Thursday, December 13, 2007

Proving a negative

From the Dec. 5, 2007 eSkeptic

You Can Prove a Negative
by Steven D. Hales

A principle of folk logic is that you can’t prove a negative. Skeptics and scientists routinely concede the point in debates about the possible existence of everything from Big Foot and Loch Ness to aliens and even God. In a recent television interview on Comedy Central’s The Colbert Report, for example, Skeptic publisher Michael Shermer admitted as much when Stephen Colbert pressed him on the point when discussing Weapons of Mass Destruction, the comedian adding that once it is admitted that scientists cannot prove the nonexistence of a thing, then belief in anything is possible. Even Richard Dawkins writes in The God Delusion that “you cannot prove God’s non-existence is accepted and trivial, if only in the sense that we can never absolutely prove the non-existence of anything.”

There is one big problem with this. Among professional logicians, guess how many think that you can’t prove a negative? That’s right, zero. Yes, Virginia, you can prove a negative, and it’s easy, too. For one thing, a real, actual law of logic is a negative, namely the law of non-contradiction. This law states that that a proposition cannot be both true and not true. Nothing is both true and false. Furthermore, you can prove this law. It can be formally derived from the empty set using provably valid rules of inference. (I’ll spare you the boring details). One of the laws of logic is a provable negative. Wait … this means we’ve just proven that it is not the case that one of the laws of logic is that you can’t prove a negative. So we’ve proven yet another negative! In fact, “you can’t prove a negative” is a negative — so if you could prove it true, it wouldn’t be true! Uh-oh.
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C2H50H said...

What is a "negative"? That's the problem.

It is trivial mathematical logic that "A implies B" is equivalent to "A cannot be true unless B is true", which is clearly identical to saying "not(B) implies not(A)" (which is the contrapositive of the original statement. Is a contrapositive a "negative"?)

It's easy to prove the nonexistence of something: all you have to do is look everywhere it might be. If this requires omniscience, that's a matter for theology, not mathematics or logic.

Proving nonexistence generally involves deeper mathematics, such as the "pigeonhole principle" -- which, roughly translated, is "everything is somewhere". (Mathematics isn't concerned about where you are, so the "and no matter where you go, there you are" follow-on is again left for theology.)

Beware of delving too deeply into mathematical logic, H.G. That way lies confusion, metatheory, metamathematics (the mathematics of mathematical inference) and, ultimately, madness.

My own interest in mathematical logic waned after model theory, specifically after we constructed a countable model in which it was possible to prove that there exists an uncountable set.

For a time it appeared to me that recursive function theory might offer some insight, but after you've looked at infinite-injury arguments, you just don't have a lot left.

Of course, there's always axiomatic set theory, but I chose not to worry about the axiom of choice, and so here I am today, with a few remaining shreds of sanity left. There's doubt about that last statement, of course, but we're comfortable with doubt, aren't we?

Hume's Ghost said...

I don't have any intention on delving into logic past the common sense level ... if someone says to me you can't prove a negative I say

"there are no unicorns in this closet" and then open the door.

Hume's Ghost said...

I've also linked to this in the past. And for the record, your reply made my head hurt.

C2H50H said...

It was not my intention to make your head hurt -- that's a negative, and, coincidentally, one I can't prove.

Your proof about the unicorn only proves that after you open the door, there are no unicorns any more -- or that they're invisible to us.

I adhere to the theory that unicorns are invisible when they want to be, so you've proved nothing to me.

If you aren't willing to go beyond common sense, we just won't go anywhere, mathematically speaking.

Again, apologies for the head, and I do love these posts of yours on this topic. And apologies for the headache. It will go away if you watch some Fox News -- the recognized antidote to logical thought.

Hume's Ghost said...

"I adhere to the theory that unicorns are invisible when they want to be, so you've proved nothing to me."

I've got an invisible garder, myself.