Friday, March 17, 2017

Some paradoxes of reference


Liar-like:
  • one plus the biggest integer that can be expressed in English in fewer than fifty words
  • one; two; three; one plus the biggest integer mentioned in this list
  • one; two; three; one plus the last integer mentioned in this list
  • one plus the last integer mentioned in this list; two; three; one plus the first integer mentioned in this list
  • one plus this integer
Truthteller-like:
  • one; two; three; the biggest integer mentioned in this list
  • one; two; three; the last integer mentioned in this list
  • the last integer mentioned in this list; two; three; the first integer mentioned in this list
  • this integer
  • the square of this integer
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3 comments:

  1. Christopher MichaelMarch 19, 2017 at 6:13 AM

    Why think that "one plus this integer" refers to an integer (or, equivalently, at all)? The Liar seems to work because "This sentence is false," at least has all the necessary syntactic features of a sentence. But "one plus this integer" does not have the syntactic features of an integer, so why think that it refers at all? Does the phrase "a house slightly larger than this one" refer, absent ostentation?

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  2. Alexander R PrussMarch 20, 2017 at 8:20 AM

    Yeah, you're right that this particular version is less compelling than the old unstrengthened liar. That said, I think the answer you offer is basically the same as the standard "nonsense" answer to the liar. In both cases, the answer is blocked by strengthened versions--in my case the earlier paradoxes in the post.

    A more explicit strengthening is something like this:
    One plus the referent of this expression if this expression refers to an integer, else zero.

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  3. Alexander R PrussMarch 20, 2017 at 8:25 AM

    There is a house version, too:
    The next bigger house than the one referred to by this expression if this expression refers to a non-largest house, else my house.

    I assume that actual houses are linearly ordered by size. (In the macroscopic world there are no exact ties.)

    Either the expression refers to a non-largest house, or not. If it refers to a non-largest house, then it refers to a house bigger than the one it refers to, contradiction. If it doesn't refer to a non-largest house, then it refers to my house, which is a non-largest house, contradiction.

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