There is a straightforward objection to any denial of the law of noncontradiction: A denial of the law of noncontradiction is self-refuting because one must presuppose the truth of the law of noncontradiction in order to deny it. That is, to deny the law of noncontradiction is to allow that it is both true and false at the same time. In other words, by affirming the statement “The law of noncontradiction is false,” one would be simultaneously conceding the denial of the same statement, and would in effect be saying “The law of noncontradiction is false…and true!”
Sadly, this simple objection, which has a pleasant tidiness to it, appears to be flawed.
The objection misunderstands what a denial of the law of noncontradiction entails. The law of non-contradiction can be formulated as follows: “Necessarily, ‘A and not-A’ is false.” (Or, put in terms of possible worlds, there is no possible world where ‘A’ and ‘not-A’ are both true at the same time.) Formulated this way, the law entails that contradictions are false in every case. A denial of this law, then, merely entails that not every instance of a contradiction is false. The objection, however, as stated above, takes a denial of the law to entail that any contradiction is true—including the contradiction between the denial itself and its negation. If that were the case, then it would be self-refuting to deny the law of noncontradiction. But to deny the law of noncontradiction is not to say that any contradiction can be true; rather it is to say that at least one contradiction is true (for example, the liar’s paradox).
Now, I think the law of noncontradiction is, in fact, true. I reject the denial. But a defense of the law, it would appear, has to be made on different grounds than that the denial is straightforwardly self-refuting.
[Many more qualifiers could and should be offered here, but, given that this is a blog post, I will excuse myself from the obligation. I should at least mention, though, that I have left it ambiguous as to how the law of noncontradiction should be understood. Is it merely a linguistic or logical claim? Or is it a metaphysical claim? How does that affect the above discussion?]
Please let me know what you think in the comments below.