Proving the Law of Non-contradiction?
The logical Law of Non-contradiction – that a proposition cannot be both true and false – enjoys a special, perhaps uniquely privileged, status in philosophy. Most philosophers think that finding a contradiction – the assertion of both P and not-P – in one's reasoning is the best possible evidence that something has gone wrong, the ultimate refutation of a position. But why should this be so? What reason do we have to believe it?
Well, what kinds of reasons could there be? Is there a convincing logically deductive proof for the Law of Non-contradiction? Maybe, but I'm not aware of any. And this is not surprising, because a convincing deductive proof, if there were such a thing, would be a sound argument that proceeded from premises that were more basic or obvious than the Law itself, and it is hard to imagine anything more basic or more obvious than that! Maybe there are inductive proofs, then, on a par with ‘All the swans I've seen so far have been white, so probably the next swan I see will be white’? Again, however, I'm not aware of any good ones like this, and, again, it should not be surprising. Such inductive proofs are only as strong as their evidential bases are representative: the more white swans we've observed (out of all the swans there are), the more probable it is that the next one will be white. But the Law of Non-contradiction applies to infinitely many propositions, so, no matter how many propositions we sample that are either true or are false (but not both), we will only ever have investigated a vanishingly small fraction of them. Now, the simple fact that we currently have no direct logical proof of the Law is not by itself very impressive. For all we know, some genius might discover one tomorrow. Even so, our current lack of direct, logical arguments may explain why some philosophers have offered indirect or pragmatic arguments for the Law of Non-contradiction, to the effect that we must believe the Law because any attempt to deny it somehow shoots itself in the foot, logically or practically.
An indirect proof?
This is the approach taken by Julian Baggini and Peter S. Fosl, e.g., who describe the Law of Non-contradiction as ‘a cornerstone of philosophical logic’ and ‘a principle that cannot be rationally criticized, because it is presupposed by all rationality.’Footnote 1 They claim this on the grounds that:
… any attempt to refute it presupposes it. To argue that the law of non-contradiction is false is to imply that it is not also true. In other words, the critic presupposes that what he or she is criticizing can be either true or false, but not both true and false. But this presupposition is just the law of non-contradiction itself – the same law the critic aims to refute. In other words, anyone who denies the principle of non-contradiction simultaneously affirms it. (Baggini & Fosl, 36–7)
As much as I like the good, old L of Non-C, and as reluctant as I would be to give it up, I do not think that this sort of argument works. In so saying, I am reminded of American philosopher Nicholas L. Sturgeon's suggestion that any critical response to a philosophical position can be classified as either an ‘Oh yeah?’ or a ‘So what?’Footnote 2 I'm not sure whether Sturgeon's two classifications exhaust all the possibilities – (I rather suspect he may have overlooked the important ‘Huh?’ criticism) – but I believe they apply rather nicely here.
‘Oh Yeah?’ is a way of questioning the truth of some claim; ‘So What?’ is a way of questioning its significance or supposed implications. I have my doubts about both the truth and the significance of the claim that anyone who denies the principle of non-contradiction simultaneously affirms it.
Oh yeah?
My doubts about the truth of this claim emerge when I reflect on the Law of Non-contradiction and what it means: a proposition cannot be both true and false. Obviously, this is not a particular claim about some particular proposition, P, and its negation. It is instead a general claim that, for any proposition whatever, it is not the case that both that proposition and its negation are true. Indeed, this way of putting it suggests that the Law can be understood equally well as a claim about pairs of propositions. (After all, every proposition and its negation form a tidy complementary pair.) Consider, for example:
1. 7 + 5 = 12; It is not the case that 7 + 5 = 12
2. I was born on July 19; It is not the case that I was born on July 19
3. The bird in my garden is a woodpecker; It is not the case that the bird in my garden is a woodpecker
4. Manchester United have never knowingly hired a Yeti; It is not the case that Manchester United have never knowingly hired a Yeti
5. Ulan Bator is the capital of Mongolia; It is not the case that Ulan Bator is the capital of Mongolia
6. The atomic weight of gold is approximately 196.97; It is not the case that the atomic weight of gold is approximately 196.97
7. Kemal Atatürk had an entire menagerie all called ‘Abdul’; It is not the case that Kemal Atatürk had an entire menagerie all called ‘Abdul’
… and so on, ad infinitum.
Anyway, in terms of such complementary pairs, the Law of Non-contradiction says that there is no pair both of whose members are true. The denial of the Law of Non-contradiction, on the other hand, says that there is at least one complementary propositional pair both of whose members are true. Which pair? It's hard to say in advance. Maybe it will be pair #85 (which is some proposition about the quantum mechanical properties of some particular sub-atomic particle), or maybe it will be pair #3782 (about the nature of God), or maybe even pair #111,064 (about Bertrand Russell's mathematical set-theory). Presumably, the opponent of the Law of Non-contradiction has some idea about which pair it is, or at least the general neighbourhood of subject matter in which it resides (otherwise it is hard to see what grounds they would have for denying the Law). Whichever pair it is, however, it need not be:
8. There is no complementary propositional pair such that both members of that pair are true; It is not the case that there is no complementary propositional pair such that both members of that pair are true
or even
9. The Law of Non-contradiction is true; It is not the case that the Law of Non-contradiction is true
That is, in the absence of further argument, there is no reason at all to suppose that one of these is the complementary propositional pair that our opponent of the Law of Non-contradiction has in mind. (It is worth noting that most thinkers who have questioned the Law of Non-contradiction have done so for reasons emerging from, say, physics, theology or set-theory – and not simple ambivalence about the Law of Non-contradiction considered in itself.) Thus, our sort of opponent holds that pairs #8 and #9 are just like pairs #1–7, in that exactly one member of the pair is true and the other is false. Regarding #9, e.g., she holds that the first member of the pair (‘The Law of Non-contradiction is true’) is false, and that the second member of the pair (‘It is not the case that the Law of Non-contradiction is true’) is true. So, to the argument that anyone who denies the principle of non-contradiction simultaneously affirms it, I say, ‘Oh yeah?’
So what?
But suppose I am wrong about this. Suppose our opponent of the Law of Non-contradiction really is committed to holding that pair #9 is one of the propositional pairs both whose members are true. What then? Well, in that case, Baggini & Fosl would be right when they say that anyone who denies the principle of non-contradiction simultaneously affirms it. But they are wrong when they say that ‘the critic presupposes that what he or she is criticizing can be either true or false, but not both true and false.’ Our opponent both affirms and denies the Law of Non-contradiction; that is she holds that it is both true and false. Baggini & Fosl may claim that the opponent thereby contradicts herself, but I say, ‘So What? What else would you expect?’
