logic

Dialetheic Solutions to the Liar Paradox

How can something be both true and false at the same time? What are dialetheic solutions? How do classical and non-classical logic differ? These and other questions are answered by Professor Emeritus of History and Philosophy of Logic in the Arché Research Centre for Logic, Language, Metaphysics and Epistemology, Stephen Read.

So I want to talk about dialetheic solutions to the Liar paradox. First of all, I need to tell you about the Liar paradox and then I need to explain this funny word ‘dialetheic’ which has been invented recently. So the Liar paradox is the paradox that arises when someone tries to say ‘I’m lying in this very utterance’ or ‘This utterance is false’. It seems rather strange, and why someone might even want to do this? But if they do, then it’s going to undermine our theory of truth or so it would seem. The argument goes like this: supposing what I say is true when I say that what I say is false. Then if it’s true, then surely that’s how things are and I’ve said that it’s false, so if it’s true, it’s false. But that means, assuming it’s true, it’s both true and false; surely nothing could be both true and false. So the assumption must be wrong: it can’t be true, it must be false. However, if we conclude that what I was saying was false, I was actually saying that what I was saying was false, then that’s how things are, so I must have been saying something true. So it seems we have an easy proof that what I’m saying now is both true and false. That’s a contradiction. It seems that nothing can be both true and false, everything is either true or false. One of those principles is called ‘the law of bivalence’: there are two truth values, and every sentence has one or other truth value.

So if I admit that the sentence I uttered was both true and not true, I also seem to be committed to say that the moon is made of green cheese and any other proposition you come up with.

The principle right at the end there that I used, ‘Either this sentence is true or the moon is made of green cheese and it’s not true, therefore the moon may be made of green cheese’, that generalizes to the general form ‘either P or Q and not P therefore Q’, that’s a standard principle of logic which is called ‘disjunctive syllogism’: you’re moving from a disjunction. And another premise to the conclusion from the denial of one of the disjuncts to the other disjunct as a conclusion of disjunctive syllogism.

A number of people have tried to give a solution to the Liar paradox by embracing contradictions but without being committed to any conclusion whatsoever. That’s where the word ‘dialetheic’ comes in: a dialetheia is any sentence which is both true and not true; ‘dialetheia’ from the Greek means ‘two truth values’: it’s got both true and not true.

So we’re looking to see, could we embrace the dialetheic solutions to the Liar paradox but without this further implication that we would then be committed to any conclusion whatsoever? The way to do it, they say, is to reject disjunctive syllogism, reject disjunctive syllogism as a means of argument.

We can see how they’re going to do it in a way because if they think that propositions can have both true values, they can be both true and not true, then we can see that the premises of the disjunctive syllogism argument are true but the conclusion is false. And an inference that has true premises and a false conclusion is obviously not a principle that we want to have in our logic. So if we take ‘P or Q and not P’, now suppose P is both true and false; then if P is both true and false then ‘P or Q’ is both true and false whatever Q is. But if P is both true and false, then ‘not P’ is also both true and false, so ‘P or Q’ and ‘not P’ are both true and false but if that’s true and the conclusion Q in general ‘the moon is made of green cheese’ is false, so if we accepted the principle of disjunctive syllogism we could be committed to moving from true premises to a false conclusion.

And there’s a funny sense in which the premises are true, because they’re not just true, they’re also false, but the point is you’re moving from truth to false and that’s a bad move to make in logic. So now it looks as if we’ve got a viable solution to the Liar paradox if we’re willing to revise our logic, go for what’s called a non-classical logic in which we reject the principle of disjunctive syllogism.