## Monday, October 11, 2010

### A Test Question I Just Wrote

5. When one is thinking about the more basic form of the Liar Paradox, called the Simple Liar—“This sentence is false”—one might think that the contradiction can be avoided by saying that the sentence is “neither true nor false.” The problem is that, when one tries to apply this solution to the version of the Liar Paradox called the Strengthened Liar—“This sentence is not true”—it generates a contradiction. How?

a. It just does.

b. If the sentence “this sentence is not true” is neither true nor false, then it’s not true, which is what it says of itself, and if what it says of itself is right, it’s true. Thus, if it’s neither true nor false, it’s both true and not true.

c. If you say that it is neither true nor false, the corner of the page on which it is written begins to smolder and burn, and deep, ominous laughter can be heard in the background.

Colin said...

Most people don't realize that (C) is the problem.

Glowing Face Man said...

"if what it says of itself is right, it’s true"

Most consistent theories of truth don't have this property.

When we informally say "this sentence is not true", formally we've used Godel's fixed point lemma to find a phi such that (say) Peano arithmetic proves "phi iff ~T(phi)". Then, if phi isn't true, i.e. if ~T(phi), then phi logically follows from this together with PA. But just because phi logically follows, doesn't mean T(phi) logically follows.

The point being, "it's neither true nor false" is a perfectly legitimate way to explain the paradox. One can build a concrete model (using e.g. Kripke's construction) where this sentence in question is indeed neither true nor false.

Ben said...

Colin,

Indeed.

Glowing Face Man,

Well, of course, the test question is asked at a fairly simple level. The sort of sophisticated formal work-arounds you're talking about are way beyond the level of what we've talked about the paradoxes in class, and I would hope that any student who was aware of the existence of such work-arounds would still take the question in he spirit in which it's asked--like, "in terms of basic intuitive Liar reasoning, what's the standard 'you still get a contradiction' objection to this standard intuitive move?" (It ought to help, too, that the instructions say to "pick the best answer.")

For what it's worth, though, on the philosophical substance....

"But just because phi logically follows, doesn't mean T(phi) logically follows."

....is where you and I part ways. That's more than I'm willing to swallow, for the sake of consistency or anything else.

Fortunately, I think consistency can be saved even within the confines of a "naive" theory of truth in which there's no possibility of separating any phi from any T(phi)... See http://blogandnot-blog.blogspot.com/2010/08/my-take-on-liar-paradox-part-i-of-iv.html and the follow-up posts.