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Knowledge of Logic Is it possible for us to know the fundamental truths of logic a priori? This question presupposes another: is it possible for us to know them at all, a priori or a posteriori? In the case of the fundamental truths of logic, there has always seemed to be a difficulty about this, one that may be vaguely glossed as follows (more below): since logic will inevitably be involved in any account of how we might be justified in believing it, how is it possible for us to be justified in our fundamental logical beliefs? In this essay, I aim to explain how we might be justified in our fundamental logical beliefs. If the explanation works, it will explain not merely how we might know logic, but how we might know it a priori. The Problem Stated To keep matters as simple as possible, let us restrict ourselves to propositional logic and let us suppose that we are working within a system in which modus ponens (MPP) is the only underived rule of inference. My question is this: is it so much as possible for us to be justified in supposing that MPP is a valid rule of inference, necessarily truth‐preserving in all its applications?1 I am not at the moment concerned with how we are actually justified, but only with whether it makes sense to suppose that we could be. We need to begin with certain distinctions. Suppose it is a fact about S that, whenever he believes that p and believes that ‘if p, then q’, he is disposed either to believe q or to reject one of the other propositions. Whenever this is so, and (p.230) putting many subtleties to one side, I shall say that S is disposed to reason according to the rule modus ponens. In addition to this disposition to reason in a certain way, it can also be a fact about S that he has the full‐blown belief that MPP is necessarily truthpreserving: he believes, that is, that if p is true and if ‘if p, then q’ is true, then q has to be true. As a number of considerations reveal, S’s disposition to reason in accordance with MPP and his belief that MPP is truth‐preserving are distinct kinds of state.2 Just as we need to distinguish between the disposition to reason and the belief, so we need to distinguish between the epistemic status of the disposition and that of the belief. I want to ask initially here about whether it is possible for us to be justified in holding the belief that MPP is valid. I will ask later about whether it is possible for us to be entitled to the disposition to reason according to MPP and about the relation between that question and the corresponding question about belief. (As my choice of language implies, I shall reserve the term ‘justification’ for the sort of warrant that a belief might have, and the term ‘entitlement’ for the sort of warrant that an employment of a rule might have.)3 Turning, then, to the question of the justifiability of the belief that MPP is truth‐preserving, the space of available answers seems relatively clear: it is perspicuously represented by a flowchart (Fig. 10.1 ). When we look at the available options, it is very tempting to say ‘No’ to the question about justification, that it is not possible for us to be justified in believing something as basic as modus ponens. For in what could such a justification consist? It would have to be either inferential or noninferential. And there look to be serious problems standing in the way of either option.Fig. 10.1 (p.231) How could our justification for MPP be non‐inferential? In any ordinary sense of ‘see’, we cannot just see that it is valid. To be sure, the idea that we possess a quasi‐perceptual faculty—going by the name of ‘rational intuition’—the exercise of which gives us direct insight into the necessary properties of the world, has been historically influential. It would be fair to say, however, that no one has succeeded in saying what this faculty really is nor how it manages to yield the relevant knowledge. ‘Intuition’ seems like a name for the mystery we are addressing, rather than a solution to it. A related thought that is often mentioned in this connection has it that we can know that MPP is truthpreserving because we cannot conceive or imagine what a counter‐example to it might be.4 But the suggestion that ‘conceiving’ or ‘imagining’ is here the name for a non‐inferential capacity to detect logical, or other necessary, properties will not withstand any scrutiny. When we say that we cannot conceive a counter‐example to some general claim—for example, that all bachelors are unmarried—we do not mean that we have some imagistic, non‐propositional ability to assess whether such a case can be coherently described. What we mean is that a more or less elementary piece of reasoning shows that there cannot be any such state of affairs: if someone is a bachelor, then he is an unmarried male. If someone is an unmarried male, then he is unmarried. Therefore, any bachelor would have to be unmarried. So, there cannot be any such thing as an unmarried bachelor. Talk of ‘conceiving’ here is just a thin disguise for a certain familiar style of logical reasoning. This is not, of course, to condemn it. But it is to emphasize that its acceptability as an epistemology for logic turns on the acceptability of an inferential account more generally. This brings us, then, to the inferential path. Here there are a number of distinct possibilities, but they would all seem to suffer from the same master difficulty: in being inferential, they would have to be rule‐circular. If MPP is the only underived rule of inference, then any inferential argument for MPP would either have to use MPP or use some other rule whose justification depends on MPP. And many philosophers have maintained that a rule‐circular justification of a rule of inference is no justification at all. Thus, it is tempting to suppose that we can give an a priori justification for modus ponens on the basis of our knowledge of the truth‐table for ‘if, then’. Suppose that p is true and that ‘if p, then q’ is also true. By the truth‐table for ‘if, then’, if p is true and if ‘if p, then q’ is true, then q is true. So q must be true, too. As is clear, however, this justification for MPP must itself take at least one step in accord with MPP. (p.232) The Ban on Rule‐Circular Justification Doubt about the cogency of rule‐circular justification was expressed most recently by Gilbert Harman. Harman’s specific target was a recent attempt of mine to defend the notion of analyticity from Quine’s famous critique and to try to show how it might be used to provide a theory of a priori knowledge, including our knowledge of fundamental logical principles.5 In finding the inferential character of the proposed account problematic, Harman follows a substantial tradition in the philosophy of logic and the theory of knowledge of assuming that one’s warrant for a principle of logic cannot consist in reasoning that employs that very principle. Since, furthermore, Harman cannot see how there might be an account of the apriority of our core logical principles that did not have to presuppose that subjects reasoned in accord with those very principles, he concludes that those principles cannot be a priori. If, however, the core logical principles are not a priori, then neither is anything that is based on them, and so it looks as though we have to concede that nothing of much interest can be a priori after all. By his own account, Harman’s worry here derives from Quine’s ‘Truth by Convention’. Quine too argues that if there is to be any sort of explanation of logic’s apriority, it would have to be based on something like what I have called implicit definition. He too argues that any such account would have to presuppose that subjects reasoned according to the principles of logic. And he too concludes that this renders such accounts useless and, hence, that logic’s apriority cannot be vindicated. Quine goes on, in later work, to present an alternative epistemology for logic, one that portrays our warrant for it as consisting in a combination of empirical and pragmatic elements. On the following point, I am in complete agreement with Quine and Harman. If we accept the ban on the use of a logical principle in reconstructing our a priori warrant for that very principle, we would have to conclude that there can be no such reconstruction. If so, we would have to give up on the idea that our warrant for logic can be a priori. An Empirical Justification for Logic? What I do not see, however, is how this point can be used to motivate an alternative epistemology for logic, one that is empirical in nature. For if we are barred (p.233) from supposing that reasoning using a given logical principle can reconstruct an a priori warrant for that very principle, are we not equally barred from supposing that it could reconstruct an empirical warrant for that principle? Yet would not any empirical account of our warrant for believing the core principles of logic inevitably involve attributing to us reasoning using those very principles? To see why, let us take a look at the only reasonably worked out empirical account of logic, namely, Quine’s.6 According to this account, warrant accrues to a logical principle in the same way that it accrues to any other empirical belief, by that principle’s playing an appropriate role in an overall explanatory and predictive theory that maximizes simplicity and minimizes the occurrence of recalcitrant experience. We start with a particular theory T with its underlying logic L and from T we derive, using L, a claim p. Next, suppose we undergo a string of experiences that are recalcitrant in that they incline us to assent to not‐p. We need to consider how T might best be modified in order to accommodate this recalcitrance, where it is understood that one of our options is to so modify the underlying logic of T that the offending claim p is no longer derivable from it. We need to consider, that is, various ordered pairs of theory and logic—, , . . . , , . . .— picking that pair that entails the best set of observation sentences. Whatever logic ends up being so selected is the logic that is maximally justified by experience. In rough schematic outline, that is the Quinean picture. But there is a problem with this picture if the ban on circular justifications is in place. For it is very difficult to see how the use of core logical principles, such as those of non‐contradiction, modus ponens, universal instantiation, and others, is to be avoided in the meta‐theory in which this comparative assessment of the various theory‐logic pairs is to take place. For instances of the following forms of reasoning are presumably unavoidable in that meta‐theory: • The best set of observation sentences is the set with property F. Set O has property F. Therefore, O is best. • , L> is that theory–logic pair that predicts O. O is the best set of observation sentences. Therefore, , L> predicts the best set of observations sentences.And so forth. A little thought should reveal that a large number of the core principles of logic will have to be used to select the logic that, according to the picture under consideration, is maximally justified by experience. I do not say that it will involve the whole of classical logic, so it is not out of the question that such a picture may be used coherently to adjudicate certain disagreements about logic— for instance, whether quantum mechanics dictates rejection of the distributive (p.234) principles. But it is clear that it will involve a sizeable number of the core principles of ordinary logic. It follows, therefore, that so long as the ban on circular justification is kept in place, there is no question of using the procedure described by the Quinean picture to generate a warrant for the core principles of logic. Scepticism About Logic The point here is, I think, perfectly general. It is very hard to see how there could be any sort of compelling empirical picture of our warrant for logic that did not rely on a very large number of the core principles of ordinary logic. Contrary to what some philosophers seem to think, then, the ban on circular justifications of logic cannot be used selectively, to knock out only a priori accounts of our warrant for logic. If it is allowed to stand, I do not see how it can be made to stop short of the very severe conclusion that we can have no warrant of any kind for our fundamental logical beliefs— whether of an a priori or an a posteriori nature. If this is correct, we find ourselves on the left‐hand path of our flowchart, answering ‘No’ to the question whether our fundamental logical beliefs can be justified. But is this itself a conclusion we can live with? Can we coherently accept the claim that our fundamental principles of reasoning are completely unjustifiable? Well, if it is impossible for the claim that MPP is truth‐preserving to be justified that would seem to imply that our use of MPP is also unjustifiable. For how could we be entitled to reason according to a given inference rule if it is impossible for the claim that that rule is truth‐preserving to be justified in the slightest? There is a principle here, linking the epistemic status of reasoning in accord with a rule and that of a belief concerning the rule, that it will be useful to spell out: (LP): We can be entitled to reason in accordance with a logical rule only if the belief that that rule is truth‐preserving can be justified. Stated contrapositively: (LP’) If it is impossible for us to be justified in believing that a certain logical rule is truthpreserving, we cannot be entitled to reason in accordance with that rule.7 The appeal of this linking principle is quite intuitive. If it is—logically or metaphysically—impossible for us ever to be justified in claiming that a particular (p.235) logical truth is truth‐preserving, it seems to follow that we could never be entitled to reason in accordance with it. Any plausible epistemology for logic should respect this link between our entitlement to reason according to a certain rule and the corresponding meta‐claim. (More on this below.) Now, however, we are in a position to see that saying that we cannot justify our fundamental rules of inference is extremely problematic. For if a claim to the effect that MPP is truth‐preserving is not justifiable, then, the linking principle tells us, neither is our use of MPP. If, however, our use of MPP is unjustifiable, then so is anything that is based on it, and that would appear to include any belief whose justification is deductive. In particular, since this sceptical conclusion is itself one of those claims that is based on deductive reasoning, the very thesis that our core logical principles are unjustifiable will itself come out unjustifiable, on such a view. Prima facie, this does not seem to be a stable platform on which to stand. What are we to do? We cannot simply accept the proffered result unadulterated. If we are to accept it at all, we have to try to embed it in some larger conception that will render it stable. The dialectical situation here is reminiscent of Kripke’s famous reconstruction of Wittgenstein’s discussion of rules, albeit concerning a different subject matter.8 A powerful argument leads to a sceptical thesis that looks to be self‐undermining. Is there a ‘straight solution’ that, by rejecting one of the assumptions that led to the sceptical conclusion, shows how it is to be avoided? Or is there, at best, a ‘sceptical solution’, one that can accommodate the sceptical conclusion while removing the taint of paradox that renders it unstable? Solutions: Sceptical and Straight As far as I can see, there are four interesting possibilities in this particular case, one sceptical and three straight. The sceptical solution would attempt to argue that it was a mistake to think of logic in a factual way to begin with, that the ‘claim’ that MPP is truth‐preserving is not a genuine claim after all and so does not need to be justified. As for the straight solutions, there look to be three possibilities, two non‐inferential and one inferential. On the non‐inferential path, I see no prospect of developing a theory in terms of the notion of rational intuition or its ilk. But an idea that has been gaining influence recently has it that there are certain beliefs that are ‘default‐justified’, reasonable to believe in the absence of any positive reasons that recommends (p.236) them. If there were such beliefs, and if belief in MPP were among them, this would constitute a straight solution to our sceptical problem about logic. A further idea derives from recent work on non‐factualist conceptions of normative concepts: it suggests a distinct way in which logical belief might be non‐inferentially justifiable, namely, if the notion of justification were itself to be treated non‐factually. Finally, on the inferential path, there is the question whether it is irremediably true that rule‐circular arguments provide no justification. Given these further options, our flowchart now looks as in Fig. 10.2 . I will begin with a discussion of the sceptical solution. Non‐Factualism About Logic How might it turn out to have been a mistake to think that our belief in MPP requires justification? The only possible answer would appear to be: if it were a mistake to think of it as a belief in the first place. What is in view, in other words, is an expressivist conception of the sentences that are used to state what we naïvely suppose are logical beliefs. How should we formulate such a view? Here is a simple version: to say that P follows logically from Q is not to state some sort of fact about the relation between P and Q; rather, it is to express one’s acceptance of a system of norms Fig. 10. 2 (p.237) that permits inferring P from Q. Since there are no facts about logical implication on this view, there is no need to justify one’s selection of the norms. Furthermore, since in saying that a particular inference is valid, one is not actually saying anything, but only expressing one’s acceptance of a system of norms that permits it, there is no question of justifying that either. It is, therefore, perfectly unproblematic that the claims of logic are not justifiable; properly understood, there was nothing to justify in the first place. Now, naturally, this view would have to be elaborated in much greater detail before it could count as an expressivist or non‐factualist theory of logical truth. But it seems to me that we can identify a fatal difficulty right at the start, one that obviates the need to consider any further refinements. To see what it is, consider the sort of conventionalist theory of logical truth, famously discussed by Quine in ‘Truth by Convention’, according to which facts about logical implication are the products of convention. If we had not conventionally stipulated that this follows from that, according to this view, there would not have been any facts about what follows from what. Against this particular metaphysical view of the source of logical truth, it seems to me, one of Quine’s objections is decisive. For as he points out, the conventional stipulations that would have to be responsible for generating the infinity of logical truths that there are would have to be general in nature. Thus, they might take the form of saying: For all x, y, and z, if x and y stand in the modus ponens relation to z, then x and y imply z. Now let us suppose that P, and ‘If P, then Q’, stand in the modus ponens relation to Q. And consider the question: do these two propositions imply Q? Well, that depends on whether it follows from the stipulated convention and the supposition, that P and ‘If P, then Q’ imply Q. That is, it is the question whether it follows from: (a) P and ‘If P, then Q’, stand in the modus ponens relation to Q and that (b) If P and ‘If P, then Q’ stand in the modus ponens relation to Q, then P and ‘If P, then Q’ imply Q (c) P and ‘If P, then Q’, imply Q? But that in turn is just the question whether it is true that if x and y stand in the modus ponens relation to z, that x and y imply z. The problem is clear. We can not have it both that what logically follows from what is determined by what follows from certain conventions and that what follows from the conventions is determined by what logically follows from what. Facts about logical implication not accounted for by conventionalism are presupposed by the model itself.(p.238) Now, although a non‐factualist expressivism about logical truth is a distinct view from a conventionalism about it, it suffers from an exactly analogous problem. The expressivist has it that to say that P follows from Q is to express one’s acceptance of a system of norms that permits inferring P from Q. Once again, though, these norms of permission will have to assume a general form: • For all x, y and z, if x and y stand in the modus ponens relation to z, then it is permissible to infer z from x and y. Does this norm allow inferring Q from P and ‘If P, then Q’? Well, the norm has it, by universal instantiation, that: • If P and ‘If P, then Q’, stand in the modus ponens relation to Q, then it is permissible to infer Q from them. If, furthermore, P and ‘If P, then Q’ do stand in the modus ponens relation to Q, then by modus ponens, it would follow that it is permitted to infer Q from the premises in question. Once again, the problem is clear. We cannot coherently have it both that whether A follows from B depends on what the system of norms permits and that what the system of norms permits depends on whether A follows from B. Just as with conventionalism, facts about logical implication not accountable for by the non‐factualism are presupposed by the model itself. A non‐factualist construal of logical implication, then, does not seem a promising strategy for blocking the paradoxical consequences of the claim that fundamental logical beliefs are incapable of being justified. If there is to be a solution to our sceptical problem, it looks as though it has to be a straight solution. I will begin with a discussion of default‐reasonable beliefs.