Have you ever heard of dialetheism
? (It has nothing to do with theism.)
"A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as one's favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false.
Dialetheism is the view that there are dialetheias. One can define a contradiction as a couple of sentences, one of which is the negation of the other, or as a conjunction of such sentences. Therefore, dialetheism amounts to the claim that there are true contradictions. As such, dialetheism opposes the so-called Law of Non-Contradiction (LNC) (sometimes also called the Law of Contradiction). The Law can, and has been, expressed in various ways, but the simplest and most perspicuous for our purposes is probably the following: for any A, it is impossible for both A and ¬A to be true."
In my view, the fatal logical problem with dialetheism is that if it is true, it is itself both true and false. That is, it is logically provable that a dialetheic logic of contradictions with a nonclassical semantics of negation is itself contradictory. Graham Priest, the godfather of dialetheism, is prepared to bite the bullet:
"It may be rational to accept that dialetheism is both true and false. In a sense, this is what I do accept: not only are some sentences of the form p & ~p true, but ~(p & ~p) is itself a logical truth. It is no refutation of one´s views to hold that it is false, i.e., has a true negation, if one´s view is precisely that some contradictions, in particular ones of this kind, may be true."
(Priest, Graham. Beyond the Limits of Thought.
2nd ed. Oxford: Oxford University Press, 2002. p. 275n10)
I think it's not rational but blatantly absurd to accept that dialethism is both true and false. I fail to see how it could be rational to accept a contradictory logic of contradictions.
-- Updated September 20th, 2016, 12:03 am to add the following --
Anything is possible…unless it is impossible.
When someone denies that the law of non-contradiction is true, I see no possibility of replying with an argument.
"No truth does have, and no truth could have, a true negation. Nothing is, and nothing could be, literally both true and false. This we know for certain, and a priori, and without any exception for especially perplexing subject matters. …
That may seem dogmatic. And it is: I am affirming the very thesis that Routley and Priest have called into question and—contrary to the rules of debate—I decline to defend it. Furthermore, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate."
(Lewis, David. "Logic for Equivocators." 1982. Reprinted in Papers in Philosophical Logic
, 97-110. Cambridge: Cambridge University Press, 1998. p. 101)
"I'm sorry; I decline to contribute to your proposed book about the 'debate' over the law of non-contradiction. My feeling is that since this debate instantly reaches deadlock, there's really nothing much to say about it. To conduct a debate, one needs common ground; principles in dispute cannot of course fairly be used as common ground; and in this case, the principles not in dispute are so very much less certain than non-contradiction itself that it matters little whether or not a successful defence of non-contradiction could be based on them."
(Lewis, David. "Letters to Beall and Priest." In The Law of Non-Contradiction: New Philosophical Essays
, edited by Graham Priest, JC Beall, and Bradley Armour-Garb, 176-177. Oxford: Oxford University Press, 2004. p. 176)
"As to the market: The reader thinks that the likely readership is limited to those who have the expertise and fancy true contradictions, plus a few who find the formal development interesting in its own right. I think this is quite wrong. And not just because there are many who do not fancy true contradictions but might readily be persuaded to. I think that there are many reasons for an implacable opponent of true contradictions—such as myself—to take great interest in what Priest and his allies are doing. If the book is a commercial success, as I think it might be, here is the form I think its success might take. The immediate, readymade market does indeed consist of the sympathizers, and is indeed small. After that, the book makes its own market. Some outraged defender of classical virtue (I have in mind here the very man for the part, but let me not name him) hears of this new heresy and decides to squash it once and for all—and it is apparent to all that his attempt is question-begging and worthless. Others set out to do the job properly: of course we all know that Priest is wrong, but you have to refute him this way. No, that won't work, it has to be this way… Then the difference splitters: you have to grant Priest this much but then you can't hold the line here… In short: a snow-balling, complicated debate among opponents about how the paraconsistent position might be resisted—and of course, the paraconsistent manifesto is required reading for participants in the debate. The increasingly obvious disarray of the opponents helps Priest to gather converts who themselves pitch in….
I premise this scenario on two beliefs. (1) Many people will think that it is an easy thing to refute Priest's position, decisively and in accordance with customary rules of debate. It is not an easy thing. I myself think that it is an impossible thing: so much is called into question that debate will bog down into question-begging and deadlock. (On this point, Priest disagrees with me: he thinks that shared principles of methodology might provide enough common ground.) I think this calls into question the very idea that philosophy always can and should proceed by debate—itself a heretical view, likely to be vigorously opposed. (2) Many philosophers hold an unprincipled and unstable position: they have been persuaded by Quine and Putnam that logic is in principle open to revision, they are prepared to contemplate revisions of logic that seem to them to require only small and esoteric changes, yet they still think it absurd to countenance true contradictions—some souls saved for staunch classicism, some lost to Priest, and doubtless some novel positions as well."
(Lewis, David. "Letter to the Publisher." Printed in: Graham Priest, In Contradiction: A Study of the Transconsistent.
2nd ed. Oxford: Oxford University Press, 2006. Pref. of 2nd ed., p. xix)
-- Updated September 20th, 2016, 12:09 am to add the following --
I wrote: "When someone denies that the law of non-contradiction is true, I see no possibility of replying with an argument."
Well, if it is an argument, the only possible argument I see is that this denial is absurd because the law of non-contradiction cannot be denied non-contradictorily