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Basic Logical Knowledge

Published online by Cambridge University Press:  12 April 2010

Bob Hale
Affiliation:
University of Glasgow

Extract

At least some of us, at least some of the time—when not in the grip of radical sceptical doubt—are inclined to believe that we know, for example, that if we infer a conclusion from two true premises, one a conditional whose consequent is that conclusion and the other the antecedent of that conditional, then our conclusion must be true, or that we know similar things about other simple patterns of inference. If we do indeed have knowledge of this sort, it is what I mean by logical knowledge. Logical knowledge is, roughly speaking, knowledge about logic—such as knowledge that a certain principle of inference necessarily preserves truth, or that every proposition of a certain form must be true—and so is not the same thing as knowledge that is gained by using logic, i.e. inferential knowledge. That is not to say, of course, that logical knowledge can't be inferential. On the contrary, it is barely open to question that—if there is any logical knowledge at all—there is a lot of inferential logical knowledge. For example, if we know that the introduction and elimination principles for the conditional are truth-preserving, we can surely get to know, by inference, that the principle of hypothetical syllogism (i.e. transitivity of the conditional) is so too, not to mention other, less obvious and more recondite, examples of putative logical knowledge.

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Copyright © The Royal Institute of Philosophy and the contributors 2002

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References

1 I am concerned exclusively with principles of deductive inference. Other properties which might be taken to be essential to good principles of deduction include, most obviously, conservativeness and harmony. On a broadly constructivist approach to logic, soundness would be understood in terms of proof-theoretic virtues such as these, rather than in terms of necessary truth preservation. In the interests of generality, it is clearly desirable to avoid essential reliance on any assumption about what is required for soundness which divides advocates of classical logic from their constructivist—or, for that matter, relevantist—opponents. I shall not always strive, in presenting the argument that follows, to maintain complete neutrality as between these opposed conceptions of soundness—in particular, I shall often take soundness to require necessary truth preservation; but I believe that wherever I do so, it would be possible to reconstruct my argument to suit any of the principal non-classical conceptions.

2 For attempts to show, contra Quine, that we must believe in logical necessity, see Wright, Crispin ‘Inventing Logical Necessity’ in Butterfield, J., (ed.) Language, Mind and Logic (Cambridge: Cambridge University Press, 1986), pp. 187209Google Scholar; McFetridge, IanLogical Necessity: Some Issues’ in his posthumously published papers, Logical Necessity & other essays, edited by Haldane, John & Scruton, Roger, Aristotelian Society Series Volume II, pp. 135–54Google Scholar; and Hale, BobOn Some Arguments for the Necessity of NecessityMind 108 (1999), pp. 2352.CrossRefGoogle Scholar

3 Cf McFetridge, op. cit. p. 136.

4 It is also open to objection, as it stands, unless a certain qualification is understood. This is because the vast majority of propositions which might plausibly be claimed to be physically necessary are not strictly speaking logical consequences of the laws of physics alone—some laws of mathematics will be required. So the glosses in the text will be acceptable as they stand only if the laws of mathematics are themselves logical consequences of the laws of logic. Whilst I believe that is a defensible view, at least as far as the laws of mathematics that are needed for physics are concerned, it is certainly not widely accepted, and it is therefore desirable to avoid presupposing it here. We can do so by simply stipulating that laws of physics are to be taken as including the the requisite laws of mathematics.

5 There is obviously a question about how epistemically modal notions may be demarcated. One simple thought would be that ψ-possibility is a species of epistemic possibility if the truth-value of statements of the form ‘It is ψ-ly possible that p’ can vary, relative to different states of information.

6 Assuming the correctness of the widely accepted view that they are indeed necessities.

7 It may, of course, be an empirical discovery that a particular word is used to express that concept.

8 Kant Critique of Pure Reason, B3.

9 This is not claim that nothing like self evidence or rational insight can have any part to play in explaining ligical knowledge. My claim is only that, pending clarification of its precise rôle and of how it is supposed to work, such an appeal is no advance. In fact, and as I shall try later to explain, I think that there must be an essentially non-inferential component, which might reasonably be viewed as a species of rational insight, in such an explain.

10 The idea makes its first appearance, as far as I know, in Dummett's British Academy Lecture ‘The justification of deduction, reprinted in his Truth and other enigmas (London: Duckworth 1991)Google Scholar, see especially Ch. 9.

11 Dummett calls this ‘pragmatic’ circularity, but I shall follow Paul Boghossian in referring to it as rule-circularity, as this label is the more informative.

12 These are Dummett's terms.

13 Cf Prior, A. N.The Runabout Inference Ticket’, Analysis 21 (1960), pp. 38–9.CrossRefGoogle Scholar

14 I could hardly do so, given the very sophisticated defence of that approach mounted by Paul Boghossian in some recent papers, including ‘Analyticity’ in Hale, Bob & Wright, Crispin, (eds) The Blackwell Companion to the Philosophy of Language (Oxford: Blackwell, 1997), pp. 331–68Google Scholar; ‘Analyticity Reconsidered’, Nous (1996) pp. 360–91; ‘Knowledge of Logic’ in Boghossian, Paul & Peacocke, Christopher, (eds) New Essays on the A Priori (Oxford: Oxford University Press, 2000), pp. 229–54CrossRefGoogle Scholar. There is also a further paper, ‘How are Objective Epistemic Reasons Possible?’, originally presented to a recent APA meeting at Albuquerque, which is not as yet, as far as I know, published. A proper assessment of Boghossian's defence of rule-circularity lies well beyond the scope of this paper—here I can only record my opinion that, for all its ingenuity, it does not adequately answer the objections adumbrated here. In addition, Boghossian's defence ultimately requires him—as he clearly perceives (‘Knowledge of Logic’, p.253)—to reject what he terms ‘the principle of the universal accessibility of reasons’, which claims, roughly, that if something is a genuine reason for believing that p, ‘its rationalizing force ought to be accessible from any epistemic standpoint’—in particular, reasons for believing that p ought to be, in principle, appreciable as such by someone who doubts or questions whether p. I am not completely certain that this principle is true, and can-not see how to argue for it from premises more likely to command assent, but like Nagel, Thomas (The Last Word(Oxford: Oxford University Press, 1997), 8—quoted by Boghossian)Google Scholar, I so not see how one can give it up without giving way to relativism or subjectivism about reasons. Boghossian acknowledhes the pull of the principle, but concludes that it must nevertheless be false. Certainly if his account of logical knowledge is correct, it is false. But one man's modus ponens…! Boghossian's position is subjected to a searching critical examination by Crispin Wright in his response to the last of the papers cited above (‘On Basic Logical Knowledge: Reflections on Paul Boghossian's “How are Objective Epistemic Reasons Possible?”’, forthcoming, along with Boghossian's paper, in Philosophical Studies). I am in agreement with the main critical points he makes there.

15 In the papers to which I've alluded, Boghossian claims to show that a non-inferential answer cannot work. However, his argument is an argument by elimination, and since he does not—so far as I have been able to see—consider the kind of non-inferential answer I'm going to propose, I do not need to discuss it here.

16 See his Meaning (Oxford: Oxford University Press, 1998), p. 8Google Scholar and ch. 6 passim, but especially sect. 2, and his paper Implicit Definition, Analytic Truth, and A Priori Knowledge’, Nous 31 (1997), pp. 423–40CrossRefGoogle Scholar. Horwich is mainly concerned to argue against what he calls the ‘standard model of implicit definition’, according to which implicit definition proceeds through stipulation of the truth of some sentence(s) containing the definiendum, and in favour of a ‘use-theoretic’ conception, according to which the meaning of the definiendum is fixed, rather, by our accepting, or taking as true, some such sentence(s). And his thought—to put it somewhat crudely—is that once we shift away from the standard model to his preferred account, we can see that implicit definition cannot ground significant a priori knowledge, simply because acceptance doesn't entail truth. I am not myself persuaded by Horwich's arguments against the standard model—for some of my reasons, see Hale, Bob & Wright, Crispin ‘Implicit Definition and the A Priori’ in Boghossian, Paul & Peacocke, Christopher, (eds) New Essays on the A Priori (Oxford: Oxford University Press, 2000), pp. 286319CrossRefGoogle Scholar, especially 290–95 and 309, fn. 40—but his point about acceptance is ungainsayable, and is all that matters here.

17 Even if there shoul prove to be serious, or even insuperable, obstacles in the way of building upon a satisfying explanation why the correctness of basic inference rules is beyond coherent doubt to get an explanation of how we can know them to be correct, it seems to me that successful completion of project B would be a significant advance.

18 But how can this possibly be so, it may be objected, given that clever people who know what they are talking about have seriously proposed counter-examples? (See, for example, McGee, VannA Counterexample to Modus VonensJournal of Philosophy, Vol. 82, Issue 9 (1985), pp. 462–71)CrossRefGoogle Scholar. Several points need to be made here. First, it is obviously immaterial whether the alleged counter-examples are genuine—it is enough for the objection that someone could intelligibly think them to be so. Second, and equally obviously, the fact—if it is one—that one can intelligibly take something to be a counterexample to modus ponens does not go against the conditional claim that, if one's understanding of the conditional is (partially) constituted by one's acceptance of modus ponens, then one cannot intelligibly take something to be a counter-example to that rule; what it would show, rather, is that one's understanding cannot be so constituted. If someone can intelligibly take something to be a counter-example, it cannot be that understanding of the conditional is (partially) constituted by acceptance of (unrestricted) modus ponens. So the supposition that a thinker may, after all, intelligibly view something as a counter-example evidently begs the question: In what does understanding of the conditional consist? How is its meaning constituted, if not through the entrenchment of the usual inference rules? Well, either through the entrenchment of some other rules (perhaps restricted versions of the usual rules), or in some other way (i.e. not through the entrenchment of any inference rules at all but, say, through the assignment of certain truth-conditions to condition-al sentences). Either way, it seems to me, something sufficiently close to my claim can be sustained. Alleged counter-examples of the kind proposed by McGee essentially involve as major premises conditionals whose consequents are themselves conditional (e.g. ‘If a Republican wins, then if Reagan loses, Anderson will win’). As McGee himself remarks, such examples have no tendency to suggest that modus ponens may be unsound in simple cases where the consequent of the conditional major premise is itself non-conditional. So one might continue to take the meaning of the conditional as (partially) constituted by acceptance of modus ponens—but in a suitably restricted version. The argument I develop in the remainder of this paper could be straightforwardly recast to suit such a restricted version of the rule. One may instead take the meaning of the conditional to be fixed by its association with certain truth-conditions, or perhaps conditions of correct or justified assertion. But however these are specified, they will surely validate at least a restricted form of modus ponens, even if they do not validate the rule in full generality. And once again, these will besuch as to preclude the possibility of intelligibly regarding something as a counter-example to whatever (possibly restricted) version of modus ponens they underwrite.

19 To borrow an example from the paper by Wright cited in note 14, suppose Frege, instead of formulating his notorious Basic Law V as an axiom, had proposed a pair of introduction and elimination rules governing the course of values operator permitting us, respectively, to infer α'F(α) =α'G(α) from ∀α(F(α)↔G(α)) and ∀α(F(α)↔G(α)) from α'F(α)= α'G(α). The inconsistency of such a stipulation would have been no less unobvious than was the inconsistency of Frege's actual proposal—it took the genius of Russell realize that not all was well.

20 It would not, of course, be sufficient, for the reason given above.

21 First made, I believe, by Belnap, Nuel in ‘Tonk, Plonk and Plink’, Analysis 22 (1962), pp. 130–-4CrossRefGoogle Scholar. Another, closely related, idea is that acceptable introduction and elimination rules should be harmonious in the sense—to put it roughly and intuitively—that the elimination rule is just as strong as it can be, given the introduction rule, or, putting it the other way around, the introduction rule is just as weak as it can be, given the elimination rule. For a more careful explanation and discussion, see Dummett The Logical Basis of Metxaphysics chs. 9,11.Google Scholar

22 I am assuming here that a doubt which admits of no conceivable vindication would be a merely idle doubt, and that we may take ourselves to be concerned with doubts that are real doubts, not merely idle ones.

23 A determined objector may retort: ‘But suppose I do, by using rule R, reach the conclusion that rule R is unsound. I accept this conclusion, and surely I am right to do so. For either R is unsound or it isn't. If it is, thenclearly I'm right. And we suppose instead that it's sound, then my reasoning to the conclusion that it's unsound stands, and I'm right again. So either way, I'm right.’ ut this just repeats the confusion in a new form—this time through not distinguishing what's true from what the envisaged proponent of reasoning using R is entitled to think. It entirely right that if the objector reasons by an unsound rule R to the conclusion that rule R is unsound, he will wind up with a true belief. But he won't be entitled to it. He would be entitled to it, if, having reasoned using R to some intermediate conclusion C (a contradiction, perhaps) on the basis of which he further infers that R is unsound, he stood back (as it were) and argued ‘By using R, I was able to reach conclusion C. But any rule that enables one to reach that conclusion must be unsound. So R is unsound.’ Once again, it is essential to keep clear the distinction between reasoning with R and reasoning about R (without using it). It is the second piece of reasoning, without R—the meta-reasoning—which entitles one to the conclusion that R isn't sound, not the (unsound) reasoning using R.

24 The obvious truth of the minor premisses—(3) and (7) should not prevent us from enquiring into their justification. Since what is in question is, in effect, our recognition that a particular inference exemplifies the general pattern sanctioned by a rule of inference, it would be not merely implausible but potentially disastrous to suggest that our knowledge of these and similar such statements is got by inference—quite apart from the difficulty of coming up with any even remotely plausible premises from which such statements might be drawn as conclusions, it seems clear that any such inferential answer would set going a vicious infinite regress, of a piece with that into which Carroll's wily Tortoise enveigles the unwary Achilles. Our recognition of their correctness must, it seems, be a non-inferential matter. That is, the right answer is just the one we should naturally give, viz. that we can just see that the particular inference is of the displayed general form. Seeing here is not—or at least not simply—a matter of visual perception. For even if visual perception is involved-as it will be, if we are confronted with an inscription of the particular inference and perhaps also of the general pattern—the recognition that the one exemplifies the other is not a purely visual matter. Nor is seeing in this case, as it no doubt is in others, a matter of there being a more or less simple piece of reasoning which we could, if called upon, articulate. I can see no alternative to acknowledging that what is involved here is a species of non-inferential intellectual recognition—which we may as well call rational insight, and which has an indispensable rôle to play whenever we operate with rules of inference. No doubt there is much more to be said about this. For now, two caveats must suffice. First, I should perhaps emphasize the very limited role I am assigning to rational insight—in particular, I am not claiming that extends to recognition of the validity of inferences, only that it is mediates recognition of particular inferences as exemplifying general rules. Second, I am not suggesting that whenever we correctly apply general rules of inference, our application must be seen as a matter of inferring that a particular transition accords with the rule. That would be disastrously regressive. Here it is, once again, crucial to remember that explicit derivation of (4) and (8), in the argument in the text, is needed because we are reasoning about the tonk rules, not using them.

25 In particular, it is desirable to formulate my claim about the essential involvement of conditionality and generality, and the consequent need for principles governing them, in a way that renders it independent of any assumption about which particular linguistic devices subserve the expression of those notions. Obviously it is inessential that we express conditional propositions using the word ‘if’, for example, rather than as disjunctions or negated conjunctions. The only essential point is that we have some means of expressing a binary sentential compound with the distinctive inferential properties of the conditional; likewise, mutatis mutandis, for generality. A fuller discussion would also investigate what other principles of inference, in addition to principles concerning conditionality and generality, might be needed in the minimal kit. It is plausible that the kit should provide for reasoning by reductio ad absurdum, and hence that some (weak, non-classical) rules for handling negation will be needed.

26 It may be suggested that, at least when fully articulated, the proposed explanation will be found to involve appications of at least some of the minimal rules, and will therefore be rule-circular. That is probably so—but it will be an objection only if, contrary to what I have claimed, rule-circularity must be vicious even in the contect of an explanation why something is the case, just as it arguably is in an explanation how it can be known to be the case or in a suasive argument.

27 See section 3 above, and the previous footnote.

28 This is, fairly obviously, too weak as it stands, since it would have it that all that is required of me, if I am to qualify as knowing some true and rationally indubitable proposition p, is that I believe p. But that can hardly suffice for knowledge—since it imposes no constraint whatever on the reasons why I so believe, or the route by which I come to believe, and so leaves it open that I may believe p for bad or even quite crazy reasons. I cannot, for example, get to know that p by fallaciously inferring it from q and if p then q (which I also believe), even if it is true and indubitable that p. It would not help to require that I believe p precisely because I believe it to be indubitable, since the same difficulty would arise over my belief that p is indubitable—if I hold that belief for irrelevant or crazy reasons, I should (still) not count as knowing. It seems clear that what is required, if my true belief in what is in fact rationally indubitable is to amount to knowledge, is that my belief should arise in the right way, or be appropriately caused: I should believe that p, not as a result of my holding any other beliefs, but simply because I cannot see how it might intelligibly be doubted that p—where, to stress the crucial point, this does not mean that I believe that p because I believe that there is no room for doubt that p, much less that I infer that p from my inability to see how one might doubt it.

29 What I mainly have in mind here is the question whether—to express the matter in terms of the Euthyphro contrast prominent in much of Crispin Wright's work on truth and objectivity—the facts which are the objects of logical knowledge are appropriately conceived as obtaining independently of, and at most ‘tracked’ by, our best opinions or judgements under optimal conditions, or whether they ought instead to be viewed as in some manner constituted or determined by such opinions or judgements. For discussion of the Ethyphro, see Wright's Truth & Objectivity (Cambridge, MA: University Press, 1992), especially pp. 108–39.Google Scholar

30 This paper grew out of several discussions with Crispin Wright, to whom Iam much indebted. Earlier versions had airings at seminars in the universities of Aarhus, Durham, Glasgow, Western Ontario and Waterloo, as well as in the Royal Institute lecture series itself. I am grateful to my audiences on all these occasions, and especially to Bill Demopoulos, Jim Edwards, Laurence Goldstein, Lars Gunderson, Bill Harper, Robin Hendry, Gary Kemp, Max Kölbel, Jimmy Lenman, Jonathan Lowe, David Papineau, Philip Percival, John Benson and Adam Rieger.

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