Joe Lau

Paradox without self-reference?

"Everyone behind me is thinking an untruth."

Readings

  • [Required] cite:yablo93
  • [Recommended] J. C. Beall. 2001. Is Yablo's Paradox Non-circular? Analysis, 61:3, July, 176-87. doi:10.1111/1467-8284.00292
  • Graham Priest. 1997. Yablo's Paradox. Analysis, 57.4, October, 236–242.
  • Roy A. Sorensen. 1998. Yablo's Paradox and kindred infinite liars. Mind, Vol. 107, No. 425, Jan, 137-155.
  • Laurence Goldstein. 2006. Fibonacci, Yablo, and the Cassationist Approach to Paradox. Mind 2006 115: 867-890. doi:10.1093/mind/fzl867

The new paradox

Kripke's example

@(Jack) Jack is short.@

  • Self-reference is not sufficient for paradox. But it is often thought necessary.
  • Stephen Yablo thinks that it is not necessary. Consider an infinite series of sentences :

@(S1) All the sentences below from (S2) onwards are not true.
(S2) All the sentences below from (S3) onwards are not true.
(S3) All the sentences below from (S4) onwards are not true.
...
(Sn) All the sentences below from (Sn+1) onwards are not true.
...@

  • What if the sequence is finite?

Significance

  • Is self-reference necessary for paradox?
  • Separation of object language from meta-language does not avoid ungroundedness.

Any circularity?

  • Yablo says that his paradox is "not in any way circular". Is this correct?
  • Consider "Everyone behind me is lying." The speaker refers to himself.
    • But not referring to its own truth / falsity / goundedness.
    • Using eternal sentences (without indexicals and context dependence) to eliminate circularity -
      "Everyone south of position X1 is lying at time T.",
      "Everyone south of position X2 is lying at time T." ...
    • Problem: How to define this list of sentences?

Sorensen

@As a finite thinker, Yablo can only generate his infinite sequence with a quantified expression of the form

(Yn) For all k greater than n, Yk is not true.

The need for this proposition is disguised by casual presentations that merely list the first few members and then recourse to a vague "etc. and so on", or This explicit (Yn) formulation is self-referential in the sense that (Yn) uses its own location in the sequence as a reference point to specify which statements are not true i.e. the statements after (Yn).@

  • It is possible that as finite beings our finite description of the paradox must involve self-reference in order to specify the infinite list of sentences.
  • But this does not imply that the paradox itself is self-referential.

@The sentence exactly the same as this sentence except that the first ten words have been deleted.@

Beall

@Priest has shown that any description we employ to pick out (or otherwise define) a Yabloesque sequence is circular; this much Sorensen concedes. From here, however, it is a small step to the circularity of the sequence itself. We are fixing the reference of ‘Yablo’s paradox’ via (attributive) description, which means that ‘Yablo’s paradox’ denotes whatever satisfies the given reference-fixing description. The situation, however, is this: that the satisfaction conditions of our available reference-fixing descriptions require a circular satisfier – a sequence that involves circularity, self-reference, a fixed point. Given all this, it follows that the reference of ‘Yablo’s paradox’ is circular. The upshot of this is apparently missed by Sorensen; the upshot is that, unless we find some other way of fixing the reference of ‘Yablo’s paradox’, we are stuck fixing it on a circular sequence – a sequence containing fixed points, self-reference, etc.@

  1. Reference is fixed either by demonstration or description.
  2. The referent of "Yablo's paradox" is not fixed by demonstration.
  3. So it is fixed by description.
  4. All possible descriptions for fixing the referent of "Yablo's paradox" require that the referent be a circular sequence.
  5. So the referent of "Yablo's paradox" is indeed circular.

Reply

  • "All possible descriptions" vs. "all humanly possible descriptions"
  • Suppose there is a supermachine that prints out instances of Sn: ∀x≥(n+1) Sx is not true.
    • S1: ∀x≥2 Sx is not true.
    • S2: ∀x≥3 Sx is not true.
    • S3: ∀x≥4 Sx is not true. ...
  • The specification of the supermachine involves self-reference, but not the completed list. (1s for S1, 0.5s for S2, 0.25 for S3 ...)
  • Issue: Is this list well-defined?

Category.LogicAndMaths