The Liar says, “This sign is false.” True?
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.
A Philosopher's Take 
Anonymous
March 7, 2016
Indeed, in certain paraconsistent logics “the law of non-contradiction is false…and true” will be correctly assertable.
It looks to me like the proponent of the argument you are attacking is the one question begging, not the questioner of LnC.
Interestingly, in my experience at least, most people denying LnC are doing so as a consequence of denying LEM but retaining something like De Morgan’s ‘laws’, not that attacking a logical consequence of a theory is a bad strategy.
Finally, it does appear to be a logical principle in this context, with the ontological analogue being either a special case, or a separate issue.
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Lage
March 7, 2016
But if the LNC were able to be violated, that would simply mean that “true and false” isn’t a true dichotomy and then can’t be applied to any other case where such a dichotomy is assumed. Which would make any claims involving the values of true or false invalid. Therefore, denying the LNC is in fact a self-refuting endeavor because it involves denying the true-false dichotomy that is required to claim that the LNC is false in the first place. That’s my take on it anyway…
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Anonymous
March 7, 2016
This depends on what you mean by denying the LnC. There are very few people, if any, that would hold that it doesn’t apply sometimes, or even often. The work for the denier of LnC is is in establishing where it holds and where it doesn’t.
It looks to me like you and Gordon’s opponent are assuming a principle of bivalence that is denied by many non-classical logicians — some propositions (or sentences) are neither true nor false or some are both. If LnC is one such, then LnC is not a logical law in the sense of always/necessarily being true, but that doesn’t mean that it’s always false.
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Gordon Hawkes
March 8, 2016
Thanks, Anonymous, for the wise words. I agree with you that in this context, it is a logical principle that is being discussed. [Of course, one could go on to inquire as to the ontological status of logical principles, or the status of propositions and their relations to one another, but for the purposes of this discussion, those questions can be ignored. Besides, I get the impression from my logician friends that they’d prefer to “leave that stuff to the metaphysicians.”]
You’re also right to point out that “my opponent”, as you call them below, possibly begs the question. In discussing this with a logician friend, he made similar points to the ones you’ve made, and pointed out that the LnC must be defined first in a non-question-begging way, and that any response to the denier of the LnC must be made without assuming the principle of bivalence (or the law of excluded middle?–I’m not well-versed on their relation), since, as you also pointed out, people who deny the LnC also deny the LEM (or bivalence).
Heady stuff (for us non-logicians), so thanks for your 2 cents.
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Lage
March 9, 2016
Yes, by denial, I simply mean anyone that denies it in any case.
“It looks to me like you and Gordon’s opponent are assuming a principle of bivalence that is denied by many non-classical logicians — some propositions (or sentences) are neither true nor false or some are both. If LnC is one such, then LnC is not a logical law in the sense of always/necessarily being true, but that doesn’t mean that it’s always false.
Yes this is true. I am assuming a principle of bivalence as I am in agreement with classical logicians on this point. It seems that propositions can’t technically be neither true nor false nor both, though sentences may. I know there are some subtleties regarding differences between what propositions are and other types of formulas or statements/sentences. It seems that saying that something is true and false at the same time is rather meaningless and likely a result of poor definitions and poor determinations of identities. So while for example paraconsistent logicians may discard bivalence along with LNC by also eliminating either disjunction intros, disjunctive syllogisms, or transitivity — in order to eliminate the principle of explosion while still allowing for “true” contradictions — I take this to be more of a pragmatic approach to avoiding paradoxes resulting from either vagueness and/or lack of proper definitions needed to determine proper truth values. So since I am interested in truth, and do not believe that truth (in any meaningful sense) can be ascertained without bivalence, I hold to the principle of bivalence in logic when trying to ascertain truth.
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Anonymous
September 19, 2016
I agree completely with the author. If you actually debate someone about the law of non-contradiction it is also an almost impossible task because you never know which contradictory terms they will happen to agree with. Nevertheless, it is not so straightforward to prove it, and it may not be possible at all. I still believe the law of non-contradiction to be true, but it is still a belief which other people may not assent to. They may be right, or I may be right, who knows? (but I happen to not believe we both are right).
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Gordon Hawkes
September 19, 2016
I’m inclined to take the law of noncontradiction as a first principle, following in the footsteps of Aristotle. The principle that a contradiction cannot be true strikes me as something so obvious, a principle assumed almost universally in human reasoning, with the exception of Graham Priest, that I don’t think we need be defensive about holding to the principle. That said, Graham Priest is still a brilliant philosopher and worth taking seriously.
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