### Yablo's paradox: Bueno & Colyvan vs. Ketland

I am looking at the debate between Jeff Ketland on one hand and Bueno & Colyvan on the other, pertaining to the consistency of Yablo sentences. (See HERE for a wider list of references compiled by Jeff. While I think B&C have a few brilliant remarks, I also think their arguments are sometimes unsatisfactory, and I want to comment on these points. This is all rather sketchy, and presumably half-baked.

1. The provability of the existence of the Yablo sequence.

B&C complain:
"it's not clear...how we know that the Yablo list exists. Priest's argument seems to presuppose the existence of the list, in order to establish that to derive a contradiction from the latter, a fixed-point construction is required." (Analysis 63, 156).
For me, it seems like a very unusal approach to the diagonal lemma (DL). DL is an existence claim: it says that given a certain formula, another formula satisfying a certain equivalence exists. My impression is that DL is used by Priest to prove the existence of Yablo sentences, and the fact that certain equivalences hold for the formulas (part of the application of DL under consideration) whose existence has been thus proved is used in the derivation of a contradiction. But let's move on.

Ketland (Analysis 64,166-167) says:
"...if one extends the language L of arithmetic by adding a 1-place prediate symbol T to form a language LT, the uniform diagonalization theorem shows that there is an LT-formula Yx() such that the Uniform Fixed-Point Yablo Principle is provable in PA."
Some notational details aside, he means that PA proves
(x)[Y(x) <-> (y>x)~T(Y(y))]
He continues:
"To stress, it is a theorem of mathematical logic that the Yablo list exists."
B&C (in an unpublished (?) ms "Yablo's paradox rides again", available HERE)  pick on this sentence by saying:
"...what is Ketland taking mathematical logic to be? Given the context, and his argument that the existence...follows from PA supplemented with a truth predicate, presumably he considers that combination to amount to mathematical logic. But, clearly, this combination is quite a bit more  than mathematical logic - unless one assumes, without argument, the truth of logicism!"
Okay, sure, Ketland was probably a bit hasty when he said the existence of Yablo sequence is a theorem of logic: for it is conditional upon PA extended with a truth predicate. (Although, this of course hinges on what you mean by mathematical logic). Anyway, his main point holds: the existence of the Yablo sequence is not assumed without argument, it is proven within a relatively uncontroversial theory which no side of the debate wants to reject (and indeed if you look at the quote I gave, Ketland's statement is conditional).

B&C continue: "Still, if the list is a theorem of Peano arithmetic suppelemented with a truth predicate, isn't that enough? Not really." The problem is that they didn't say what it is not enough FOR. I think it is definitely enough to respond to their qualm about the existence of the sequence.

To be fair, we find out what it is not supposed to be sufficient for soon enough. B&C, on the assumption that the use of DL is a mark of self-reference, point out that using DL for proving the existence of the sequence doesn't show the sequence to be self-referential, at least unless you also show that any way of specifying the sequence has to use a fixed point construction. (They move on to sketching a way of specifying the sequence without a fixed point; evaluate it on your own).

2. The consistency of the Yablo sequence.

One of Ketland's main claims is that the sequence isn't really inconsistent: rather, it's omega-inconsistent (for a proof see his Sythese paper from 2005, vol. 145, 295-302). B&C attack this claim. They focus on an argument given by Ketland earlier (Analysis 64, 165-172), where he relied on the compactness theorem and used the claim that every finite subset of Yablo sentences is consistent. They oppose:
"There is a very interesting mistake in this argument, though. While it is true that each finite subset of Yablo sentences is not paradoxical, it is not true that each subset is satisfiable. Consider, for example, the subset consisting of just the first two sentences ... Since there is no s_k for k>2 in this set, s_2 is vacuously true. But this means that s_1 is straightforwardly false. That is, [the set] is not satisfiable. [but it] is not paradoxical ...: there is a consistent valuation function." (B&C, "Yablo's paradox rides again")
I'm not convinced. The fact that no S_k with k>2 is in the set doesn't make s_2 vacuously true: such s_{k}s still exist (well, we can prove that on uncontroversial assumptions, see point 1), only NOT IN THIS SET. Just like the fact that you're temporarily considering the set of even numbers doesn't make the odd numbers come out of existence.

3. The inconsistency of Yablo sequence.

B&C still claim that the sequence is inconsistent. They start with assuming each instance of a Yablo biconditional:
Y_i <-> (m>i) ~T(Y_m)
and each instance of the local disquotation (for i in omega):
T(Y_i) <-> Y_i
This gives them each instance of:
T(Y_i)<-> (m>i)~T(Y_m)
Then, the crucial move occurs: they say that "since each instance was arbitrary", the uniform homogeneus Yablo Principle is derivable:
(n)[T(Y_n)<-> (m>n)~T(Y_m)]
And this one is clearly inconsistent (see Ketland 2005 for a proof).

But here is a problem: the uniform homogeneus Yablo principle doesn't follow directly from the local instances. It follows on the assumption that the model of arithmetic you're looking at is standard (and indeed, the instances weren't completely arbitrary: they were restricted to standard numerals). But this is exactly what Ketland's 2005 consistency proof hinges on: the Yablo sequence can be satisfied in a non-standard model!

4. Non-standard numbers

In their 2003 (alleged) derivation of contradiction B&C move from the untruth of s_1 to the truth of some sentence s_k with k>1. In his 2004 response Ketland claims from the untruth of s_1 that it doesn't deductively follow that "there is at least one true sentence in the Yablo list" (2004:171). B&C insist that the negation of s_1 simply is equivalent to the claim that for some k>1 s_k is true (and they're right), from which they infer that there is a true sentence on the Yablo list. B&C rejoin (ms):
"if Ketland is right, and k might stand for a non-standard natural number, so what? The nonstandard models of the natural numbers are also well-ordered, and that's all that matters here. There will still be a first true sentence in the list independent of whether the index ranges over standard or non-standard numbers. And our argument goes through either way."
Well, Ketland's response is slightly misleading, for it doesn't emphasize the (possible) difference between being on the Yablo list, and being a Yablo sentence. On one interpretation (interpretation A) the Yablo list is the list of Yablo sentences obtained for natural numbers as indices. In another (B) the Yablo list is just the set of all Yablo sentences, ordered by the less-then relation, no matter whether the numbers involved are standard or not.

Now, on interpretation (A) Ketland is right: just because for some k>1 s_k is true, it doesn't follow that some sentence on Yablo list is true, because k, the witness, might be non-standard.

But on interpetation (B), Ketland's response is inadequate, and B&C's point partially holds. It does follow that for some k>1 s_k is true.

This doesn't mean they can derive a contradiction, though! In their reasoning they relied on the local Yablo disquotation principle which for n in omega says:
T(Y_n) <-> Y_n
The thing is, if k is non-standard, the local disquotation principle doesn't apply to it, and this blocks the inference.

Of course, in response you might start with the uniform Yablo disquotation principle:
(x)[T(Y(x)) <-> Y(x)]
and this will allow you to deduce a contradiction (Ketland 2005 admits that much).

Anyway, Ketland's main point still holds: the sequence BY ITSELF is consistent, as long as you don't use uniform Yablo fixed-point principle, or uniform Yablo disquotation principle. Whether we should go for the uniform or only for the local is a separate question I don't intend to tackle here. 