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Published online by Cambridge University Press: 01 January 2020
A tempting solution to problems of semantic vagueness and to the Liar Paradox is an appeal to truth-value gaps. It is tempting to say, for example, that, where Harry is a borderline case of bald, the sentence
(1) Harry is bald
is neither true nor false: it is in the ‘gap’ between these two values, and perhaps deserves a third truth-value. Similarly with the Liar Paradox. Consider the following Liar sentence:
(2) (2) is false.
That is, sentence (2) says of itself that it is false. If we accept the Tarskian schema
(T) S is true iff p
where ‘S’ is a name of a sentence ‘p,’ we are led into paradox. Both the assumption that (2) is true, and the assumption that (2) is false lead us, via (T), to
(3) (2) is true if and only if (2) is false.
Given this result, a natural reaction is to place (2) in a ‘gap’ between true and false.
1 Kripke, Saul ‘Outline of a Theory of Truth,’ Journal of Philosophy 72 (1975) 690–716CrossRefGoogle Scholar
2 Fine, Kit ‘Vagueness, Truth and Logic,’ Synthese 30 (1975) 265–300CrossRefGoogle Scholar
3 McGee, Vann Truth, Vagueness, and Paradox (Indianapolis: Hackett 1990)Google Scholar (henceforth, TVP) and ‘Applying Kripke's Theory of Truth,’ Journal of Philosophy 86 (1989) 530-9 (henceforth, KTT)
4 I am also of the opinion that the arguments presented here against McGee will extend, more or less intact, to any vagueness-based approach to the Liar Paradox. Defending this claim, however, is a much larger project that I will not undertake here.
5 It should be noted that on Kripke's approach, the ‘gap’ is not treated as a third semantic category.
6 KTT, 535. Though McGee is not altogether clear on this point, it seems to be the case that once a sentence is settled, it becomes definitely true, not merely true. This must be the case, as I see it, for McGee's system only involves three categories: definitely true, definitely false, and unsettled. Our ordinary conception of ‘true’ is being refined, it seems, into a more precise concept, that of definite truth. (At least for the purposes of science. McGee is not advocating revisionism with respect to our 'everyday’ notion of truth.)
7 In McGee's terminology we are assured of a classical logic because there are only two settled categories: definitely true and definitely false. If something is in the unsettled category, then it is not — indeed cannot — be settled that this is so; to say that it is, is to elevate the ‘unsettled’ category to a third semantic value.
8 ‘That “true” is a vague predicate should come as no surprise. Intuitively, when we assert or deny that “Harry is bald” is true, we are saying the same thing as when we assert or deny that Harry is bald. If that is so, then, if the linguistic conventions that govern the use of the vague term “bald” leave it unsettled whether or not Harry is bald, the linguistic conventions that govern the use of the term “true” likewise leave it unsettled whether or not “Harry is bald” is true …. Thus, “true” inherits the vagueness of all the vague nonsemantical predicates of our language’ (TVP, 216-17).
9 Two remarks are apropos here. First, though one might (as I do) find the inheritance claim puzzling, I do not think that it is essential to McGee's project. What is essential is the upshot of that claim, viz., that what is responsible for the unsettled character of ordinary predicates and semantic predicates is the same in each case (and, the corollary, that the reasons for preserving or expunging the vagueness is the same in both the ordinary and the semantic case).
Second, one might, of course, take issue with the claim that ‘true’ is a vague predicate. (See for example Haack, Susan Philosophy of Logics [Cambridge: Cambridge University Press 1978].)CrossRefGoogle Scholar One might say that the vagueness of ‘bald’ and other non-semantic terms does not transfer to the predicate ‘true’ when that predicate is appended to sentences containing a non-semantic vague term. ‘True’ (as well as 'false’) is a perfectly sharp predicate, but it is not properly applied to ‘Harry is bald.' It is not, for the very reason that ‘bald’ is a vague term. In order to apply ‘true’ to 'Harry is bald,’ we must be more explicit: ‘Harry is definitely bald’ is true, or ‘Harry is slightly bald’ is true, where ‘true’ is a sharp predicate. However, I shall not pursue this objection here. Rather, I will show that even if one admits that ‘true’ is vague (something I do not think we should do), serious problems remain.
10 TVP, 7; emphasis added. Though we will examine this passage in more detail below, it is easy to see the relation of this passage to the remark in n. 6. Saying that it is not settled that sentences are unsettled is tantamount to denying the third category a semantic status on a par with ‘definitely true’ and ‘definitely false.’
11 Cf. Quine, W.V. ‘Ontological Relativity,’ in Ontological Relativity and Other Essays (New York: Columbia University Press 1969).CrossRefGoogle Scholar
12 This is McGee's example of an organism, Jocko, on the borderline between animal and vegetable.
13 We shall examine the case of ‘true’ below.
14 Dummett, Michael The Interpretation of Frege's Philosophy (Cambridge, MA: Harvard University Press 1981), 440Google Scholar
15 But see section VI below. The identity conditions of concepts is an issue that McGee will have to deal with if he adopts this reading. The question of whether a concept can ‘survive’ sharpening is an important one. It might seem that if being vague is essential to a concept, then sharpening it is tantamount to replacing it.
16 I suspect this last reading is the most plausible in the case of the non-semantic vague terms. An inherent gappiness in nature might offend against certain favored physical theories, and moreover that nature has’ decided’ upon an upper limit of hairs in the case of ‘bald’ or an upper limit of stones in the case of ‘heap’ is a difficult notion to swallow. So, thereby, is the epistemological reading of ‘unsettled’ — at least in the case of these predicates. Such a reading might, however, seem plausible in the case of ‘protozoan.’ Of course, one might want to ‘mix and match,’ opting for one reading of ‘unsettled’ for one predicate and another reading of ‘unsettled’ for another. I see no reason aside from simplicity against such a position. Our concern below, however, will be with the proper reading for one predicate in particular: ‘true.’
17 TVP, 217. There is also a hint of the conceptual reading of ‘unsettled’ here: note McGee's talk of ‘the concept protozoan’ and ‘the concept of truth.’ We shall consider this reading below.
18 TVP, 7. Note here that McGee speaks of settling unsettled sentences by adopting linguistic conventions, but the point I am making is a more general one, that does not depend on any particular means of settling unsettled sentences. My point is that if ‘unsettled’ is read in the metaphysical way, then unsettled sentences cannot be settled (on pain of misrepresenting the world), no matter how the settling is supposed to take place.
19 If this is not the case, then in an important sense we have not settled a sentence, but merely replaced it with another. That is, we resolve the Liar Paradox by agreeing not to utter the problematic sentences, and this does not seem much of a resolution.
20 This point is also made by Yablo, Stephen in his ‘Truth, Definite Truth, and Paradox,’ Journal of Philosophy 86 (1989) 539–41.CrossRefGoogle Scholar Simmons, Keith makes a number of similar points in his Universality and the Liar (Cambridge: Cambridge University Press 1993).CrossRefGoogle Scholar
21 I am indebted to an anonymous referee for this journal for bringing this distinction to my attention.
22 A predecessor of this paper was presented to the Graduate Philosophy Club at the University of North Carolina at Chapel Hill. I am indebted to that audience, Simon Blackburn, an anonymous referee for this journal, and most especially to Keith Simmons for helpful comments on earlier drafts of this paper.