If you've ever taken an intro logic class and you asked your professor how they know there are no true contradictions, you probably get some variation on the proof that anything follows from a contradiction. This is a neat little proof, and a lot of very smart people have taken it to settle the question. Sadly, it doesn't.
Sure, taken as an argument that "if there are true contradictions, using classical logic is probably not a good way to model how they work," there's nothing wrong with it--it is, after all, a perfectly valid proof--buf if it is taken as an argument against the possibility of true contradictions, it is (a) somewhat beside the point, and (b) radically question-begging.
In case anyone reads this who missed the first two posts, let me emphasize that I don't believe in the possibility of true contradictions. If it were really this easy to refute dialetheism, I'd be a happy man...although I suppose I'd also have wander off and find another topic to worry about. (I can imagine various theistic philosophers thinking the same thoughts when it occurred to them that the ontological argument might really work, that it might be *just that easy* to come up with a knock-down argument for the existence of God.) For better or for worse, it isn't this easy.
John Depoe, who I went to grad school with for a few semesters when we were getting our MAs at Western Michigan, is one of the very smart people I referred to above who apparently takes it as settling the question. I'm going to pick on him, not because he says anything unusual about it, but because he blogged about it not so long ago, so he's easily quotable and because his post is one of the clearest statements of explosion-as-an-argument-against-true-contradictions. Here's what he had to say:
"The possibility of affirming logical contradictions is strongly unintuitive, but this is not the only reason to reject the possibility of affirming logical contradictions. For those who lack this intuition (or perhaps the intuition isn’t very strong), you can consider the following proof, which shows that if you accept any logical contradiction, you can prove anything.." After giving the proof, he concludes:
"...so you can literally prove anything given the assumption of a logical contradiction. This is sometimes called “explosion” because once you grant a logical contradiction, you can repeat this proof over and over indefinitely, potentially proving every possible proposition is true. So, even [if] you lack the inuition that logical contradictions are impossible, this little exercise is a reminder why contradictions cannot ever be affirmed."
This sounds like a pretty devestating argument. I mean, if someone told you that if some other claim P were true, everything else would be true too, that would be a problem. We we all know that there are at least some falsehoods, right? If P entails everything, P must be false itself.
First of all, hold on. As alluded to above--point (a)--I'm worried this may be somewhat beside the point. OK, so if you can derive any Q (or ~Q) from (P&~P), then everything is both true and false. That might be embarassing for a regular, moderate dialetheist who wants at least some things to come out as just-true or just-false, but I'm not sure how much comfort it can really provide those of us who believe reality to be consistent. (This is somewhat analogous to the way that some people treat the fact that some versions of global skepticism are hard to coherently formulate as a great victory for global realism. Does that really give the rest of us any more confidence that we are justified in *our* beliefs about the external world?) Maybe all it proves is that reality really is globally inconsistent.
Perhaps (a) doesn't bother you. You say, "if I can be sure of anything on the basis of my immediate experience of the world, it's that some statements are false (and not also true.) If dialetheism generates the possibility that I'm wrong about that, then dialetheism is wrong."
Fair enough. There is, however, a much more serious objection, the one alluded to as point (b) above. This argument doesn't actually show that if there really were true contradictions, everything would follow from them. Why not? Well, let's take a look at the proof itself.
1. P & ~P
2. P (from 1 and &-elimination)
3. P v Q (from 2 and v-addition)
4. ~P (from 1 and &-elimination)
5. Q (from 3, 4 and Disjunctive Syllogism)
The problem is that step 5 begs the question. If there are any true contradictions, any at all--even if out of the infinite list of contradictory propositions, only one of them is true--then it follows that Disjunctive Syllogism is not a valid rule of inference. Remember, whether your logic is classical or paraconsistent, a disjunction is true as long as at least one of the disjuncts is true. If P is true and Q is false, or if P is false and Q is true, or if both P and Q are true, then (P v Q) is true. So far, so good. But what about the possibility that P is both true and false, while Q is just false? If that's a possibility, then the disjunction of P and Q could be true, ~P could be true and Q could still be false. This is *not* a matter of arbitrarily tweaking the rules or piling on a second absurdity to save the first. Quite the opposite. This result simply follows from the assumption that it is possible that some statements of the form (P&~P) could be true. In that scenario, plugging in Disjunctive Syllogism would at least sometimes result in generating a false (but not true) conclusion from true premises.
Now, of course, you might object that Disjunctive Syllogism *is* a valid rule of inference. After all, since it is impossible for a statement of the form (P & ~P) to be true, there is no possible scenario where (P v Q) will be true, ~P will be true and Q will be false. I agree. This is, however, to assume precisely what is supposed to be proved...that there are no true contradictions. It's not that impressive that, starting from the premise that there are no true contradictions, you get the result that absurd things would be true if there were any true contradictions.
The result could be put like this. It is false that anything would follow from a contradiction. There is, however, one (and only one) contradiction from which anything follows...the contradiction involved in assuming that the Law of Non-Contradiction is true by treating Disjunctive Syllogism as a valid rule of inference, and assuming that it is false by starting the proof with (P & ~P). This is, sadly, a terribly uninteresting result.