This is an updated version of a paper I gave at a British Undergraduate Philosophy Society conference in Durham, 2007. My thanks go to my respondent, Ed Grefenstette, Professor Robert Hale and everyone else who offered feedback and criticism. Responsibility for any remaining inaccuracies and omissions are of course my own.

I. Introduction

According to the logical principle of bivalence, all utterances that express a proposition must possess one of two possible truth values: true or false (Williamson 1994: 187). However, in cases of borderline vagueness, it seems intuitive to characterise certain utterances as neither true nor false on the grounds that it is unclear which, if either, of these truth values obtains. The conception of truth that is generally taken to apply to such cases is represented by Tarski’s Convention-T, which may be symbolised as follows:

(T1)    T[‘P’] ↔ P

(T2)    T[‘¬P’] ↔ ¬P

where T[] signifies the metalinguistic truth predicate (Gómez-Torrente 2006). According to T1, the sentence ‘This rose is red’ is true iff the rose in question is actually red. But if the rose was a borderline case of red and pink, it would not be true that the rose was red, but neither would it be determinately false—or at least not straightforwardly so. If we accept this linguistic intuition at face value, then we must allow that some utterances can be neither true nor false but rather possess a third ‘indeterminate’ truth value, thus contravening bivalence.

II. Williamson’s Reductio

Timothy Williamson (1992) presents a reductio ad absurdum argument against such a denial of bivalence on the grounds that it entails denying the Law of Excluded Middle (LEM) in the metalanguage, i.e:

(W1)    ¬(T[‘P’] ∨ T[‘¬P’])

By replacing the two disjuncts with the right-hand sides of the Tarskian biconditionals, T1 and T2, this yields:

(W2)    ¬(P ∨ ¬P)

which, by De Morgan’s Law, entails:

(W3)    ¬P & ¬¬P

This conclusion is clearly unacceptable to the opponent of bivalence as it would render their position incoherent. The source of this contradiction, however, may be traced back to Williamson’s use of a classical (i.e. bivalent) metalanguage. The assumption of bivalence in the metalanguage is then ‘forced through’ to the object language via Tarski’s Convention-T, with disastrous consequences. If this diagnosis is correct, then in order to refute the argument one must either adopt a non-classical metalanguage in W1, or reject T1 and T2, the basis of Tarski’s semantic conception of truth. In the remainder of this essay, I will argue that the opponent of bivalence can and should overcome the above argument by rejecting both of these assumptions.

III. Multivalence and Assertibility

Pelletier and Stainton (2003: 372) deny that LEM necessarily fails in a three-valued (i.e. true, false and indeterminate) logic provided that the negation operator only ever yields determinate truth or falsity, even in borderline cases.1 On this view, the second disjunct of W1 comes out as true where P is indeterminate, and Williamson’s argument never even gets off the ground. Furthermore, they claim, it is doubtful whether both directions of T1 and T2 hold in any such three-valued logic. Although the left-to-right direction is uncontentious, if P is indeterminate—i.e. neither true nor false—then it is not true that P, and so P → T[‘P’] is either false or indeterminate. Regardless of which option one takes, the right-to-left direction of T1 and T2 will fail in cases of borderline vagueness, and so these biconditionals do not constitute logical truths in the resulting multivalent logic. Thus, Pelletier and Stainton conclude that T1 and T2 ‘already presuppose bivalence, and so they are not the appropriate disquotational schemas for a multivalued logic’ (ibid. 374).

Williamson (op. cit. 268–9) justifies his use of Convention-T on the grounds that any predicates that fail to refer to sharply defined properties fail to say anything at all, and so we should refrain from their use as in other cases of reference failure. Given that most concepts contain at least some element of vagueness, this would effectively render the majority of utterances meaningless, thus constituting a further argument against the denial of bivalence—or else in favour of some sort of linguistic nihilism (cf. Unger 1979). However, Williamson’s claim is too strong. In order for a proposition to say something, it is arguably sufficient that there are cases in which it is determinately true or determinately false. In cases of borderline vagueness, its use might be considered incorrect or inappropriate, but it would not be strictly meaningless (we might, for example, meaningfully assert it as a form of exaggeration, or to make a point, even though the resulting utterance would be literally false). On this view, whilst the meaning of the sentence may still be fixed by its (determinate) truth conditions, its use is governed by a separate set of assertibility conditions—the conditions under which it may truthfully be asserted. In the case of vague predicates, these two sets of conditions come apart, leaving gaps in our assignment of determinate truth and falsity, and resulting in what Pelletier and Stainton (op. cit. 371) call a ‘truth-value gap’ theory. The notion of truth employed by this theory may be symbolised as:

(T1*)    T[‘P’] ↔ ∆P

(T2*)    T[‘¬P’] ↔ ∆¬P

where the ∆ operator represents the sentence modifier ‘It is determinately true that …’, effectively ‘collapsing’ the indeterminacy. Substituting T1* and T2* into Williamson’s original argument, we obtain the entirely plausible and non-contradictory conclusion:

(W3*)    ¬∆P & ¬∆¬P

or ‘it is not the case that determinately P, nor is the case that determinately not-P’—precisely what we would expect an opponent of bivalence to say in borderline cases of vagueness (cf. Williamson 1994: 194).

IV. Higher-Order Vagueness

The use of the determinately operator, however, introduces another problem: at what point does a proposition become determinately true or determinately false? Although we can cite paradigm cases of determinate truth and falsity (e.g. ‘snow is white’, ‘the moon is made of green cheese’), it is unclear exactly where the boundaries lie. In other words, determinate truth is itself a vague concept, thus giving rise to the problem of ‘higher-order vagueness’ (Williamson 2003: 8). As Williamson points out, any analysis of vagueness that cannot account for higher-order vagueness is implausible, since we cannot deny that the concept of vagueness itself permits of borderline cases (ibid.). Furthermore, we must be able to assign some particular meaning to the concept of determinacy that is distinct from our original unqualified conception of truth (Williamson 1994: 194).

One possible solution to this problem, based upon a proposal by McGee and McLaughlin (1995: 229–30), is to explain higher-order vagueness in terms of higher-order instances of the (non-bivalent) metalanguage. On this account, determinacy is a metalinguistic concept, comparable to T[], by which the semantics of the object language are defined. First-order vagueness is governed by the constraints upon the object language along with the allowable models that assign truth and falsity to each of its sentences. Since the object language is vague, there will be many such models, each of which assigns different truth values to the vague sentences in the language. Determinate truth and falsity are just those cases where all of the models agree upon a sentence’s truth value (i.e. the ∆ operator acts as a sort of quantifier over allowable models). Second-order vagueness is defined in terms of allowable models of the metalanguage, with a second-order metalanguage (the metametalanguage) governing the formal semantics of the first-order constraints, including the concept of determinate truth. Since both truth and determinate truth inherit the vagueness of terms in the object language, this yields a series of modified biconditionals including:

(T1**)   ∆T[‘P’] ↔ ∆P  (ibid.)

each of which define the (vague) semantics of the previous level’s constraints, and whose vagueness is itself resolved at the subsequent level. This pattern may be repeated indefinitely to cope with any number of orders of vagueness without fear of contradiction—or as McGee and McLaughlin put it, ‘[a]s we ascend the hierarchy of metalanguages, we find vagueness all the way up’ (ibid. 230).

V. Conclusion

Williamson’s reductio represents a forceful attack on the denial of bivalence. However, its assumption of a classically bivalent metalanguage, along with the use of Tarski’s Convention-T, are clearly question begging. When combined with an appropriate three-valued logic plus a modified conception of truth that takes both higher-order vagueness and determinacy into account, the alleged absurdity gives way to mere complexity. Thus we can conclude that whilst it is not absurd to deny bivalence, such a move loses much of the simplicity and elegance of classical logic, as well as requiring serious revision to the conception of truth represented by the Tarskian biconditionals. Whether this in itself constitutes an argument against bivalence, as Williamson (1992: 279–80) claims, remains open to question. Although there are obvious advantages to retaining the classical approach in cases where vagueness is not an issue or can safely be ignored, the ‘truth value-gap’ theory is arguably a more accurate reflection of our actual linguistic practices as it avoids the need for implausible assumptions about the precise nature of either meaning (Williamson 1992) or interpretation (Fine 1975), both of which remain irreducibly vague on the non-bivalent account.

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1 They call this ‘exclusion negation’ on the basis that only one of the three possible truth values (determinate truth) is being denied or excluded.

References

Fine, Kit 1975: ‘Vagueness, Truth and Logic’. In R. Keefe and P. Smith (eds.), pp. 119–50.

Gómez-Torrente, Mario 2006: ‘Alfred Tarski’. The Stanford Encyclopedia of Philosophy (Winter 2006 Edition), Edward N. Zalta (ed.). <http://plato.stanford.edu/archives/win2006/entries/tarski/>.

McGee, Vann and Brian P. McLaughlin 1995: ‘Distinctions Without a Difference’. Southern Journal of Philosophy, 32 (supplemental), pp. 203–51.

Pelletier, Jeffrey and Robert J. Stainton 2003: ‘On “The Denial of Bivalence is Absurd”’. Australasian Journal of Philosophy, 81 (3), pp. 369–82.

Unger, Peter 1979: ‘There Are No Ordinary Things’. Synthese, 4. pp. 117-54

Williamson, Timothy 1992: ‘Vagueness and Ignorance’. In R. Keefe and P. Smith (eds.), pp. 264–80.

—————  1994: Vagueness. London: Routledge.

—————  2003: ‘Vagueness in Reality’. In M. Loux and D. Zimmerman (eds.). <http://www.philosophy.ox.ac.uk/faculty/members/docs/handbook.pdf> (accessed 14/3/07).