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CHINESE UNIVERSITY OF HONG KONG MARCH 5, 2012 Revising Our Logic

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Outline 2 1. Paradoxes of truth, definability, … (e.g. Liar) Why they aren’t idle puzzles How they make a case for revising logic (= fundamental rules of reasoning) 2. But can we rationally revise logic (fundamental rules of reasoning)? 3. The nature of logic. How the right account of this helps with (but doesn’t by itself fully answer) the puzzles about rational revision of logic.

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The Liar paradox 3 “What I’m saying is false”. Seems that (1) what he says (S) is true if and only if it’s false. (2) it’s false if and only if it’s not false. (1) is surprising: it seems to imply that S is either both true and false, or else neither true nor false. (2) is more than surprising, it seems contradictory: If S is false, it’s not false, so both false and not false. If S is not false, it’s false, so again both false and not false. So either way, S is both false and not false!

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Not take it seriously? 4 This is “the Liar paradox”. It’s been around for 2000 years, with no agreement as to how best to deal with it. One might not be inclined to take it seriously, on one of two grounds: (a) that it only arises for artificial examples that could never arise in practice (b) that it can never affect anything we’re really interested in. Both grounds are mistaken.

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(a) Non-artificial examples 5 Two candidates for the same office, on election night. Polls have closed. Initial returns strongly support Candidate A. Candidate B is on TV, denouncing Candidate A. Candidate A, watching the TV, says “What the idiot who lost the election is now saying is false.” The initial returns were wrong: Candidate B won. And of course both candidates are idiots. So Candidate A’s remark is equivalent (given the facts) to the claim that it itself is false. So again, it’s false if and only if it’s not false.

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Why it’s a paradox 6 This is a paradox: elementary assumptions about truth and falsity, together with very simple principles of standard logic, have led to inconsistency. The point isn’t that the politician was being inconsistent. That wouldn’t be paradoxical at all: then what he’d be saying is false. Rather, the point is that whatever we say to describe the politician’s utterance inevitably leads us into inconsistency. If we call it false, we seem committed to also calling it not false, and conversely. The politician story is an illustration of how paradox can arise out of normal conversation in which people don’t intend to be speaking paradoxically.

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(b) Do such paradoxes matter? 7 Dismiss the paradoxes on the ground that they never affect anything we’re really interested in? That would be mistaken. There are infinitely many other paradoxes that turn on the same principles that the Liar paradox turns on. A. Even very good mathematicians and logicians are sometimes led to serious errors by using reasoning which on closer inspection turns on some paradox or other. B. We need a notion of truth, and will be led to error if we don’t get its principles consistent, and indeed correct.

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Dangers of inadvertent paradoxical reasoning 8 A. People sometimes are led to serious errors by using reasoning which on closer inspection turns on paradoxes. Examples: 1) König’s purported disproof of the Axiom of Choice. 2) Soundness arguments in logic: arguments that certain systems of derivation can never lead from truth to non-truth. More on (1):

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König 9 Axiom of Choice: a once-controversial mathematical axiom, now a cornerstone of mathematics. König (prominent early 20 th century set theorist) thought he’d disproved it, by arguing that it implies that there are real numbers that both are and are not definable in a given language. The argument is extremely persuasive: indeed, there is no agreement as to where it goes wrong! People came to agree that it’s wrong only when Berry came up with a similar argument for contradiction not relying on controversial assumptions like AC.

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Other examples where we’re fooled 10 König’s paradox, like the Liar, is a paradox of truth. (Reason: definability is explained in terms of an expression being true of an object.) It is just one of many examples where it’s easy to be taken in by proofs that, on closer analysis, turn on principles about truth that jointly lead to paradox. Soundness arguments in logic---arguments that certain systems of derivation can never lead from truth to non-truth---fool many philosophers today.

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REPRISE: Do such paradoxes matter? 11 Dismiss the paradoxes on the ground that they never affect anything we’re really interested in? Mistaken. A. Even very good mathematicians and logicians are sometimes led to serious errors by using reasoning which on closer inspection turns on paradoxes B. We need a notion of truth, and will be led to error if we don’t get it’s principles consistent and indeed correct.

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Need of truth (1) 12 We need a notion of truth, in daily life and in mathematics. Within the standard framework of mathematics (Zermelo-Fraenkel set theory) we can define restricted notions of truth, but not the full thing. This limits natural reasoning within ZF set theory in many ways: many informal mathematical arguments that everyone wants to make simply can’t be formalized within the official ZF.

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Need of truth (2) 13 We get a much more flexible system if we add a notion of truth to the set theoretic framework. But different assumptions about truth yield different expansions of the framework. Some mathematical sentences not settled in ZF set theory are settled in such expansions. But we want the right expansion, to settle the questions in the right way. For this, we need to know the right morals of the Liar and similar paradoxes. ///

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REPRISE: Outline 14 1. Paradoxes of truth, definability, … Why they aren’t idle puzzles How they make a case for revising logic (= fundamental rules of reasoning) 2. But can we rationally revise logic (fundamental rules of reasoning)? 3. Understanding the nature of logic, and how it helps with (though doesn’t by itself fully answer) the puzzles about rational revision of logic.

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Back to Liar paradox (1) 15 Simpler version: a sentence L that says of itself that it is not true (rather than false). (Such sentences can arise naturally in conversation.) It’s puzzling what to say about L: If we declare L not true, then that declaration is equivalent to L itself: so we’re asserting L while in the same breath declaring it not true. Very odd! And if we declare L true, then our declaration is equivalent to a denial of L itself: so we’re denying L while in the same breath declaring it true. Also very odd!

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Back to Liar paradox (2) 16 What if we neither declare it not true nor declare it true? If we accept standard logic, we must at least declare that it’s one or the other. But if it’s absurd to declare it true, and absurd to declare it not true, isn’t it equally absurd to say that it’s either true or not true, even if I don’t say which? (If I say “Either there are square circles or triangular squares”, I’ve said something absurd, even if I don’t say which one.)

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Yogi Berra There are people who think it’s OK to assert that either the Liar sentence is true or it’s not true, even while holding that each is absurd. Yogi Berra once offered the following advice: “If you come to a fork in the road, take it”. These people reject that advice. 17

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Incoherence Principles 18 Upshot: If we don’t restrict classical logic, we must violate at least one of the following Incoherence Principles: First Incoherence Principle: it is incoherent to accept A while accepting A isn’t true. Second Incoherence Principle: it is incoherent to accept A is true while accepting not-A. Third Incoherence Principle: If accepting either of two claims (B, C) would be incoherent, then it is incoherent to accept that either B or C. Each of these principles seems compelling. For this and other reasons, my preferred approach is to weaken classical logic slightly, in such a way that the problem doesn’t arise.

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My approach 19 The “law of excluded middle” is the principle p or not-p (for any p you like). The idea is to reject that it holds generally. In particular, we reject: (**)Either L is true or it isn’t true. Without (**), we can accept that it’s absurd to call the Liar sentence true and that it’s absurd to call it not true, as the first two Incoherence Principles require, without violating the Third Incoherence Principle. We can restrict excluded middle here in a way that allows for unrestricted classical logic within mathematics etc.

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Contrast to intuitionism 20 LEM (Law of excluded middle): p or not-p (for any p you like). Famously criticized by the Dutch intuitionists (Brouwer, Heyting), who proposed doing math without it. Hilbert: Depriving the mathematician of LEM would be like depriving the boxer the use of his fists. I have a way of restricting logic that’s immune to the critique: it leaves logic unaffected in math, physics, etc. Also: intuitionist logic doesn’t ultimately help with the paradoxes.

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Results of my approach 21 An adequate treatment of the Liar must deal also with far more complicated paradoxes (infinitely many). One of my main areas of research over the last few years was to work out such an account. I showed how one can get a logic that is in some sense close to classical logic, reduces to classical logic in contexts where paradoxes don’t threaten, e.g. math and physics, where even in paradoxical contexts, the claim that a given sentence is true is equivalent to that sentence. allows for all three incoherence principles.

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Costs and benefits 22 Is restricting LEM to handle the paradoxes worth the cost? Requires detailed cost/benefit analysis of this and competing approaches. (Given in my recent book Saving Truth From Paradox.) Some people think a cost/benefit analysis inappropriate: they think it intrinsically wrong to suggest an alteration of the laws of classical logic. Similar arguments have been given in the case of other cases of radical change in view: Gauss/Riemann on weakening Euclidean views of geometric structure Einstein on weakening Newtonian views of simultaneity.

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Physics as precedent (1) 23 Before Einstein, it was hard to see how Newtonian views of simultaneity could fail. (Similarly for Gauss and Riemann re Euclidean geometry.) It seemed part of our conceptual scheme that there is an objective simultaneity relation satisfying Newtonian assumptions. No one even made these assumptions explicit until Einstein questioned them. (Similarly Gauss/Riemann, with variable-curvature geometry.)

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Physics as precedent (2) 24 Gauss/Riemann and Einstein developed generalized theories that accommodated the successes of the earlier theories, while allowing exceptions to handle contrary observations. Even after their work, some conservatives argued that there’s something intrinsically wrong with trying to change such basic conceptual apparatus. Generally agreed to be disreputable there. But if there, why not equally so in the case of logic?

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Summary so far 25 Logical change shouldn’t be made lightly. Perhaps it shouldn’t be made at all. But if a way is proposed to avoid serious anomalies by an alteration of logic that will accommodate ordinary reasoning in most circumstances, and allow for something close to ordinary reasoning everywhere, then it should not be dismissed out of hand. ///

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REPRISE: Outline 26 1. Paradoxes of truth, definability, … Why they aren’t idle puzzles How they make a case for revising logic (= fundamental rules of reasoning) 2. But can we rationally revise logic (fundamental rules of reasoning)? 3. Understanding the nature of logic, and how it helps with (though doesn’t by itself fully answer) the puzzles about rational revision of logic.

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Can we change logic rationally? 27 Revision of logic does raise some special epistemological problems. We need an account of how rational change in logic is possible. I’ve said: it’s rational to change our logic when we have an alternative that is enough like the old logic to serve ordinary purposes just as well, but is just different enough to avoid certain anomalies that arise only in special circumstances. But this is too vague to answer all worries.

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An apparent obstacle 28 A precise account would be something like a computer (or probabilistic automaton) model of an agent rationally changing her logic. Or anyway, a sketch of one. An apparent obstacle to providing one: you need logic in arguing for the change in logic. This may seem to raise a threat of some kind of vicious circularity. To elaborate:

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Fundamental rules 29 It is natural (inevitable?) to think of rational thought in terms of the employment of fundamental rules: “To [ascertain what’s true] we rely on a set of epistemic rules, or norms, that tell us in some general way what it would be most rational to believe under different epistemic circumstances.” (Boghossian 2008) The idea isn’t just of rules, but of fundamental ones: general rules under which all our rational practices can be subsumed, and which in particular dictate all rational revision.

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Rational change of fundamental rules? 30 Fundamental rules that dictate all rational revision? If so, it’s hard to see how these fundamental rules can rationally change. Do the rules say: don’t follow me, follow some rules that conflict with me? Then following them would require not following them! But if the (alleged) fundamental rules include rules of logic (as apparently they must), then it’s hard to see how the logic could rationally change.

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Representative quotes 31 “Not everything can be revised, because something must be used to determine whether a revision is warranted….” Tom Nagel 1997, p. 65 “There is no intellectual position we can occupy from which it is possible to scrutinize [our logical beliefs] without presupposing them. That is why they are exempt from skepticism. They cannot be put into question by an imaginative process that essentially relies on them.” Ibid. p 64 “The only way to revise one’s logic is by brain surgery” (Louise Anthony)

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Core vs. peripheral? 32 Solution? Distinguish between core principles of logic, built into the fundamental epistemic rules, and peripheral principles not part of the rules. Allow for rational change only in the peripheral principles. But I doubt that this is adequate.

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Not just after-the-fact account 33 I concede that given any serious proposal for a change of logic, one could find, after the proposal is made, a common core between that and the old logic sufficient to reconstruct some epistemic principles that would have licensed the change. But the fundamental rule picture requires more: a single set of rules that could be used to evaluate any proposal for an altered logic (and handle ordinary reasoning tasks too). It should be possible to give the rules for handling all possible reasonable proposals for altering our logic, in advance of any particular proposed alteration.

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Alternative to rules? Mental models (1) 34 A standard alternative to rules in the psychology of logic: Johnson-Laird on mental models in place of rules of deduction. This doesn’t help much: one seems to need rules for the construction and interpretation of the models. If so, those rules need to be revised to accommodate new logics. More explicitly:

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Mental models (2) 35 The generation of models for classical reasoning seems to go by rules, according to which in any describable circumstances, a given sentence is either true or not true. Only by giving this up and constructing a more general conception of model could the paradoxes be adequately accommodated. The rules of model-generation aren’t rules of deduction, but present the same difficulty: it isn’t clear how to model their change. I’ve put this in terms of rules for generating models. Maybe there’s an alternative to the use of rules. But talk of models doesn’t provide one.

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Default program 36 A better approach is to take logic to be part of a “default program”, rewritable under exceptional circumstances. The procedure for rewriting it is assumed unrevisable (a non-rewritable program or part of basic architecture). Its unrevisability doesn’t raise the same problems, because it doesn’t need to carry out the burden of ordinary reasoning---that’s in the default logic.

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A Direction 37 But we need a version of this that is detailed enough to make substantive and correct predictions about the circumstances under which one might rationally change one’s logic. I think that recent work by others on how we handle inconsistent information provides a clue. Our frequent need to handle inconsistent information and our occasional need to revise logic are both aspects of our logical imperfection, something that philosophers unfortunately tend to idealize away. But to see how to implement this, we need to become clearer about the nature of logic itself. ///

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REPRISE: Outline 38 1. Paradoxes of truth, definability, … Why they aren’t idle puzzles How they make a case for revising logic (= fundamental rules of reasoning) 2. But can we rationally revise logic (fundamental rules of reasoning)? 3. Understanding the nature of logic, and how it helps with (though doesn’t by itself fully answer) the puzzles about rational revision of logic.

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The nature of logic (1) 39 A common view is that logic is the science of which forms of inference necessarily preserve truth. The usual alternative explains logical validity in normative terms: in terms of what we ought to believe and not believe. Both views seriously misconceive the nature of logic. The paradoxes can be shown to decisively refute the first. The second would sully the precision of logic by bringing in imprecise normative notions.

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The nature of logic (2) 40 There’s a more subtle normative view of logic. On it, a logic is a description of a component of a possible set of norms for believing. A good logic is a description of a component of a good set of norms. (Purpose-relative, but truth-oriented purposes are especially important.) We can theorize about which possible norms are good, relative to the purposes we have. For this, we use our current logic. If we come to think an alternative better, we can try to train ourselves to employ it.

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Connection to rational change (1) 41 This doesn’t by itself give the sort of account of rational change of logic I’d like: a (sketch of) a computer or probabilistic automaton model of an agent rationally changing her logic. But I think it gives the proper framework for that. It divides the problem into 1. a theoretical investigation (using currently accepted logic) into (i) what possible systems of logic there are and (ii) what life would be like if we employed them 2. a practical problem of training ourselves to alter our inferential practices.

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Connection to rational change (2) 42 This split lessens the worries about using logic to revise logic: the use of logic is mostly confined to the theoretical task at stage 1, of figuring out which logic is best; The revision comes at the practical level of retraining ourselves, in stage 2.

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Disallow Question Begging. (How?) 43 This doesn’t completely remove all circularity worries, since there are arguments available at stage 1, using the old logic, that would undermine any change. But such arguments would seem question-begging. When change of logic and other change of central practices (e.g. observational practices) is at issue, we need a way to block the use of question-begging reasoning. I think we can adapt models of our use of inconsistent information to do this.

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Summary of this part 44 So my claim is: 1. Logic is not the science of what preserves truth. (I think the paradoxes decisively refute that view, though I didn’t argue that today.) 2. It is not the science of what one ought to believe. (One shouldn’t bring normativity into the subject matter of logic.) 3. The normativity is external: a logic is a partial description of norms for believing, and the normativity comes into the question of what makes a logic good. This helps clear the ground for a proper approach to the subject of the rational revision of logic, and gives some clues as to the direction of a solution.

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FINAL REPRISE: Outline 45 1. Paradoxes of truth, definability, … Why they aren’t idle puzzles How they make a case for revising logic (= fundamental rules of reasoning) 2. But can we rationally revise logic (fundamental rules of reasoning)? 3. Understanding the nature of logic, and how it helps with (though doesn’t by itself fully answer) the puzzles about rational revision of logic.

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