So “Topics in Logic” yesterday was fun. Dr. Bueno likes to play devil's advocate, taking fringe positions and making a rigorous case for them. Since I'm a metaphysical stick-in-the-mud, liking my picture of reality straightforward and intuitive, I make a good straight-man foil for him in discussions, a fact he shamelessly took advantage of to move forward the discussion today. ("Ben, you don't look happy." "No, I'm not." "OK, what's your objection to that argument?" At which point, of course, he plays with my counter-argument much as cat plays with a mouse before eating it.) One of the more interesting bits concerned the following, which is a classical proof for how anything follows from a contradiciton (hence, its a good idea to disallow contradictions from your logical system.)

The only bits of symbolism important for this proof are that letters represent propositions (a proposition is just a statement that can be true or false, like, "the cat is on the mat" or "the sky is blue" or "Bueno is wrong about contradictions"), the v symbol represents "or", the & symbol represents, obviously, "and", and the ~ symbol represents "not", hence ~A represents "not A."

Only three deductive logic rules are appealed to in it, all of which are pretty intuitive: (a) &-elimination, sometimes called simplification, whereby you can deduce p from p & q and deal with it separately (e.g. if its true that "the sky is blue and the grass is green", then it's true that "the sky is blue"), (b) v-introduction, sometimes called addition, whereby if p is true, you can deduce p v q (e.g. if its true that "the sky is blue", it's also true that "either the sky is blue or [fill in the blank...any proposition whether true or false can be stuck in here), and finally (c) v-elimination (I don't remember the synonym), where from the premises p v q and ~p, you can deduce q (e.g. if it's true that "either the sky is blue or the grass is purple" and it's false that the grass is purple, then its true that the sky is blue.)

Got it? Good. Here's the proof (from Duns Scotus or somebody, originally, I think):

1. p & ~p
2. p (from 1 and the &-elimination rule)
3. p v r (from 2 and the v-introduction rule)
4. ~p (from 1 and the &-elimination rule)
5. r (from 3, 4 and the v-elimination rule)

Hence, you can derive anything whatsoever from p & ~p.

Now, I've seen this in various classes before, but, as Zaphod Beelebrox would say "the clever bit", is the following counter-example that Bueno pulled out today from Graham Priest's book, "In Contradiction," which is a defense of paraconsistent logics (i.e. logical systems that allow for "real" contradictions.)

Now, to backtrack a bit, in deductive logic "valid" is defined in a specific way that doesn't rely on whether the premises or the conclusion of the argument is true or not. E.g. the following argument.

1. If werewolves are real, then vampires are real.
2. Werewolves are real.
3. Therefore, vampires are real.

...is totally and completely deductively valid. Its not sound (a sound argument is a deductive argument with all true premises), so we don't need to assent to the conclusion, but it is valid.

A valid argument is just an argument which has the right sort of structure, meaning that *if* the premises are all true, *then* the conclusion cannot be false. E.g. its not logically possible for the statement "if werewolves are real, then vampires are real" to be true and the statement "werewolves are real" to be true and the statement "vampires are real" to be false. If vampires aren't real, then either one or both of those premises must be false.

Hence, one of the best ways to prove that an argument form is invalid is to come up with a counter-example for it, an instance where the structure is the same, the premises are entirely true and the conclusion is false.

e.g.

1. If A, then B.
2. B
3. Therefore, A.

Is invalid, and this can be proven by coming up with arguments with that structure and true premises and a false conclusion, e.g.

1. If Ralph Nader won the 2004 Presidential election, then John Kerry lost the election.
2. John Kerry lost the election.
3. Therefore, Ralph Nader won.

Now, take the proof that anything follows from a contradiction above. As it turns out, the persuasive power of the argument at least partially rests on the assumption that the only possible truth values for propositions are "true" and "false." Graham Priest argues that on a three-valued logic where you include a third category of "both true and false", a good counter-example can be constructed to show that this is no longer a valid argument.

Look at the argument that anything follows from a contradiction again:

1. p & ~p
2. p (from 1 and the &-elimination rule)
3. p v r (from 2 and the v-introduction rule)
4. ~p (from 1 and the &-elimination rule)
5. r (from 3, 4 and the v-elimination rule)

The move from (4) to (5) relies on the assumption that if p is not true, it must be false. Stick in the assumption that "true and false" is an option, and this just doesn't follow. If p is in fact truefalse and r is just false, then the negation of p doesn't make r true, and all of the premises could be true (even if some of them are also false (-: ) and the conclusion would be false.

Now, of course, this does nothing to establish that contradictions or OK, or that the assertion (p & ~p) is anything other than intrinsically meaningless nonsense no matter what the value of p, but it does establish that there's a sense that the "from a contradiction anything follows" argument begs the question by assuming the very assertion--that the only possible truth values for propositions are (just) true and (just) false that's the bone of contention in the first place.

Again, what to make of this is a-whole-nother issue. I'd argue that the possibility that anything whatsoever can be derived is no more absurd or counter-intuitive sounding than the possibility of self-contradictory-but-true compound propositions. It's as if Person A says "all dogs are purple" and Person B says, "but if all dogs are purple, then all cats are green, and we all know that all cats aren't green" and Person A comes back with a good, solid counter-argument showing that it doesn't follow that "all cats are green." That's only a point in favor of Person A if Person B agrees that it's *more* ridiculous to argue that all cats are green than to argue that all dogs are purple.

...but still, for its limited purpose, I think the Graham Priest thing is a good counter-argument, and certainly one that'll give me pause in teaching the original argument in classes of my own without qualifying it a little.