Global Indeterminacy

Kit Fine gave an interesting talk last Friday on globalism about vagueness, i.e. the claim that indeterminacy occurs in the first place over a range of cases (e.g. men in various stages of balding) rather than in any one intrinsically 'borderline' case. In logical terms: we are not to deny any particular instance of the law of excluded middle ("either the nth guy is bald or not") but we may deny the conjunction of two or more such claims, and so avoid commitment to determinate relations ("both are bald, or neither, or one is and the other isn't") over a range of cases.

Importantly, Fine's account no longer licenses the inference [A, ~(A & ~B), therefore B]. This means that we can accept the intuitive principle that there's no sharp boundary between the bald and the not bald -- that is, ~[BALD(i) & ~BALD(i+1)] -- without licensing the sorites inference ("given that the first guy is bald, so is the next").

Doesn't this commit us to the claim that there's some guy, maybe that latter one we just talked about, who is neither bald nor not bald? Here's where things get tricky. Fine claims that this alleged 'commitment' only follows if we accept a 'transcendental' conception of truth according to which we can look down on the entire sorites sequence from on high and assign truth values all at once (at which point we will find that some particular instance must be neither T nor F). From the 'ground level', we can't do this.

[Fine draws an analogy here to the mistake behind naive set theory. By his diagnosis, the intuitive principle -
∃⁺y ∀x (x∈y ≡ φ(x))
- is just fine. The mistake is to think we can apply it "transcendentally", to quantify over absolutely everything, so the ∃⁺ cannot further extend the ∀ quantifier.]

I'm not sure I really understand how the details of Fine's account are supposed to go. On the ground level, shown a lineup of men in various stages of balding, we might assess the sequence of truth values for the propositions of type "the nth guy is bald" as something like the following -
T, T, T, T, ?, ?, F, F, F, F, F
- where the '?' is not a positive answer of any kind (not even "neither T nor F"!) but simply a gap, a silence, where we are unable to answer the question. Speaking "transcendentally", Fine says, we may say that those cases are neither T nor F, but that's a partial truth of our theoretical modeling, not a strictly accurate description of how things are on the ground.

It's puzzling, but apparently not straightforwardly incoherent. At least, Fine offered a semantics that can support his logical claims. Central to it is the following semantic rule for negation:

~A is true under a given use iff A is not true under any compatible use.

In particular, note that if ~A is false, it doesn't follow that A is true under the given use, but merely that A is true under some compatible use. Fine illustrates the general idea by analogy: let 'p' and 'q' be two men planning a dinner party with their spouses, where they follow the British rule that one is not to sit by one's own spouse. We model this by saying that the 'compatibility' relation is here identical to the 'next to' relation, and that p's spouse, for example, is to sit in the ~p spot (similarly for q and ~q):

Either spot linked by red lines is 'compatible' with (read: 'next to') p, so the only place for ~p (i.e. where 'p is not true under any compatible use') is directly opposite. Similarly for q. Now, we see that the denial of LEM: ~(p v ~p) fails, for every spot on the table has some compatible use where either p or ~p holds. But the denied conjunction ~[(p v ~p) & (q v ~q)] does hold, because there is no spot on the table (or 'compatible use') where both one of p or ~p, and one of q or ~q, positively hold.

Still, confusing.