Maudlin s Truth and Paradox Hartry Field

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2 To explain this, I ll first describe an alternative type of solution (which I ll call a Full T- Inference solution) that unproblematically counts as one on which the T-inferences are valid and the classical rules just mentioned aren t. (Like Maudlin s, it doesn t allow accepting contradictions but does accept reasoning by cases). On a Full T-inference solution, the T- inferences are inferentially valid, in the sense that believing A commits you to believing that A is true (i.e. believing T<A>) and vice versa. (On typical Full T-inference solutions, believing A also commits you to believing T<A>, and vice versa. Indeed on many such solutions, we have full intersubstitutivity between T<A> and A: if X is any sentence in which A occurs only extensionally, and X* results from X by substituting T<A> for one or more occurrences of A, then one is committed to having the same attitude toward X and X*.) By the argument above, any Full T-inference solution that doesn t allow belief in contradictions and retains reasoning by cases cannot accept excluded middle: for it cannot accept that either the Liar sentence is true or it isn t true. 2 Maudlin, by contrast, does believe that either the Liar sentence is true or it isn t true. Indeed, he believes the latter, i.e. he believes T< Tλ>. But this is just Tλ, so he simultaneously believes Tλ and T< Tλ>; i.e. he believes Tλ while believing that this belief of his is not true. (He devotes considerable energy to trying to defuse the prima facie implausibility of this: more on that later.) Moreover, since he disallows accepting contradictions, he does not also believe that his belief is true; i.e. he believes that Tλ while not believing that < Tλ> is true. So Upward T-Inference is not inferentially valid. When he calls it valid, what he means is that it s semantically valid in the sense of being (necessarily) truthpreserving. On his account we can have semantic validity without inferential validity. The reason: on that account we must believe certain sentences that we don t believe to be true (and indeed, believe untrue); when these believed sentences are the premises of a truth-preserving argument, there is no reason to believe the conclusion of the argument true, or to believe the conclusion itself. 3 If we regard the T-inferences as inferentially valid, as in the naive theory of truth and in Full T-inference solutions to the paradoxes, then there is no real difference between (i) the inferential validity of the inference from A to B and (ii) the inferential validity of the inference from T<A> to T<B>. But on Maudlin s theory, the fact that T<A> and T<B> aren t inferentially equivalent to A and B opens up the possibility of a difference between (i) and (ii); and the difference, for Maudlin, is huge. The semantic validity of the inference from A to B is equivalent in his theory to (ii), not to (i). So when Maudlin says that the Upward T-inference is valid, what this amounts to is that for any A, if you accept T<A> then you should accept T<T<A>>; it doesn t mean that if you accept A then you should accept T<A>. The upshot: while Maudlin s solution to the inferential Liar paradox has it that the T- inferences are semantically valid but classical rules like reductio and excluded middle aren t, it also has it that those classical rules are inferentially valid while upward T-inference isn t. (Downward T-inference is inferentially valid on Maudlin s view, though this doesn t follow from its semantic validity.) When Maudlin s theory is put in terms of inferential validity, it turns out to be (an extension of) what s often called the Kripke-Feferman theory (KF): a classical logic 2

3 theory that allows Downward T-Inference but not Upward T-Inference (but contains the weaker rule that T<A> implies T<T<A>>). 4 Turning from exposition, the substantive question I raise at this point is whether we shouldn t regard Upward T-Inference as inferentially valid rather than merely semantically valid. Some brief comments: 1. The one disappointing feature of the book is that it doesn t even mention any Full T- inference solutions: a discussion of why he thinks his solution preferable to them would have been extremely helpful. 2. Maudlin s initial motivation for the Upward and Downward T-Inference rules (pp. 9-10) seems based on the thought that T<A> is little more than a notational variant of A. But if that s so, it supports not only that the inference from the latter to the former must preserve truth, but also that we ought to have the same attitudes (belief, disbelief, etc.) toward each; i.e. it motivates the inferential validity of the rules, not just their semantic validity. 3. Denying the inferential validity of the Upward T-Inference cripples the ordinary use of true. Suppose Jones disagrees with the overall body of theory that Maudlin espoused on Monday, but can t decide which specific claims of Maudlin s are faulty (or can t remember exactly what Maudlin said that was wrong). The standard story about the truth predicate (Quine 1970, Leeds 1978) is that it is precisely in circumstances like these that true is useful: Jones can express his disagreement by saying Something Maudlin said on Monday isn t true. But this loses its effect unless both Upward and Downward T- Inference (and related rules involving negation) are in force: indeed, since Maudlin himself thinks that important parts of his own theory aren t true, it is clear that Jones hasn t succeeded in expressing disagreement. 5 But let s turn to Maudlin s views on the metaphysics of truth, which supply an intriguing and novel justification for rejecting the inferential validity of the Upward T-Inference. 2. Maudlin justifies his approach to the paradoxes by means of a fairly appealing picture of how semantics works. In this picture, truth and falsity are primarily properties only of boundary sentences: (parameterized) atomic sentences with predicates other than True. (A parameterized atomic sentence is an atomic formula with objects assigned to its free variables.) And if we ignore vagueness (as Maudlin mostly does, saying it has nothing to do with the paradoxes), then all boundary sentences are either true or false. Other sentences can inherit truth or falsity from the boundary sentences, but this can happen only if these other sentences are grounded in a sense equivalent to Kripke s (i.e., members of the minimal Kleene-based fixed point). 6 So we get a threefold partition: the true, the false, and the ungrounded. The claim that ungrounded sentences such as the Liar aren t true or false is not a claim that appears in the minimal fixed point. But it is part of Maudlin s theory, albeit a part of the theory that the theory itself asserts is untrue. A Full T-Inference theory, by contrast, contains only those sentences that appear in the fixed point (at least, this is so for sentences with no 6 ). Such a theory thus agrees with Maudlin s in holding that you shouldn t assert that ungrounded sentences are true or false, but it says that you also shouldn t assert that they aren t true or false 3

5 Tarskian levels solution. 8 Maudlin justifies this by his claim that permissibility, unlike truth, is not objective, because dependent on norms of which none is uniquely best. But it s unobvious that permissibility in his sense should be viewed as less objective than truth. Maudlin makes the non-objectivity of permissibility seem more obvious than it ought to seem by sometimes talking of permissibility as based on norms for what it s permissible to assert (e.g. 165, especially discussion of the Monastic Rule of Silence); but it s clear that he really needs to think of the norms as governing belief, e.g. as licensing the beliefs that the Liar sentence isn t true and that the T-Inferences preserve truth. Still, Maudlin could plausibly argue that rules governing beliefs, such as inductive rules, also lack objectivity, since there are multiple reasonable policies one might have for extrapolating from the observed to the unobserved. But that isn t enough for Maudlin s point. The rules for belief that Maudlin is concerned with aren t like inductive rules, for they don t take into account our epistemological situation, e.g. which evidence is available: they are rules governing a being who knows all the truths (but chooses to go beyond them in forming his beliefs). Indeed, Maudlin regards permissibility as completely objective when it comes to sentences that assert truth: T<A> is permissible when and only when A is true, and that is an entirely objective matter. It is only for other kinds of sentences that legitimate norms of permissibility can diverge from each other. Moreover, it s clear that Maudlin thinks all good norms agree even about many things he regards as untrue: for instance, he clearly thinks that all good norms agree in licensing belief in the untruth of the Liar, in the truth-preservingness of the T-Inferences, and in many other of the claims that Maudlin vigorously asserts in his book but regards as untrue. In short, Maudlin s notion of permissibility is not nearly as dependent on arbitrary norms as his discussion suggests. Indeed, his notion of permissibility seems at least as much like the ordinary notion of truth as does his notion of truth. (For instance, in the ordinary sense of true, Not everything he said is true expresses disagreement; in Maudlin s theory neither Not everything he said is true nor Not everything he said is permissible does that, but the latter comes somewhat closer to doing so.) It seems best to adopt a view on which truth and permissibility don t diverge a view on which one can assert the truth of A when and only when one can assert A. The Full T-Inference views mentioned earlier have this feature; indeed, the best of them allow for the intersubstitutivity of T<A> with A in all extensional contexts, and for a conditional which validates the Tarski conditionals T<A>6A and A6T<A>. Despite the ingenuity of Maudlin s discussion, I see little to favor his view over a view of that kind. Still, for anyone interested in the paradoxes this is a must-read book. References Brady 1989, The Non-Triviality of Dialectical Set Theory. In Priest, Routley and Norman (eds.), Paraconsistent Logic. Philosophia Verlag. Field 2003, A Revenge-Immune Solution to the Semantic Paradoxes, Journal of Philosophical Logic. Gupta 1982, Truth and Paradox, Journal of Philosophical Logic. 5

7 6. It s the groundedness intuition that s crucial for Maudlin: fixed points other than the minimal Kleene-based one play no role in his account, unlike Kripke s. 7. I ll just say that truth-table definitions get bite only from the logic one uses to apply them; Maudlin implicitly assumes that one can use excluded middle for this (e.g. by assuming that the rows of a truth table are exhaustive), which begs important questions about how the defined connectives behave. 8. The view is that a sentence asserting its own impermissibility according to norms N is true, and that it s impermissible according to N but permissible according to other norms N*. Maudlin argues that unlike the levels solution, such N* shouldn t be viewed as higher norms, but simply as different norms. 7

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