We talked about some semantic paradoxes and their attempted resolutions, which Michael McCourt will continue with next week. Pretty much everything was based on this paper. Please read it if you weren’t there — it’s not long, but it’ll take some thinking about, so probably budget 20-30 minutes. Try talking each paradox out to yourself as you go along, and reference the notes below as needed. The slideshow (follows pretty directly from the paper) is here.

The paper mentions two interesting paradoxes that it doesn’t go into much detail on, but we talked about a little more. One of them is Yablo’s paradox, which you can read about in this (single-paragraph) journal article.

The other is Curry’s paradox. Take this sentence (call it S): “If this sentence is true, then ducks are blue.” How would we go about deciding if S is true?

Well, S is a statement of if-then form; it’s saying A -> B, where A is “this sentence is true” and B is “ducks are blue.” Whenever we want to prove an if-then statement, we assume that the antecedent (A) is true and try to prove the consequent (B). For example, if we were proving “If 2x + 2 > 8, then x > 3”, we would assume 2x + 2 > 8, then reason that 2x > 6, and therefore that x > 3.

So let’s assume A is true. Then we’re assuming “this sentence is true”. That means “A -> B” is also true, since that’s what the sentence means. Therefore, B is true. We’ve found that if we assume A, then B is true; therefore, A implies B; therefore the whole sentence is true.

So “If this sentence is true, then ducks are blue” is true. The sentence is true, so ducks are blue.

The paradox there is that we can use this logic to prove any sentence whatsoever, which can’t be right — but none of the steps we used were weird; they’re all ideas that are used all the times in proofs. So what’s the problem?