The Universe of Discourse

Tue, 23 Apr 2024

Well, I guess I believe everything now!

The principle of explosion is that in an inconsistent system everything is provable: if you prove both !!P!! and not-!!P!! for any !!P!!, you can then conclude !!Q!! for any !!Q!!:

$$(P \land \lnot P) \to Q.$$

This is, to put it briefly, not intuitive. But it is awfully hard to get rid of because it appears to follow immediately from two principles that are intuitive:

  1. If we can prove that !!A!! is true, then we can prove that at least one of !!A!! or !!B!! is true. (In symbols, !!A\to(A\lor B)!!.)

  2. If we can prove that at least one of !!A!! or !!B!! is true, and we can prove that !!A!! is false, then we may conclude that that !!B!! is true. (Symbolically, !!(A\lor B) \to (\lnot A\to B)!!.).

Then suppose that we have proved that !!P!! is both true and false. Since we have proved !!P!! true, we have proved that at least one of !!P!! or !!Q!! is true. But because we have also proved that !!P!! is false, we may conclude that !!Q!! is true. Q.E.D.

This proof is as simple as can be. If you want to get rid of this, you have a hard road ahead of you. You have to follow Graham Priest into the wilderness of paraconsistent logic.

Raymond Smullyan observes that although logic is supposed to model ordinary reasoning, it really falls down here. Nobody, on discovering the fact that they hold contradictory beliefs, or even a false one, concludes that therefore they must believe everything. In fact, says Smullyan, almost everyone does hold contradictory beliefs. His argument goes like this:

  1. Consider all the things I believe individually, !!B_1, B_2, \ldots!!. I believe each of these, considered separately, is true.

  2. However, I also believe that I'm not infallible, and that at least one of !!B_1, B_2, \ldots!! is false, although I don't know which ones.

  3. Therefore I believe both !!\bigwedge B_i!! (because I believe each of the !!B_i!! separately) and !!\lnot\bigwedge B_i!! (because I believe that not all the !!B_i!! are true).

And therefore, by the principle of explosion, I ought to believe that I believe absolutely everything.

Well anyway, none of that was exactly what I planned to write about. I was pleased because I noticed a very simple, specific example of something I believed that was clearly inconsistent. Today I learned that K2, the second-highest mountain in the world, is in Asia, near the border of Pakistan and westernmost China. I was surprised by this, because I had thought that K2 was in Kenya somewhere.

But I also knew that the highest mountain in Africa was Kilimanjaro. So my simultaneous beliefs were flatly contradictory:

  1. K2 is the second-highest mountain in the world.
  2. Kilimanjaro is not the highest mountain in the world, but it is the highest mountain in Africa
  3. K2 is in Africa

Well, I guess until this morning I must have believed everything!

[Other articles in category /math/logic] permanent link