Thursday, November 20, 2003

Yes, Virginia, there PROVABLY IS a Santa Claus

I learned a new paradox this morning in Brian Weatherson's logic class. (By "new", I mean a paradox I hadn't encountered before; in fact, I'm told it was first presented in 1942.) It's one of those fun paradoxes that arise out of fairly straight-forward self-referential ideas, along the lines of the Liar paradox ("this sentence is false" or Russell's paradox (the set consisting of all sets that are not members of themselves). I have an extraordinary number of college friends who are not strictly speaking philosophers but who are likely to be interested in the paradox (and other philosophical issues), so I'm happy to share. That's part of the reason this blog is fun. I'll present the Curry paradox here pretty much exactly as Brian did in class. It's based on this self-referential sentence: C: If this sentence (C) is true, then Santa Claus exists. Consider the following argument, where (1) is an obviously true premise:
(1) If C is true, then If C is true, then Santa Claus exists.
Or, to spell it out,
(1') If if C is true, then Santa Claus exists is true, then (if C is true, then Santa Claus exists).
Note that the parenthetical antecedent is true when C is true (after all, that's what C means), so (1) pretty clearly entails by modus ponens,
(2) If if C is true, then Santa Claus exists is true, then Santa Claus exists.
But since the italic claim in (2) above just is C, (2) is pretty clearly also equivalent to C. So let's sub it in:
(3) If C is true, then Santa Claus exists.
(3), of course, is equivalent to C. So it follows from modus ponens on (3) and (3),
(4) Santa Claus exists.
Of course, you should feel free to sub in your favorite implausible claim for "Santa Claus exists". This argument would be equally effective at proving that God exists, or that the St. Louis Rams are worthy of praise. There's a lot of big logic-y words in my explanation, but it's actually all very intuitive... if talk of modus ponens and equivalence and entailment makes your eyes glaze over, then just look at the numbered sentences -- you should be able to see that they follow logically from one another. UPDATE 11/21: I had no idea that "Santa Claus" didn't have an "e" in it. How very, very strange.

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