[FOM] Yablo's Paradox

Jeffrey Ketland ketland at ketland.fsnet.co.uk
Fri Sep 13 15:16:35 EDT 2002

Richard Heck:

>There is indeed much controversy about whether Yablo's paradox involves
>self-reference. There was a good deal of debate about this matter in
>Analysis shortly after he published it there. Perhaps it is worth
>looking at one classical (in the sense of "standard") analysis of the
>paradox. (So far as I know, this kind of analysis has not appeared in
>print. If it has, I'd be happy to acknowledge that fact. The analysis,
>though, is independently mine.)

Thanks for that. I think there's a paper by Graham Priest in Analysis from
97-ish (?) sketching
the diagonal construction and claiming that Yablo is _therefore_
self-referential. There's a bit more discussion more recently, by JC Beall,
I think. I don't have Analysis at hand.

The argument that Yablo's paradox is self-referential is roughly that each
Yn says "Each sentence after _this one_ is false". So, although the
particular _set_ of sentences that Yn talks about doesn't contain Yn, still
the _specification_ of the set of sentences talked about is self-referential
(c.f., "Everyone behind _me_ is lying"). I'm still not sure whether this
counts as self-referential or not. I heard Roy Cook give an excellent paper
on this recently, and perhaps it'll appear in print soon.

I had a go at looking at this about two years ago, before I saw Priest's
paper. Given that theories of truth with the full T-scheme are inconsistent,
the obvious thing to do is to try and figure out what happens in some
consistent truth theory: e.g., a version of PA+truth predicate (e.g.,
PA+restricted or uniform T-scheme, or the theories known as Tr(PA) and KF).

On one approach, which doesn't use diagonalization, you add Yn's as
primitive sentence letters, and add biconditionals as axioms such that Yn
<=> "all sentences after Yn are not true". It led to an omega-inconsistency,
which seemed kind of interesting. But I can't find my write-up of this.

One the other approach, using diagonalization inside the consistent truth
theory PA+restricted T-sentences or in Tr(PA), all the Yablo sentences are
provable but (trivially) not true sentences in the language of
arithmetic. I.e., they're true in the richer structure, (N,E), where E is
the set of arithmetic truths.

I didn't check then what happened with KF. But this seems to be how it goes
for a Kripkean fixed-point (E,A). Yablo reasoning shows that each Yn is in
neither E nor A (all Yablo sentences are ungrounded). Strengthened or
revenge reasoning then suggests that each Yn should be true. Indeed, just
scribbling this out, this reasoning goes through in KF, yielding proofs of
each Yn, plus proofs of ~T([Y(n)]) (c.f., the usual strengthened liar
sentence is provable in KF). I.e., for each n, KF |- Yn and KF |- ~T([Y(n)])
I'm assuming the usual compositional KF axioms plus the axiom CONS (i.e.,
"There is no sentence which is both true and false").

--- Jeff

Jeffrey Ketland
Dept of Philosophy, Kings College,
jeffrey.ketland at kcl.ac.uk

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