In chapter 2, on pages 16-17, Williamson introduces the sentence operator Δ, read as ‘definitely’ and argues as follows:
1) ‘Δ’ has the same kind of semantics as ‘~’, both are given by a many-valued truth table.
2) If (1), then (3).
3) ‘Δ’ is no more a metalinguistic symbol than ‘~’ is.
4) ~P is about whatever P is about (which is not metalinguistic).
5) If (3)&(4), then (6).
6) ΔP is about whatever P is about (which is not metalinguistic).
7) P is indefinite iff ~ ΔP.
8) If (4)&(6)&(7), then (C).
C) Replying to the question ‘Is Mars dry?’ by saying ‘indefinite’ is not a metalinguistic response; it is a response about Mars.
I deny (1). Though it is true that the semantics of both Δ and ~ are given by a many-valued truth table, it is false that Δ has the same kind of semantics as ~. Accepting Δ into the object language leads to semantic paradox, whereas accepting ~ alone into the object language is innocuous. To see why, consider the following sentence:
i) ‘The proposition expressed by (i) is indefinite.’
Now, by (7) to assert (~ ΔP) is to assert (P is indefinite) so the sentence (i) is equivalent to ‘~ Δ the proposition expressed by (i)’.
But, on Williamson’s proposed semantics for ‘Δ’,
if (i) is true, then (i) is false;
if (i) is false, then (i) is true;
and if (i) is intermediately valued, then (i) is false.
In all three cases, whether one adheres to a bivalent semantics or one of the many-valued semantics canvassed by Williamson, having ‘Δ’ as a symbol in the object language allows one to construct paradoxical sentences that appear to be both true and not-true. The standard solution here would be to say that we can only use ‘Δ’ metalinguistically. However, this approach is not open to Williamson given his argument. Another possible save would be to accept dileathism, the view that sentences/propositions can be both true and false. But this seems like an extreme concession. He could also argue that sentences like (i) are in some sense illegitimate sentences. While this move seems initially attractive it is notoriously hard to support, as has been demonstrated in the discussion surrounding the traditional liar’s paradox. In sum, if Williamson wants to retain his point when formulating his argument in terms of 'Δ', it appears that he will have to make some serious foundational concessions.