I always assumed that there could be no contradictions -- that the principle of non-contradiction was absolute, so to say. Recently, however, I read about dialetheism and paraconsistent logic and realized that some philosophers disagreed. It seems all of logic falls apart if contradictions are permitted. I fail to understand how their position makes any sense (which could admittedly be just a failure on my part). So is it possible someone could better explain their viewpoint? Surely none of them believe that, say, one could simultaneously open and close a book, right?

It's not so much that some logicians believe that there are no contradictions as it is that there are different ways of dealing with them. There are different kinds of paraconsistent logics. Many (e.g., "relevance logics") got their start by trying to handle the so-called paradoxes of the material conditional (e.g., that from a contradiction, anything can be derived). There are also situations in which it makes sense to allow for propositions that can be both true and false as well as propositions that are neither true nor false, in addition to ones that are either true or else false (see Belnap's paper, cited below). (Just for a quick example: "This sentence is false" might be both true and false, whereas "Colorless green ideas sleep furiously" might be neither true nor false.) In artificial intelligence, there have been applications of relevance logics to deductive knowledge bases (see the Martins & Shapiro paper, cited below): Suppose you have a deductive knowledge base and that person A tells it a proposition P, person B tells it a proposition R that implies not-P, and person C asks it a question Q. If the knowledge base used classical logic, then it might tell C that Q is true and it might tell person D that Q is false; after all, anything follows from a contradiction, and the system contains a contradiction, namely P and not-P. What some such knowledge bases do is "contain" the contradiction, so that when C asks whether Q, the knowledge base will tell it that answering that involves an inference from a contradiction, and the system will ask the user which of the contradictory propositions it wants to remove in order to make the knowledge base consistent.

For more information, see:

Belnap, Nuel (1992), "A useful four-valued logic: How a computer should think", in A.R. Anderson et al. (eds.), Entailment (Princeton University Press), Vol. 2.

Mares, Edwin, "Relevance Logic", The Stanford Encyclopedia of Philosophy (Spring 2009 Edition), Edward N. Zalta (ed.).

J. P. Martins and S. C. Shapiro. A model for belief revision. Artificial Intelligence, 35(1):25-79, 1988.

Priest, Graham and Koji Tanaka, "Paraconsistent Logic", The Stanford Encyclopedia of Philosophy (Spring 2009 Edition), Edward N. Zalta (ed.).

Those who believe that there are contradictions which are true don't think that all contradictions are true. They don't think that "ordinary" contradictions like "the book is fully open and the book is fully closed" can be true. It is only special cases, like Liar propositions and other paradoxical propositions, and perhaps some others, that are claimed to be true and false at the same time. ['Fully open' vs 'fully closed' here, for the reasons that Richard gives in his next posting!]

So far as I know, no "dialethists" believe that all contradictions are true. But there is a significant disagreement about whether it's just weird cases, like the liar, that give rise to contradictions, or whether there might be contradictions that are in some sense observable. Graham Priest thinks there are; moderates like J.C. Beall think there aren't.

The case of simultaneously opening and closing a book leads naturally to issues about vagueness. It's natural to think that it's vague whether a book is closed. Take an obviously closed book and then "open" it a nanometer. Surely a nanometer can't make a difference to whether the book is closed, can it? (If you think it can, try a picometer. Or something even smaller.) But then, lots of nanometers add up to a centimeter, which surely can make a difference. So, a dialethist might say, if we take a "borderline case", that will be a case where the book is both open and closed. (If you're inclined instead to say that it's a case where it's neither open nor closed, then consider the fact that that is also a case whether it's neither not closed nor not open.) So the person who got it into that state may both have opened it and closed it.

If you think that's nuts, then you're in the majority, for sure. But there are other applications of paraconsistent logic, as William mentioned. One can think of paraconsistent logic as explaining not so much what follows from what, so that contradictions might be true, but rather as describing how one should reason in the presence of possibly contradictory beliefs, which, probably, we all have.

Read another response by William Rapaport, Peter Smith, Richard Heck
Read another response about Logic