we would spontaneously adopt a homophonic translation for sentences such as “The box is red and the book is blue” uttered by Mary; and wouldn’t that be a way of acknowledging that, in Mary’s mouth, the word “and” has the same meaning it is has for us? (407) So, the way Paolo’s recipe is supposed to work is that, for any rule, R, and constant, C, we can cook up a counterexample to the claim that inferring according to R is required for possession of C, by imagining someone who lacks the ability to infer with R but who, we might make plausible, possesses C because she uses it competently in sentences held true. I am not convinced that there is a good recipe here for generating counterexamples to inferentialism. There are two ways in which we can develop the Mary example between which Paolo’s description does not distinguish and, however we develop the example, I do not see that we get a convincing counterexample to the necessity of either conjunction elimination or conjunction introduction for possession of and. On the first way of developing the example, we may claim that Mary can think each of the atomic sentences “The box is red” and “The book is blue” separately, reliably asserting the first in the presence of a red box and the second in the presence of a blue book. Furthermore, we may claim that she can also think the compound sentence “The box is red and the book is blue”, reliably asserting it in the presence of a state of affairs that contains both a red box and a blue book. What, then, is she lacking? What we are told is that she cannot infer from the atomic sentences taken together to the compound sentence, or from the compound sentence to either of the atomics. Confronted with a red box and blue book she is prepared to assert “The box is red and the book is blue”, but if you ask her right after she has asserted that conjunction (perhaps this has to take place in a different room), “So, is the box red?” she might say “No”. Similarly, for a question about the book. Having shown Mary a red box and gotten her to assert “The box is red”, and having shown her a blue book in a different room and gotten her to assert “The book is blue”, she then refuses to assent to “The box is red and the book is blue”, although, by hypothesis she is willing to assent to the compound sentence when she is shown the box and book together. I find all this mystifying and certainly do not feel inclined to say: Clearly, Mary retains the ordinary concept of conjunction that ordinary people have, even as she fails to make various inferences that ordinary people would make. We can also flesh out Paolo’s example by stipulating that Mary can think only the compound sentence and cannot think the atomic ones separately. This, I think, would make it even less plausible that Mary means and by ‘and’.