Let’s start by presenting a Williamson-style argument in a slightly different way.
On an interpretative truth-theory for a language L, as we said, we’ll have a clause for a monadic L-predicate P along the lines of ‘for all o, P is true-of o iff Fo‘. But we are now in the business of imagining running through various different possible interpretations for P, which will result in clauses in definitions of different true-of relations, i.e. different relations ….. is true-of ….. on interpretation I. Now, it might well on the one hand seem that we needn’t think of the different interpretations that are in play here as ‘objects’ (whatever exactly that means). But, on the other hand, we might reasonably suppose that the different true-of relations could at least be indexed by some suitably big collection of objects (some class of numbers perhaps, or more generously some sets, for example).
So the clause in a definition for an indexed true-of relation true-ofα will be given in the form ‘for all o, P is true-ofα o iff Fo‘. But now, since the indexing objects are by hypothesis kosher objects, we can unproblematically define a property R which is had by an object o just in case o is an indexing object and not-(P is true-ofo o).
We can then ask: is there an index κ such that for all o, P is true-ofκ o iff Ro? The familiar argument shows that there can be no such index κ (assuming, that is, that κ falls into the range of the universal quantifier ‘for all o‘). But what should we conclude from that?
Well, we could conclude that, after all, the universal quantifier somehow manages to miss including the object κ in its range. But that is hardly the most natural lesson to draw! Rather, the natural moral is a Tarskian hierarchical one, that given some truth-predicates true-ofκ, we can ‘diagonalize out’ and define another truth-predicate which is not one of them.
Now, Parsons almost makes the point. But, what he actually says is that, if you resist the idea that the universal quantifier must fail to cover absolutely everything, then this “forces us to take the Tarskian view now about the predicate ‘P is true of x according to I‘. That amounts again to saying that we have determinate quantification over absolutely all interpretations but do not have an equally general notion of truth under an interpretation.” (Which Parsons suggests is a troublesome line for the believer in absolutely general quantification to manage.) But in fact that doesn’t seem quite right. For there is, we are supposing, a determinate quantification hereabouts, but we are not required to think of it as a being over ‘all interpretations’, so much as being over all the objects that index some initial bunch of interpretations. The claim, however, is that we can always diagonalize out and define a further interpretation.
And now the question arises why, in this setting when we are generalizing about Davidsonian interpretations, we can’t echo the line that Parsons took about one-off interpretations. He said, you’ll recall, that (in the case of unrelativized truth-theories) ‘true of’ had better not be in the language being interpreted on pain of paradox. So, as he put it, “the interpretation does require ‘ideology’ not present in the language interpreted, but it does not require an expansion of ontology”. Now we are going up a level and talking about different definitions of ‘true of’ on different possible interpretations. And again on pain of paradox there will be a ‘true of’ that isn’t already among those different definitions. But why can’t we say again, “this new interpretation does require ‘ideology’ not already present, but it does not require an expansion of ontology”?
So in the end, I’m not sure that Parsons has firmly put his finger on a problem for the defender of absolute quantification, or at any rate a problem that comes from ideas about ‘interpretation’.
Let me add just a quick footnote harking back to Linnebo’s paper which we skipped over. One thing I did note was that he takes the strongest response to the Williamson line of argument to be a type-theoretic one — but Linnebo goes on to discuss a simple theory of types, and in a way this seems now to be going off in the wrong direction. For what we have just seen, in the case of a hierarchy of true-of relations is a ramification into levels and it is that which is doing the paradox-avoiding. But I’ll try to return to this observation.