The Liar Paradox

More on paradoxes, and getting back to the Greek philosophers, we have Epimenides, from Crete in the sixth century BC: “All Cretans are liars.” At least that’s what he’s purported to have said, and that statement, or ones like it, have formed the basis for logic puzzles for centuries (“On an island live two tribes. Members of the Tutu tribe always tell the truth, while members of the Lolo tribe always lie. ...”).

Of course, the statement attributed to Epimenides doesn’t cause any contradiction: it’s not nailed down well enough. It’s possible that Epimenides is a liar, and that there’s another Cretan who isn’t. And, of course, a liar doesn’t have to lie all the time. But here’s the classic, bullet-proof formulation of the liar paradox:

This sentence is false.

It should be clear that if the sentence is true, then it’s true that it’s false. And if it’s false, that means it’s true. So we have a nice, tight contradiction.

The problem here is that we’re trying to fit well-defined rules onto human language, which is inherently not well defined — not well defined at all. If we try to put something like it into symbolic logic, we’ll wind up with something obviously nonsensical (something like P := ~P). In natural language, it appears to make sense, but it’s really just as nonsensical.

In other words, what the Liar Paradox points out to us is that it’s possible to write sentences in plain English that look fine, but make no sense.

It’s quite easy, in fact: George Bush has written a lot of them.