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.

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## 4 comments:

couldn't the same argument be run for '~' in a bivalent logic? I mean, that IS the liar paradox. Consider (ii):

ii) the propositions expressed by (ii) is false

If (ii) is true then it is false. If it is false then it is true.

In a multi-valued logic it would be easy enough just to grant (ii) and indefinite status. But assuming bivalence, the problem stands. My point being, in a bivalent system the liar paradox arises with the addition of '~', in 3 valued logic it arises with the addition of 'Δ' (as you rightly pointed out). If you reject the 'Δ' operator on these grounds in 3 valued logic, I think you owe some motivation for allowing '~' in two valued logic.

One response could be that two valued semantics doesn't accurately model the world (or language usage, or whatever logic is supposed to do) which is why we had to revert to multi-valued logic in the first place. To that one could say (sort of tongue in cheek) that even so, your argument didn't show that the definitely operator was poorly defined, rather that multi-valued logic is a poor model as well.

On what is probably a more contentious note, I'd like to point out what exactly the dilema is here. Let's sketch Williamson's semantics for 'Δ' here.

P ΔP

T T

I F

F F

That looks perfectly fine as a definition of the 'Δ' operator. That combined with:

i) ‘The proposition expressed by (i) is indefinite.’

Leads to semantic paradox. Which should we reject as ill-formed and banish to the meta-language? Expelling either one on non ad-hoc grounds would be a tricky matter.

Awesome post Dan. I realized the '~' analogy when I posted, but I think that I can now better diagnose the problem (or lack thereof). The problem is that as I had phrased the argument against Williamson I had assumed that he was explicitly using a truth *predicate*. We evaluate the truth value of a predicate by looking at its extension. In classical logic this is just the set of true things and so we have it that the liar-sentence is a member of the set iff the liar-sentence is not a member of the set. A Russell’s paradox if ever there was one. The case is roughly analogous for the logics we are considering where extensions are defined in terms of functions rather than being explicitly set theoretically defined. However, operators like ~ and Δ work a little bit differently. To find out the truth values of sentences with truth-functional operators appended, we look at the thing to which the operator is attached, find out the value that the valuation function assigns it, and then look to the truth rules for that operator. NO PARADOX ARRISES AT THIS LEVEL. I hope this clarifies where my post went wrong.

Given the above, Williamson only needs to reject my formulation of (7) in order to save his argument.

I should add that I believe Justin endorsed the original argument as a result of being misled by me.

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