Thursday, February 7, 2008

The twist

Consider the Truth-Teller Paradox:

  1. This sentence is true.
You may not think this is a paradox at all. But ask yourself: is (1) true or false? No contradiction arises from either supposition, but it also should become plausible that there is no way of settling the question, and one may start to think that there just is no answer. We could stipulate a truth value, but stipulating a truth value will not give a meaning to the sentence. (I can stipulate that it is true that mimsy were the borogoves, but I have not thereby given the sentence meaning.) The sentence seems to, as it were, pull its content out of itself, and as such has no content. Now maybe you're not convinced by this—maybe you only feel a vague discomfort at (1). So now I apply the twist, by changing (1) slightly to something that is clearly paradoxical:
  1. This sentence is false.
That (2) is paradoxical is clear (it's true if and only if it is false). The twist that took us from (1) to (2) made clearer that there is something fishy about the kind of self-dependence that (1) involves.

This kind of twist can be found in other cases. One might be vaguely worried about the set of all sets that contain themselves, and then make the twist and get the clearly paradoxical set of all sets that do not contain themselves. Or one might be worried about the causal loops that time travel would permit, say one's getting the plans for the time machine from one's future self, and then after building the time machine going back in time to hand those plans to one's then-past self. There is something fishy about such a causal loop. So you give it a twist, and you turn it into a clearly paradoxical story about shooting your grandfather before his children are conceived. I think one can argue that in this same way, Thomson's Lamp Paradox is a twist on Zeno's Achilles Paradox, and the Grim Reaper Paradox is a twist on Zeno's Dichotomy.

I wonder if there is anything interesting and general one can say about the logical structure of the twist. (There may be something in the literature.) In particular, I am curious whether one can infer the impossibility of the untwisted situation from the impossibility of the twisted situation.

4 comments:

Heath White said...

I don't know if there's any general formula for doing this in all "twisted" cases, but you can say something about (1) and (2). Suppose you think (1) might have a meaning or truth value after all. Then consider:

(1) iff not not (1), so
(1) iff not (2).

But now suppose (1) is true. Then (2) is false, so (2) is true, so (1) is false. Mutatis mutandis for supposing (1) is false. Therefore, (1) is just as paradoxical as (2).

Alexander R Pruss said...

Heath,

Are you assuming that (2) is the negation of (1)? I am a bit confused.

Heath said...

Yes:

(1) This sentence is true.

Then not-(1), i.e.

It is not the case that this sentence is true

is just

(2) It is the case that this sentence is false.

That inference assumes bivalence, but anyone who thought (1) had a meaning or truth value would be committed to that.

Alexander R Pruss said...

Heath:

I don't think so. Consider:

(1) This is the first displayed sentence in the comment.

What is the negation of (1)? By your suggestion, it should be:

(2) This is not the first displayed sentence in the comment.

But (2) is not the negation of (1). In fact, both (1) and (2) are true.