Law of Non-Contradiction

I don't necessarily take the position that the Law of Non-Contradiction doesn't hold, but I also don't necessarily accept a common refutation of this position, which involves reductio ad absurdum. That is, when someone says "if the Law of Non-Contradiction doesn't hold, then it does hold, doesn't it?". The reason I don't accept this is two fold:

1) The refutation takes the form of reductio ad absurdum and attempts to use proof by contradiction. Essentially, the contradiction that is presented is that you are both right and wrong at the same time, meaning the argument is reduced to absurdity. However, the emergence of a contradiction does not refute the position that the Law of Non-Contradiction doesn't hold, because the very nature of this position allows for contradictions.


2) In applying proof by contradiction, the refutation already accepts that the law of non-contradiction holds in order to show that it holds. Thus, the argument is circular and so doesn't refute the original position.


I have blogged about this subject before, and am currently in the middle of a conversation about it with John Fraser (JF) on the Premier forums, which I present below the fold. It started by JF sharing this anecdote about his philosophy professor:

JF: My seminary philosophy professor (a student of Plantinga's) had a standard response when some smart-alec in the back would try to dispute the law of non-contradiction (usually with some kind of half-baked appeal to quantum theory like what you guys are doing). He would say, "well, then I'm right, aren't I?" How can you contradict that (having already denied the law of non-contradiction)?

.
.
.

Rhiggs (RH): Perhaps I'm missing something, but can someone please explain why a legitimate answer to John Fraser's oft mentioned philosophy professor can't simply be...

"Well, then you're wrong, aren't you?"

...seeing as the professor, in order to give his response, has accepted the student's proposition that the law of non-contradiction doesn't hold? That is, under the terms that the law doesn't hold, if he's wrong he's actually right (as he states). but it also means that if he's right he's actually wrong. As I see it, the back and forth could go on like this ad infinitum, without actually affiriming either position.

.
.
.

JF: Earth to Rhiggs: the point of the statement is to show that BY THE STUDENT'S OWN STANDARDS he has no way to object that the professor is wrong. Get it? It's not an admission by the professor that the law of non-contradiction doesn't hold!

This is a form of reductio ad absurdum. In a reductio argument, the one making the reductio does not have to accept the conclusion of the argument he is trying to refute - rather, he shows that it leads to an absurdity.

.
.
.

RH: I know that he is not admitting that the law of non-contradiction fails, but in order to show that the student's argument leads to absurdity, he has to show what happens when the law fails. In order to do this, his argument relies on adopting the student's position.

That is, he is really saying, "well, if the law of non-contradiction doesn't hold, then I'm right, aren't I?". But the student can just turn this back on the professor and say "well, if the law of non-contradiction doesn't hold, a position you needed to adopt in order to make your point, then you're wrong, aren't you?". The point being that the professor's argument itself is also reduced to absurdity. I'm not saying the student is correct in the first place, but that the professor's response isn't sufficient to refute it.

In addition, the professor is simply showing that if the law of non-contradiction doesn't hold, then it leads to contradictions, and thus absurdity. But of course it does - that is the very claim! By using the emergence of contradictions, the professor is using the law of non-contradiction to show that the law of non-contradiction holds, which seems circular.

.
.
.

JF: Rhiggs,

That is, he is really saying, "well, if the law of non-contradiction doesn't hold, then I'm right, aren't I?". But the student can just turn this back on the professor and say "well, if the law of non-contradiction doesn't hold, a position you needed to adopt in order to make your point, then you're wrong, aren't you?".

Well, he could say those words perhaps. The problem is, in saying this he's saying that the professor is NOT right, but he's already denied the law of non-contradiction. But in saying, "then you're wrong, aren't you?" he would be AFFIRMING the law of non-contradiction - unless he wants to say that the professor is both right and not right about the same point. So actually this would be playing exactly into the trap which the professor has set for him. He can't deny the professor's claim to be right, he could only add another claim, namely that he is not right. And then he can try to affirm that those two propositions are perfectly compatible. But that would be sheer nonsense as even the befuddled student would hopefully be forced to realize.

In addition, the professor is simply showing that if the law of non-contradiction doesn't hold, then it leads to contradictions, and thus absurdity. But of course it does - that is the very claim! By using the emergence of contradictions, the professor is using the law of non-contradiction to show that the law of non-contradiction holds, which seems circular.

The problem is much deeper than that. It means the law of non-contradiction is necessary even for the student to make any claims about the law of non-contradiction. Note that I'm not saying that this proves the existence of God (a'la presuppositionalism) - just that in denying the law of non-contradiction, you are essentially denying the ability to deny anything. And yes, that is absurd. You might as well say, "I deny the truth claim that says that a truth claim can be denied." It's as self-referentially incoherent as saying that there are no absolute truth - which is itself an absolute truth claim!

.
.
.

RH: Let me be clear here. I'm not agreeing with the student, I'm just disagreeing with the professor's refutation.

But in saying, "then you're wrong, aren't you?" he would be AFFIRMING the law of non-contradiction ...

I don't see how this is so in two different ways. If he says the professor is wrong, then he is denying the law, not affirming it, as the professor's position is that the law holds. Also, he is denying the law by the very act of claiming that the professor's position is both right and wrong at the same time.

...unless he wants to say that the professor is both right and not right about the same point

Which is exactly the point. The student could say this if the law of non-contradiction doesn't hold. He could say the same about any statement, even his own. The professor might claim that this reduces the student's argument to absurdity, but in doing so he is relying on proof by contradiction, i.e. it is absurd because you can't be both right and wrong.

This fails for two reasons:

1) Showing the emergence of a contradiction does not refute the student's position, as this is his position, that contradictions are allowed.

2) In applying proof by contradiction, the professor is already accepting that the law of non-contradiction holds in order to show that it holds. His argument is circular and so doesn't refute the student's position.

just that in denying the law of non-contradiction, you are essentially denying the ability to deny anything. And yes, that is absurd.

As I see it, by denying the law of non-contradiction, you are actually allowing for any particular thing to be both denied and affirmed at the same time, not just denied. I agree that it is absurd - in that it is of no apparent practical use to humans - but it is not refuted using the professor's method due to the reasons presented above.

Interesting topic.

.
.
.

JF: Rhiggs,

I don't think you get the point of the refutation. Sure, the student could take the position, "you're both right AND wrong, as am I, because I don't believe the law of non-contradiction holds," but then even as he says this, he is implicitly affirming that the law of non-contradiction DOES hold - otherwise he would not be able to say the law of non-contradiction DOESN'T hold. Get it?

So actually his statement would have to be modified to, "you're right and wrong just as I am right and wrong because the law of non-contradiction both does and does not hold." But hopefully the light will come on at some point and the student will realize that this is all just sheer nonsensical gibberish with no content or truth value whatsoever.

.
.
.

RH:

I don't think you get the point of the refutation.

I'm not trying to be difficult here, but you're right, I don't get it. By the student's standards, if the law doesn't hold, anything that anyone says can mean what they say and the opposite all at once. It's absurd! But I still don't see how it has been refuted. The professor is simply applying the law to refute the denial of the law, and that is just circular.

Sure, the student could take the position, "you're both right AND wrong, as am I, because I don't believe the law of non-contradiction holds," but then even as he says this, he is implicitly affirming that the law of non-contradiction DOES hold - otherwise he would not be able to say the law of non-contradiction DOESN'T hold. Get it?

No. Why does the student, by his own standards, need to implicitly affirm the law in order to deny it?

So actually his statement would have to be modified to, "you're right and wrong just as I am right and wrong because the law of non-contradiction both does and does not hold." But hopefully the light will come on at some point and the student will realize that this is all just sheer nonsensical gibberish with no content or truth value whatsoever.

You say that the student's position is gibberish because of the multiple contradictions that arise, but that still doesn't actually refute the student's position due to my points above about why the professor's refutation is flawed.

Firstly,
- The student's position is that contradictions are allowed
- The professor's refutation is that the student's position leads to a contradiction (you can't be both right and wrong at the same time)
- But this fails, because the student's position is that contradictions are allowed (if the student is right, then he most certainly can be both right and wrong at the same time)

Secondly,
- The student's position is that contradictions are allowed
- The professor's refutation relies on proof by contradiction (reductio ad absurdum), which requires the law of non-contradiction to hold
- Thus, the professor is assuming that the law holds in order to show that the law holds
- The professor's argument is circular and, therefore, is not sufficient to refute the student (pointing out a logical fallacy with the use of another logical fallacy isn't very impressive)

If you can address these two points then I might be convinced.

.
.
.

JF: Rhiggs,

If you can address these two points then I might be convinced.

You're really making me earn my paycheck on this one. Oh, wait, I do this for free. It just feels like a job sometimes.

You want me to address these "two" points - with 3 and 4 subpoints, respectively?

Probably this won't work after everything else I've already said, but basically the problem is this. You keep considering the whole thing from the standpoint that the student's position is that contradictions are allowed. Right? The problem is, as the professor shows, the student's position also leads to the conclusion that contradictions are NOT allowed. So are contradictions allowed or aren't they? You can't just say, "well, that's a contradiction but it doesn't matter because contradictions are allowed," because by his own position contradictions are NOT allowed. You have to apply the rule at the meta-level, not just the sub-level. In other words, you can't just apply the student's rule to everything below the level of the rule itself, because it also has to apply to the rule.

The point of all of this, hopefully, is that the student will see his position leads to a situation which even he doesn't find desirable. That his position amounts to nothing more than meaningless word games and makes it impossible to talk about anything coherently. He could, I suppose, just say, "why talk about anything coherently?" The problem, though, is that in saying that the law of non-contradiction does not obtain, he presumably takes himself to be saying something coherent. If he isn't, then who cares?

.
.
.

RH: As I suspected, you are still simply saying that a contradiction disproves the position that contradictions are allowed. This would be a perfectly acceptable way of disproving most positions, but not this particular one due to the very nature of what is being claimed. Put simply, proof by contradiction cannot be used to prove the law of non-contradiction, or to disprove its negation. In doing so, you are saying nothing more than it's wrong because it's wrong.

So are contradictions allowed or aren't they?

This is where you are going wrong. You can't just decide the answer to this and then subsequently apply it to the problem, because this is the very thing that is being debated. Your approach is completely circular.

You have to apply the rule at the meta-level, not just the sub-level. In other words, you can't just apply the student's rule to everything below the level of the rule itself, because it also has to apply to the rule.

That is exactly what I am doing. I am applying the position that 'contradictions are allowed' to every level in this. If the rule holds, then it both holds and doesn't hold, so any apparent disproof is not a disproof at all, because the position implies that the rule both holds and doesn't hold at the same time, and of course it also doesn't imply that! Call the position nonsense or gibberish or meaningless word games if you like, but that is irrelevant to the legitimacy of the refutation, which I still contend is flawed.

.
.
.