Sunday, May 29, 2005

The Law of Non-Contradiction

"How do you respond to someone who denies the law of noncontradiction? Some logicians suggest hitting the person with a stick. A better idea is to pretend to agree. Whenever you assert something, also assert the opposite. Soon your opponent will want to hit you with a stick!"

That's Harry Gensler, in Formal Ethics, p.36. He then offers an amusing dialogue between two Hegelians:
A: Are you still a follower of Hegel?

B: Of course! I believe everything he wrote. Since he denied the law of noncontradiction, I deny this too. On my view, P is entirely compatible with not-P.

A: I'm a fan of Hegel myself. But he didn't deny the law of noncontradiction! You read the wrong commentators!

B: You're wrong, he did deny this! Let me get my copy of The Science of Logic.

A: Don't get so upset! You said that he did deny the law, and I said that he didn't. Aren't these compatible on your view? After all, you think that P is compatible with not-P.

B: Yes, I guess they're compatible.

A: No they aren't!

B: Yes they are!

A: Don't get so upset! You said that they are compatible, and I said that they aren't. Aren't these two compatible on your view? Recall that you think that P is compatible with not-P.

B: Yes, I guess they're compatible. I'm getting confused.

A: And you're also not getting confused, right?



  1. The law of non-contradiction says, I guess, that for any P, P is incompatible with not-P. But if you deny this law, you aren't committed to say that

    (1) for any P, P is compatible with not-P.

    Instead, you're only committed to say that

    (2) there is some P which is compatible with not-P.

    Of course, (2) _is_ really bad; I doubt anyone can sincerely believe (2). Nevertheless, (2) is clearly not as bad as (1), since (2) allows you to say (for instance) that for the vast majority of P's, P is incompatible with not-P. So you could affirm (2) and still _almost always_ follow the law of noncontradiction, as long as you thought there was an exception to the law somewhere.

    In the quoted passages, Gensler seems to miss this. For instance, Gensler seems to think that the person who has denied the law of noncontradiction has given you permission to follow the policy: "Whenever you assert something, also assert the opposite." However, this policy is not justified by (2); it requires (1). Yet (1) does not follow from the denial of the law in question.

  2. Fair point. But the denier is going to need to provide some new 'law' or account of when contradictions are disallowed. Since he denies the general law of noncontradiction, he no longer has general grounds for objecting to contradictions. So if he does so object to a specific contradiction, we might legitimately ask on what grounds he does so. Without the LNC to appeal to, I'm not sure how he could respond.

    P.S. Some people do believe (2), e.g. of the liar sentence. Some philosophers even adopted a "paraconsistent logic" which allows them to deal formally with such contradictions.

  3. It's easy to come up with paradoxes using perfectly ordinary logic. The point behind paraconsistent logics is to prevent this from causing widespread problems. Most logic breaks down under contradictions because you can prove anything from it.

    The most basic reason why you would want to hold both P and not-P is that you have good evidence for both P and not-P.

    I really believe that you can have true contradictions - in fact I believe there are many of them.

    For example, I believe you can be both inside and outside of a room.

    Paraconsistent logic merely allows the inclusion of additional or modified logical laws to cope with contradictory evidence. This seems pretty sensible to me, really.


  4. I've added a post on my blog with a list of things that I think are both true and false.


  5. I agree with MP,

    I don't care about if it is going to make logical syllogisms easier or not, it is about what we regard as true as a matter of reason.

    Just because human cognition has an inherent tendency to categorise in an 'either/or' fashion, (it's either p or it's not) doesn't mean that this is the way of the universe.

    I also think that one can deny the law of non-contradiction without supposing that it must therefore be necessary that all contradictions are true.

  6. Rejecting the law of noncontradiction is incredibly stupid. But permitting {P,~P} (both P and ~P simultaneously) seems reasonable in certain cases. A theory might prove both while not proving their conjunction--i.e. (P & ~P). A intuitive case in doxastic theories is when our beliefs are compartmentalized. In such cases we may believe both P and ~P, though not make the connection that those two beliefs are inconsistent. (This is essentially Kripke's puzzle about belief.) But it would be ridiculous to main that (P & ~P).

    Paradoxes do not justify that (P & ~P) is consistent. (E.g. in the case of the barber paradox, it simply shows that no such barber exists.) If someone has a *consistent* argument to the contrary, I would like to hear it.

  7. MP, most of your examples don't explicitly break the law of noncontradiction at all. Your examples are not explicitly of the form P & ~P (e.g. "I am inside a room and it is not the case that I am inside a room"). Rather, they are of the form P & Q, where we would normally consider that Q implies ~P, but it need not be so on loose interpretations (e.g. if you're standing in the doorway with just one leg in the room and one leg out of it).

    What you're pointing to are not true contradictions, but rather unclarity and sloppy descriptions.

    Lumpy - we might have inconsistent beliefs without realizing it, granted. They might even both be justified (separately, as you note). But surely not both those beliefs could be true.

    Illusive Mind, you're confusing the law of noncontradiction with that of the excluded middle. Denying that either P or not-P must be true, is quite a different matter from saying that both might be true simultaneously!

    "I also think that one can deny the law of non-contradiction without supposing that it must therefore be necessary that all contradictions are true."

    See my earlier response to David.

  8. Regarding sloppy descriptions, while that is fair, one might also say that under some analysis of vagueness all descriptions of a certain sort of vagueness are sloppy. The case of the bald man seems relevant where we can say he is and isn't bald because of the inherent nature of when someone is bald.

  9. "But surely not both those beliefs could be true."

    That depends. We might say that p and ~p are simultaneously true for S if

    (1)S is disposed to believe both of them under at least two different circumstances. (It wouldn't make sense to say S is disposed to believe both of them under the same circumstance.)

    (2)p is a particular type of proposition involving e.g. vague predicates, aesthetic notions, etc.

  10. When I say it is an example of either or thinking, I mean we tend to think, “either it is p or not p because it can’t both be p and not p”

    With regards to denying the LNC I can say that the law is false as it applies to Schrodinger's cat but not as it applies to whether or not an apple appears red to me. I think philosophers like laws of logic, but they should be wary of making them absolute laws. I don’t think any of them are.

  11. Illusive Mind - without LNC, on what grounds do you object to (some) contradictions?

  12. Richard, I think that's sometimes true and sometimes not true :)

    I would have said that any logical construction of "insideness" would be mutually exclusive with being outside. I am quite willing to claim that "It's true than I'm inside a circle and it's not true that I'm inside a circle".

    I certainly willing to propose that "Condemning Schapelle Corby to 20 years jail is both right and not right".

    The universe both had a beginning and did not have a beginning is obviously a P&~P.

    I think a proposition which proposes two things which seem mutually exclusive can be resolved in *two* ways. The proposal can be more tightly defined, or a paraconsistent logic can be used.

    Natural language, for example, is clearly *not* used in a sufficiently technical way as to avoid contradiction. How then do you model language?

    In software development, it is common practice to "trap errors". Paraconsistent logic seems to be a similar tool for logic - they allow you to sandbox dodgy evidence or dodgy implications without ruining all your other reasoning processes.

    I think that, for example with the inside-a-circle debate, a tighter definition only pushes the problem away one degree of separation.

    One pertinent problem is whether the universe is fundamentally discrete, or whether it is continuous. If it is continuous, you may have no genuine physical truth about whether one is inside the circle or not inside the circle.

    If a particular act displays aspects of both morality and immorality, or moral and nonmoral aspects, then both can easily be true. You would likely argue this is not a true contradiction, but Aristotle would disagree with you. Through the principle of conflict, he would try to break the act into *two* things - one of which is a moral thing, and the other which is not, because it is a contradiction to claim that something is moral and not moral at the same time.

    This breaks down in cases of infinite regress, or where some kind of dualism is proposed merely to escape the conclusion of inconsistency.



  13. If P is completely true I don't see how -P can also be even partially true without compromising P (making it less than completely true). Unfortunately the law of non-contradiction doesn't help much with questions that deal with gray areas.

  14. Of course, the law of non-contradiction "helps" with questions that: “deal with gray areas."

    The law 'helps' by telling you that if you are thinking about "gray areas" you are thinking about them; and that you are not going to be able to answer any question about said "gray area" subject matter if you try to predicate an attribute of a “non-gray area subject matter” to the “gray area” subject matter from which your question arises.

    In other words, the law of non-contradiction says, if you try to “connect” an attribute or characteristic, which has nothing to do with your “gray area” subject matter, you will fail.

    Or to put the point positively: You are only going to be able to answer your question about your “gray area” subject matter, if you look for answers which have to do with your “gray area” subject matter, and don't try to “connect” non-gray area characteristics to the subject.

    Contradictions can not exist. But, I'm assuming your “gray area” subject does exist, so that if you try to say something about your gray area that does not exist, i.e., you make a contradiction, you will never get anywhere with regard to finding the answer to your problem. :-) manquaman


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