Wednesday, November 4, 2009

What The Explosion Proof Isn't

The following is an extract from the first chapter of the rough draft of my dissertation.

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In classical logic, anything and everything follows from any contradiction. To make things concrete, take the following example. If we start by assuming that the Russell Set (R={x : x ∉ x}) is both a member of itself and not a member of itself, we can, in a few easy steps, derive the result that John McCain is a 400-year-old vampire. After all, if the Russell Set is both a member of itself and not a member of itself, then by conjunction-elimination we derive the consequence that the Russell Set is a member of itself. From there, by Disjunction-Addition, we conclude that either the Russell Set is a member of itself or John McCain is a 400-year-old vampire. From a second application of conjunction-elimination to the first premise, we conclude that the Russell Set is not a member of itself. From the previous two results and Disjunctive Syllogism, we get the startling result that John McCain is indeed a 400-year-old vampire.

Formally,

1. α ∧ ¬α A
2. α 1, Conjunction-Elimination
3. α ∨ β 2, Disjunction-Addition
4. ¬α 1, Conjunction-Elimination
5. β 3,4, Disjunctive Sylloigsm

Now, this proof is a familiar one that, in one form or another, most introductory logic students are exposed to in their first course, but it is odd in light of that that it is terribly unclear what sort of proof it is supposed to be. It is, obviously, not supposed to be the sort of proof that functions to convince anyone of the conclusion, since the typical logic instructor showing it to her students does not herself believe that any statement of the form (α ∧ ¬α) is true, and she believes still less that any and every possible β is true. It is often referred to as a reductio proof, but strictly speaking, it is nothing of the sort. A reductio proof that the first premise is not true would be spectacularly short and uninteresting:

1. α ∧ ¬α A for RAA
2. ¬(α ∧ ¬α) 1, RAA

Perhaps it is a conditional proof, to get us the result that (α ∧ ¬α)→β. If so, again, it is a bit like using a sledgehammer to crack a nut. The falsehood of (α ∧ ¬α) is alone sufficient to guarantee the truth of that conditional in classical logic.

We are running out of formal options. Turning to the informal ones, we could speculate that it is supposed to show that there are not and could not be any true statements of the form (α ∧ ¬α) because, if any such statement were true, everything would be true, and at least some things pretty clearly fail to be true. Or, more weakly than this counterpossible claim, it could be seen as showing that anyone who believed that some such statement was true would be rationally compelled to believe that absolutely everything is true. The problem is that if it is supposed to be an argument for either of these things, it is a terribly unconvincing one. As an argument against the possibility of true contradictions, it begs the question by employing a rule (Disjunctive Syllogism) that is only universally truth-preserving if there are no true contradictions. After all, if for some α and some β, α is both true and false, whereas β is just false, then (α ∨ β) will be true, ¬α will be true and β will nevertheless fail to be true. For the same reason, someone who believed some statement (α ∧ ¬α) was true would not be rationally compelled to conclude any and every random β, because they would have no rational reason to believe that Disjunctive Syllogism was a valid rule of inference.

Reflection on this point shows that, even from a classical monist perspective, ‘from a contradiction, anything follows’ is no more interesting a logical principle than ‘from Hitler winning World War II, anything follows,’ which could be proven as follows. ‘If Hitler won World War II, then β’ is true for any and every β, because all conditionals with false antecedents come out as true in classical logic. Now, assume that Hitler won World War II. By Modus Ponens, we are able to conclude any and every random β.

The obvious objection is that we have switched positions midway through the argument on the subject of whether Hitler won World War II. Quite so. However, this is not a bit different in principle from the way we have switched positions on the possibility of true contradictions when we write down (α ∨ ¬α) as our first premise in the explosion proof, and then proceeded to use Disjunctive Syllogism as if it were a valid rule of inference, which it would not be if the first premise was true.

As such, while the chain of inferences connecting the Russell Set’s membership conditions to John McCain’s age and supernatural status is perfectly valid as far as it goes (if classical logic is right), the answer to what the proof tells us one way or the other about any remotely philosophically interesting subject whatsoever is, I think, ‘not a damned thing.’ It’s valid if monaletheism is correct, and by helping ourselves to explosion in order to argue against dialetheism, we would be putting the cart before the horse and egregiously begging the question. Dialetheism cannot be batted away with the blunt instrument of triviality-avoidance. It needs to be engaged on the arguments.

4 comments:

ben nelson said...

"Or, more weakly than this counterpossible claim, it could be seen as showing that anyone who believed that some such statement was true would be rationally compelled to believe that absolutely everything is true."

I think a more direct informal way of making sense of it is by saying that once you seriously arrive at a contradiction, you must either reject some part of the offending argument, or reject logic itself. An explosion is the use of logic by rote after you've abandoned it for all meaningful purposes.

"The obvious objection is that we have switched positions midway through the argument on the subject of whether Hitler won World War II. Quite so."

I don't understand. You're confirming the antecedent in this example, not denying it, so you're not illustrating the point you wanted to make. If Hitler won, then β follows without event. Did you mean to say "Assume Hitler did not win"? Even so, though the whole if-then statement would be T, the consequent could still be either T or F, which is not a contradiction.

But the next paragraph indicates that maybe you meant to say that we are putting an obvious contradiction in our premises, which is akin to changing our minds. If so, then a classical logician might just make the informal remark that we're not doing logic. It seems to me that the law of identity, etc., are not just formal tools, they're also informal demarcation criteria that tell us when we're just putting empty words to paper for purposes of personal amusement.

Bob said...

You say "The falsehood of β is alone sufficient to guarantee the truth of that conditional in classical logic." This is not correct. The falsehood of (α ∧ ¬α) would be sufficient, but this doesn't make your point since we can't simply assume it is false in this case.

I happen to think that the argument in question actually shows quite a lot. For example, it shows that there is an argument for the explosion principle that relies on apparently quite weak premises and not necessarily on something "purely definitional" like a model-based definition of entailment.

You also point out that if there are any true contradictions, then assuming conjunction elimination is truth preserving, disjunctive syllogism isn't. I agree this is very close to begging the question, but it does show that explosion isn't just one separable thing and you can take it or leave it. Upon reflection, this argument easily transforms into others that show that modus tollens, contraposition, etc. are also invalid.

Joel Velasco (can't seem to change the "Bob" name - not sure where that comes from)

Ben said...

Ben

"...once you seriously arrive at a contradiction, you must either reject some part of the offending argument, or reject logic itself."

Why reject logic itself, as opposed to just rejecting classical logic?

On the face of it, since (counterpossibly) if there were any true contradictions, then Disjunctive Syllogism wouldn't be truth-preserving, then abandoning classica logic in favor of paraconsistent logic seems like exactly the right move.

The Hitler point is that "if Hitler won, B" is true for any B *on the assumption that Hitler didn't win*, but if we throw out that assumption by having "Hitler won the war" as the second premise, so we can derive triviality from Modus Ponens, we've illegitimately switched positions on Hitler winning halfway through the argument. We haven't shown, in any interesting sense, that "from Hitler winning, anything follows."

By analogy, in the contradiction case, we've equally switched positions midway through the argument if we affirm the truth of a contradiction in the first premise, but then treat a rule that wouldn't be valid if contradictions were ever true as a valid rule of inference when reasoning about that contradiction, and we've equally failed to show that "from a contradiction, anything follows" in any particularly interesting sense.

Ben said...

Joel,

Thanks for the correction. Fixed it.

As far as Modus Tollens, contraposition, etc., I'm not sure I see your point on that just yet. After all, in, e.g. Graham Priest's logic LP, those things *are* considered invalid, for the same reason.