Suppose (for reductio) that it is indeterminate whether a = b. Then b has the property of "being an x such that it is indefinite whether x is identical to a" (p.154). But a surely lacks this property. It is perfectly determinate that a is identical to a! So a and b have different properties -- they differ in whether they have the property of being determinately identical to a -- and hence (by Leibniz's Law) they are non-identical, contradicting our initial assumption.
Tye adds: "for the case in which 'a' and 'b' are singular terms for persons, the conclusion to which we seem driven is that there cannot be vagueness in the identity of persons." (p.155)
The first step of the argument is invalid if we hold vagueness to be a purely semantic phenomenon. As Lewis expresses the view:
The reason why 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 term 'outback'.
On this view, it is vague what objects and properties we are talking about at each step of Evans' proof. So it is false that "b has the property of being an x such that it is indefinite whether x is identical to a". There is no vagueness in the real properties of the objects. The only indeterminacy is in what our terms denote. The indeterminacy of "a = b" is due to language rather than the world. So Evans' argument fails.
But this response is not open to Tye, because he thinks that vagueness is metaphysical. He considers 'a' and 'b' to be semantically precise designators for inherently vague objects. So Tye accepts Evans' argument, but nevertheless insists that persons may be vague objects (i.e. objects with vague boundaries and persistence conditions) without this giving rise to problematic vague identity statements.
It's hard to see how this could be, and Tye doesn't really explain it. Suppose that Adam is undergoing a sci-fi process of total material replacement, from which Bertha will emerge. At each step of the process (after 20% replacement, 40%, etc.) we can point to the person undergoing transformation and ask, "is that person (identical to) Adam?" Tye accepts that names like 'Adam' may precisely designate a vague object, and the demonstrative 'that person' seems potentially as unambiguous as anything, so it's hard to see how Tye can avoid being committed to the indeterminacy of "that person = Adam" as asserted at some stage of the transformative process.
Now, Tye wants to hold that what's indeterminate about persons is never their identities, but merely "where [their] temporal boundaries... lie." (p.162) His reliance on four-dimensionalist perdurantism is further brought out in the footnotes:
Existence at a time is not the same as existence simpliciter. Existence at a time t is a property (expressible in the predicate 'x exists at t'), and, as the case of Fred [a dissipating cloud] shows, it does indeed admit of borderline cases. But it does not follow from this that it is indeterminate whether Fred exists, period. After all, there is an object that 'Fred' denotes, namely, Fred. So there is no indeterminacy either in whether Fred is identical to one of the things that exist. (pp.184-5)
This is terribly unsatisfying, since Tye doesn't go on to address the obvious response, which is to ask whether Fred is identical to any thing that exists at time t. Or, more generally, if it is indeterminate whether S has property P, then it appears to be likewise indeterminate whether S identical to any existing object that exemplifies P. Right?
I'm not sure how he can deal with the latter problem. But let's go back to my earlier "Adam and Bertha" objection, and I'll offer a 4-D response on Tye's behalf. Persons are temporally extended objects, rather than being wholly present at a time. So the demonstrative "that person" is less clear than initially thought. We can point to a present person-stage, but it may be unclear what non-present stages are also to be included as parts of the ostended person. However, one thing is clear: the denoted person definitely possesses the present person-stage. But (we may suppose) it is indeterminate whether Adam extends to this stage. So (by Leibniz's Law) the denoted person is not Adam.
According to this response, there's no vague identity here. There's definite non-identity, and what's vague is simply whether the different people might overlap in their temporal parts. Tye is thus committed to the possibility of one body constituting multiple people -- which he already accepts in case of Multiple Personality Disorder (pp.142-3), but seems a bit odd in this context.