Can One Get Bivalence from (Tarskian) Truth and Falsity?

According to one understanding of them, Tarskian principles about truth (and falsity) aim to explicate the core of the classical conception of truth (and falsity), as endorsed by Aristotle and others:

To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true, so that he who says of anything that it is, or that it is not, will say either what is true or what is false. Metaphysics

(Book IV.7, 1011ª: 26-8)

Timothy Williamson famously offered an argument from these Tarskian principles in favour of bivalence - the contention that whatever says something is either true or false - to the effect that denying bivalence in particular cases classically¹ entails a contradiction. This has played a crucial role in the reception of Williamson's case against the main alternative classical view of the nature of vagueness, supervaluationism, and thus in favor of his own epistemic view.

I begin by providing some background on the significance of this issue vis-à-vis the philosophical debate between competing views on the nature of vagueness (section I). I then rehearse Williamson's original argument (section II). Dwelling on (Andjelković & Williamson, 2000), I show that this argument depends on a contentious formulation of the Tarskian principles about truth (and falsity), a formulation which the supervaluationist can reject without jeopardizing the Tarskian insight (section III). In the paper in question, Miroslava Andjelković and Timothy Williamson argue that, even if the appropriate formulation seems to allow for failure of bivalence in borderline cases, this appearance is illusory, once one grants a further (independent) principle involving biconditionals. Finally, I argue that such a formulation is, however, contentious in a similar manner (section IV).

I conclude that the supervaluationist is in a position to block the argument from Tarskian truth (and falsity) in favour of bivalence.

I Bivalence and the Nature of Vagueness

One of the main views on the nature of vagueness - supervaluationism - has it that vagueness is a phenomenon of semantic indecision: (roughly) whatever it is that in the thoughts, experiences and practices of language users determines the meaning of expressions, it fails to determine, for vague expressions, any single one from a given range of similarly natural candidates. Each way of ('arbitrarily') fixing what is left semantically indeterminate gives rise to a precisification or sharpening of the original vague expression. Although all such sharpenings are, by essence, arbitrary to a certain extent, not all of them are admissible. In the case of predicates, admissible ones should preserve clear cases, both of application and of non-application - Yul Brynner should count for 'is bald,' while Andy García cannot - and they should also preserve penumbral connections - 'Whoever is bald is bald,' 'If someone is bald, then so is anyone who is balder,' and so on - .² What one says by means of a vague expression is true, according to this view, if it would be true however one (admissibly) precisifies it - or, as I will put it, if it counts as true according to all admissible sharpenings. And it is false if it counts as false according to all of admissible sharpenings. Otherwise, if there are admissible ways of precisifying it which give rise to truths, but also admissible ways of precisifying it which give rise to falsehood, the vague sentence is indeterminate: neither true nor false.

That is indeed the situation with respect to borderline cases, as the view has it. Take Harry, a borderline case with respect to 'is bald,' having exactly 3,833 hairs on his scalp. Whatever it is that in the thoughts, experiences, and practices of language users determines the meaning of expressions, it fails to determine whether someone with this very number of hairs does or does not fall under 'is bald.' Thus 'is bald' can be admissibly precisified by (let us assume) 'has at most 3,832 hairs on his scalp,' but also by 'has at most 3,834 hairs...