bigideas

The “truth value” of a proposition is just whether it’s true or not. The truth value of “2+2 is 4″ is true, and the truth value of “2+3 is 6″ is false. The truth value of “Hilary Clinton will be the 44th president of the United States” is currently unknown; but it is certainly either true or false, and in November we’ll know which.

It’s tempting to think that all sentences of a certain kind can have a definite truth value assigned to them, but reading Roy Sorensen’s lively Vagueness and Contradiction I came across some illustrations of simple cases where this fails. One’s a familiar old favourite, but it comes with a pair of interesting variants I hadn’t seen before.

The familiar one is a variant on the “liar paradox“:

The other sentence is false.
The other sentence is true.

Certainly the first sentence can’t be true, for if it were then the second sentence would be false, which would make the first sentence false after all. And by a similar vicious circle, the first sentence can’t be false either. In the same way, the second sentence can’t be true or false. However you assign truth values to these two sentences you end up with a contradiction.

(If you’re interested, and know a little logic and set theory, John Barwise’s excellent book on the subject, Vicious Circles, is available free online, along with some other interesting things).
I’m guessing you’ve probably seen that one before, so try this one, a variant on the paradox known as the “truth-teller”:

The other sentence is true.
The other sentence is true.

They’re both true, right? Or — wait — are they both false? It seems we can assign either truth-value to the two sentences, as long as they both have the same truth-value. So is the first sentence true or false? You can’t need more information to decide — what information could possibly make the difference? (Show Me the Argument recently posted a similar paradox in syllogism form).

The sentences in the “looped liar” can’t be assigned any truth values; in this case, they can be either, and we have no way to decide which. The logician Melvin Fitting has proposed a kind of bias for falsehood to resolve this problem; the idea is that if a sentence can take either truth value, the default is false. So in the truth-teller, both sentences are in fact untrue. That solves the problem, although it might seem a bit expensive, since Fitting’s version is built from a non-classical four-valued logic.

The other example Sorensen gives is what he calls the “no-no” paradox:

The other sentence is false.
The other sentence is false.

In this case, as you can see if you try the various possibilities for yourself, one of the sentences must be true and the other must be false. But which is which? And doesn’t it seem wrong to you to say the first sentence is true and the second false even though they’re identical sentences, and you have no good reason for choosing that rather than the other assignment of truth values?

How can we avoid these paradoxes? It doesn’t do to say a sentence can’t refer to itself, which is a typical move against the liar paradox (“This sentence is false”). For a start, that rules out harmless sentences like

This sentence contains five words.

which is unproblematically true, and

This sentence contains six words.

which is just as unproblematically false. (As an aside, these remind me of Douglas Hofstadter’s delightful “This sentence no verb”). What’s more, it doesn’t even solve the problem; the sentences in the examples don’t refer to themselves but to other sentences. Not being allowed to refer to sentences (or, of course, the propositions they express) at all is much too restrictive.

Maybe we could prevent sentences from referring to truth values, but again that rules out things that are useful and cause no trouble:

All apples are oranges

The sentence above is false.

Indeed, we now have no way to say that something is true or false any more without stepping outside the language into some kind of separate metalanguage. And if you want to try to make your language rigorous, you’ll need a meta-metalanguage, and a meta-meta-metalanguage, and so on without end.