# Vagueness and supervaluations

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1 Vagueness and supervaluations UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Supervaluations We saw two problems with the three-valued approach: 1. sharp boundaries 2. counterintuitive consequences of truth functionality (or degree functionality) The degree approach fixes (1) but still has (2). Supervaluationism is a way to fix (2), though perhaps it still has (1). The basic idea: A valuation is an assignment of classical extensions to each nonlogical term in the language. An interpretation of the language is a set of valuations. (Not, as in classical semantics, a single valuation.) An interpretation embodies a set of constraints on valuations: some assignments of classical extensions are ruled out as inadmissible. A valuation v is admissible (under an interpretation) if it satisfies the following constraints: the extension of a expression on v includes all the things that definitely fall into the extension of that expression (on the interpretation). the extensions of expressions on v respect penumbral connections. Penumbral connections are, essentially, relations between the meanings of different vague terms. So, for example, it might be required that nothing be in the extension of both orange and red. And t might be required that certain connections hold betw the extensions of tall, short, and taller than. We can represent these constraints using meaning postulates: valuation interpretation admissible penumbral connections meaning postulates x (Red(x) Orange(x)) x, y(tall(y) TallerThan(x, y) Tall(x)) Supervaluation is often motivated by the idea that vagueness arises from semantic indecision: The reason it s vague where the outback begins is not that there s this thing, the outback, with imprecise borders; rather there are many things, with different borders, and nobody has been fool enough to try to enforce a choice of one of them as the official referent of the word outback. Vagueness is semantic indecision. [5, p. 213] April 26,

2 2. Application to sorites The admissible valuations represent the different candidate extensions we re undecided about. A sentence is true on a valuation v iff it is true on v s assignment of classical extensions to its terms. A sentence is true on an interpretation I iff it is supertrue true on every valuation in I. A sentence is false on an interpretation I iff it is superfalse false on every valuation in I. (Equivalently, a sentence is false on an interpretation I if its negation is true in I.) Some sentences will be neither supertrue nor superfalse on an interpretation. Since we identify truth and falsity with supertruth and superfalsity, this means that we give up bivalence, the assumption that every sentence is either true or false (on a given interpretation). Example: true on a valuation v true on an interpretation I supertrue false on an interpretation I superfalse bivalence (1) Joe is bald. when the admissible interpretations contain some valuations that put Joe in the extension of bald and some that do not. But consider (2) Either Joe is bald or he isn t. This comes out true on the interpretation described, despite that fact that neither disjunct is true! So despite giving up bivalence, we keep the law of excluded middle. This is a distinctive feature of supervaluationism. (Contrast intuitionism, which gives up both bivalence and the law of excluded middle.) Indeed, (2) is true on every interpretation: it is logically true. Generalizing: every sentence that is a logically true in classical logic will be logically true (true in all interpretations) in supervaluational semantics as well. To see why, note that every sentence that is logically true in classical logic must be true on every admissible valuation, since admissible valuations are classical interpretations. It follows immediately that every such sentence will be supertrue on all supervaluational interpretations. We have a combination of classical logic and nonclassical semantics. Is that bad? You might wonder whether a sign can mean disjunction if A B is allowed to be true when neither A nor B is true. On the other hand, it seems an attractive feature of supervaluationism that it lets us continue to use classical logic without a commitment to bivalence, which might seem implausible for vague language. law of excluded middle 2 Application to sorites What does supervaluationism say about the sorites paradox? April 26,

3 3. Higher-order vagueness Remember, the fundamental problem was that if we reject the universal premise of the sorites, (3) For any n, if someone with n pennies is rich, then someone with n 1 pennies is rich, which it seems we have to to block the paradox, we seem forced to accept its negation, which is a sharp-boundaries claim: (4) For some n, someone with n pennies is rich and someone with n 1 pennies is not rich. Multivalued approaches solve the puzzle by saying: you can reject (6) (or the conditionals that are its instances) as less than fully true, without accepting its negation (4). What does supervaluationism say about the universal premise (6)? Like multivalued approaches, it says you should reject it! (It s superfalse.) But, unlike multivalued approaches, it says you should accept its negation (4). But isn t this what we found problematic with classical semantics? That rejecting (6) would require accepting (4), a sharp-boundaries claim? So how does supervaluationism improve over classical semantics? The key difference is that, for the supervaluationist, accepting (4) does not commit one to accepting that there is a true witness to the existentially quantified claim: that is, a true instance. In classical semantics, if (4) is true, then for some (perhaps unknown) n, the following instance must be true: witness (5) Someone with n pennies is rich and someone with n 1 pennies is not rich. But the supervaluationist can deny that any sentence of the form (5) is true, while still accepting (4). This is just the extension to quantification of the point we already saw: that there can be a true disjunction with neither disjunct true. So the idea is that we take the sting out of accepting (4) by making it possible to accept (4) while rejecting all of its instances. We can accept that there is a number of pennies such that a person with that many pennies is rich and a person with one fewer penny is not rich, while rejecting, for every particular number n, the claim that a person with n pennies is rich and a person with n 1 pennies is not rich. 3 Higher-order vagueness We started with a notion of admissible valuations. An admissible valuation, we said, must include all the definite F s in the extension of F. So this account presupposes a sharp boundary between the people who are definitely rich and those who are not. And this, you might think, is just as objectionable as having a sharp boundary between the people who are rich and those who aren t. If the worry was about having sharp, unknowable boundaries, we still have them. April 26,

4 3. Higher-order vagueness Suppose we add a definitely operator D that works as follows: Dφ is true on a valuation just in case φ is true on all admissible valuations. (Note: this is a bit like our standard modal.) Then supervaluationism begins to look bad...or so Williamson [9] argues. First, there is the problem of higher-order vagueness. We can just run the sorites with definitely rich instead of bald. The sorites premise seems almost as compelling: D (6) For any n, if someone with n pennies is definitely rich, then someone with n 1 pennies is definitely rich, The usual supervaluationist response (e.g., in [4]) is to say that the notion of admissibility is vague. If that is right, then we are doing our semantics for vague terms in a vague metalanguage. But if we re going to do this, why not just do it at the beginning, and say (7) The extension of bald is the set of bald things. Williamson also points to some logical problems with the introduction of the D operator. Note first that there are two ways of defining validity in a supervaluational semantics. An inference is locally valid if any valuation that makes the premises true makes the conclusion true. An inference is globally valid if any interpretation that makse the premises supertrue makes the conclusion supertrue. locally valid globally valid Williamson argues that local validity can t be the important notion of validity for supervaluationists. Since they identify truth with supertruth, the proper extension of the normal classical notion of validity to supervaluationism is preservation of supertruth. But on global notion of validity, (8) φ, therefore Dφ is valid! (Confirm this for yourself.) But (9) φ Dφ isn t a logical truth. When φ is borderline, (9) is false on some valuations, so it is not supertrue. So, Williamson argues, once we introduce D into our language, we re going to lose some properties of classical validity; If ψ is a logical consequence of φ, then φ ψ is a logical truth. If ψ is a logical consequence of φ, φ is a logical consequence of ψ. April 26,

5 REFERENCES REFERENCES (According to supervaluationism, an argument can be truth-preserving without being non-falsity preserving.) How bad are these results? Williamson says that supervaluations invalidate our natural mode of deductive thinking, such as Conditional Proof (conditional introduction), Dilemma (disjunction elimination), and Reductio (negation introduction). But this isn t clear. We have already seen examples of restricting what can be done inside a subproof for conditional proof ( -introduction), in relevance logics and modal logics. Could we not impose a similar restriction here? Instead of allowing all valid forms of argument inside a subproof for conditional proof, we could allow just locally valid forms of argument. (See [8, pp ], [7], [6, 4].) Further reading The locus classicus for supervaluationist approaches to vagueness is [2]. A recent defense against some of the objections in Williamson can be found in [4, ch. 7]. For some criticism of Keefe, see [1]. For an entirely different line of criticism, see [3]. References [1] Delia Graff Fara. Scope Confusions and Unsatisfiable Disjuncts: Two Problems for Supervaluationism. In: ed. by Richard Dietz and Sebastiano Moruzzi. Oxford University Press: Oxford, [2] Kit Fine. Vagueness, Truth and Logic. In: Vagueness: A Reader. Ed. by Rosanna Keefe and Peter Smith. Cambridge, MA: MIT, 1997, pp [3] Jerry A Fodor and Ernest Lepore. What Cannot Be Evaluated Cannot Be Evaluated and It Cannot Be Supervalued Either. In: The Journal of philosophy (1996), pp [4] Rosanna Keefe. Theories of Vagueness. Cambridge: Cambridge University Press, [5] David Lewis. On the Plurality of Worlds. Oxford: Basil Blackwell, [6] John MacFarlane. Truth in the Garden of Forking Paths. In: Relative Truth. Oxford: Oxford University Press, 2008, pp [7] Vann McGee and Brian McLaughlin. Logical Commitment and Semantic Indeterminacy: A Reply to Williamson. In: Linguistics and Philosophy 27 (), pp [8] Vann McGee and Brian McLaughlin. Review of Timothy Williamson, Vagueness. In: Linguistics and Philosophy 21 (1998), pp [9] Timothy Williamson. Vagueness. London: Routledge, April 26,

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