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Wednesday, August 11, 2010

Games and The Liar Paradox

Over at Blog & ~Blog, Ben says that the sentence "this sentence is false" (which I will refer to as P), and similar sentences, are meaningless. Ben says that the Liar Paradox (which occurs whenever we try to decide whether P is true or false) disappears once we accept that P is meaningless. I'm not convinced, which is not to say I think the Liar Paradox poses a real problem. I just prefer a different approach.

Ben's view is that the predicate "is true" does not add any content to a sentence, and therefore, a sentence which only has "is true" as its predicate cannot be meaningful. While it may be true that "'Snow is white' is true" means the same as "Snow is white," this analysis (called disquotationalism) does not clearly apply to all cases where "is true" is the predicate of a sentence. I think it only applies to cases where "is true" is predicated of a sentence. Thus, we may find semantic equivalence between "This sentence is true" and "'This sentence is true' is true." I see no reason to claim that either of these sentences are meaningless.

(As an aside, I wonder about disquotationalism in general. We might say that "Colorless green ideas sleep furiously" is a meaningless sentence, yet, "'Colorless green ideas sleep furiously' is true" might be meaningful. It might be the expression of a false belief.)

In fact, there are situations in which P is clearly meaningful. Consider pointing to a sentence which reads, "the earth is five days old," and saying P to a colleague. That would be a meaningful statement. So P can be used to make meaningful statements. We need some reason to think the relevant uses of P are meaningless.

We might say that sentences cannot state their own truth or falsity; however, there is no clear motivation for this view. If we just say, "you can't do that!", then we are winning by fiat. Alternatively, we might say that sentences cannot be used to make statements about themselves. However, "this is an example of a sentence in English" seems meaningful even when self-referential. None of these approaches seem to work.

To get at my alternative approach, first consider writing the Liar Paradox this way: On one side of a piece of paper, write "the sentence on the other side of this paper is true," and on the other side, "the sentence on the other side of this paper is false." For another interesting variation, we could write "the sentence on the other side of this paper is false" on both sides. In this variation, there might not obviously be a paradox at first; we might just think that the first side we look at is true and the second side is false. The second side we look at tells us that the first sentence is false, but since we believe the second sentence itself is false, we are only affirmed in our original belief that the first sentence is true. This is only a problem when we realize that we could have looked at the other side first and come to the opposite conclusion about which was true and which was false. Now we have a paradox.

These variations on the Liar Paradox are reminiscent of a trick children sometimes play on each other. They give a friend a piece of paper with the words "read the other side" written on both sides. The way adults respond to the Liar Paradox is much the way children respond to the "read the other side" game: They find it amusing and play with it a little, until they realize how it works and lose interest in repeating the same procedure over and over again.

The Liar Paradox is a game in which we look for a meaningful statement which might be true or false, but we never find one. At no point are the sentences we considering meaningless, however, because we understand them just as the game requires.

We might suppose that it is not the sentences, but the game itself, which is meaningless. Perhaps the game is meaningless in the sense that it is of no consequence to anything outside of itself. The sentences in the game do not refer to anything outside of the game. This is part of learning the game: realizing that there is nothing to make the sentences true or false apart from the play of the game. The only way to win the game is to figure out the rules of the game, at which point there is no reason to play. However, the fact that we can learn the rules suggests that the game is meaningful. The trick works. The sentences direct our behavior in intended ways, leading us consistently towards conflicting notions of truth and falsity.

The logic of the game is easy enough to understand: To judge that the statement is true, we must judge that a negation of the statement is also true, and this judgment cannot be made without negating that negation, ad infinitum. The only way to end the series is to stop playing. The sequential negations can occur by trying to evaluate a single, self-negating sentence, or with a pair of sentences, or through any arbitrarily large series of sentences. The logic seems obvious. What is more interesting, perhaps, is why we enter the game in the first place. We enter because we want to find out the truth value of a statement, just as the child turns over the card to find out what they are supposed to read.

[Updated Jan. 11, 2011, 23:08 GMT: As has been pointed out in the comments section, I need to more clearly spell out my answer to this paradox. My answer is that (1) sentences themselves are never true or false, but can be used to make true or false statements; (2) not every utterance of a sentence is a true or false statement; and (3) the liar's paradox is a game in which we think we are making a true or false statement, but only find ourselves saying sentences which the rules of the game do not permit. We never say anything true or false while we're playing this game. We're trying to, but the game won't let us.]

(I may not be the first person to come up with this approach, but there isn't anything quite like it mentioned in the Internet Encyclopedia of Philosophy entry. P. F. Strawson's approach is very similar, though. There are obvious parallels between my approach and Wittgenstein's philosophy, as well.)

Updated again on Jan. 21, 2011, 10:40 GMT to clarify some issues raised by Martin in the comments section.