A proof about hypocrisy.

I mentioned this in my last post and present it for you here. I stole it from Raymond Smullyan, a puzzle enthusiast and recreational mathematician, and the author of What is the name of this book?, The Riddle of Scheherezade, To Mock a Mockingbird, Alice in Puzzle-Land, and The Lady or the Tiger?, among others. (His version, incidentally, is a lot shorter than mine is, because I'd like you to absorb this and not simply treat it as an interesting game. And also I run off on weird tangents sometimes. If I were to simply cut and paste it, I would already be done by now.)

Let's start with our definition of hypocrisy, so we all know exactly what we're talking about.

1: Any person who does not believe what he claims to believe is a hypocrite.

Not a lot to build on, I admit. We'll need a premise that we can agree is true, to use as a basis.

2: Everyone believes things.

I hope this is something we can agree on! Even without getting into onerous philosophical questions, I'm sure you have certain beliefs about, oh, the shape of world, or the usual color of plants in the spring, or what you had for dinner last night, or whether you will still be alive tomorrow morning. Anytime you say "I think", "I feel", or "I know", you're expressing a belief you have.

3: Any given belief is either true or false.

I'm losing a bit of accuracy here for the sake of clarity, because if I sincerely believe that colorless green ideas sleep furiously we can argue for a long time about whether that is true, false, poorly defined, or even meaningful (and, if it's not meaningful can I really believe it?) - so for the sake of getting on with it we'll assume that all beliefs can be clearly expressed in a way that is either true or false, and if there are some that can't we don't care about them in this argument anyway, so there. :P

4: Each of your beliefs is either true or false.

This follows as a syllogism from 2 and 3, so hopefully there's no disputing it.

5: You believe each of your beliefs is true.

Does this seem obvious? It's not. There's a whole convoluted discussion over Moore's Paradox, addressing sentences like "It's raining, but I don't believe that it's raining." The statement seems absurd, but there's no reason that the two halves of the statement can't both be true - maybe it's raining outside, but you're indoors and away from the windows and can't hear the drops. Moreover, both placing the situation in the past tense ("It rained, but I did not believe that it rained") and shifting the subject ("It is raining, but you don't believe it is raining") result in perfectly reasonable statements!

That conjunction must be playing tricks, yes? Actually, to somewhat oversimplify the situation, it's in the word "belief". Someone who does not believe that it's raining cannot honestly claim that it is raining, and vice versa. If our supposed speaker says that it is raining, and believes it, then they are lying when they say they believe it is not - and, as a person who lies about what they believe, is therefore a hypocrite. On the other hand, though, if they say it is raining but are truthfully claiming they don't believe it, then even if it really is raining they are still lying about the situation as they understand it! In either case, the statement indicates that the speaker is not trustworthy, either by malice or simple stupidity, regardless of whether the statement as a whole is true or false.*

Alas, I digress. The only real explanation is to understand that, when you make a statement, you are implicitly claiming to believe that statement is true. All those beliefs you've stored up in your head can be brought out whenever you like, and whenever you do, you're making a truth claim. We could create a list of all your beliefs, to make this an actuality rather than just a potentiality, but that would take far too much time.

6: At least one of your beliefs is false.

Here we come to a bit of a potential impasse, because it is not possible to prove this, universally, without actually systematically running through a list of all your beliefs and verifying them one by one. So this claim is supported merely by probability and psychology.

If you take that theoretical list of your uncountably huge number of beliefs and approach it without prior judgment - how likely is it that every single one of those beliefs is true? You can think of it as flipping a coin for each statement, if you like - tails for true and heads for false. Or, if you think your system of generating beliefs is a little hardier than that, roll a hundred-sided die for each statement, and only mark it false if you roll a 1. Even if you use a dice with twice as many sides as there are beliefs on your list because your judgment is just that sound, the odds only go down to 50-50 that there are no false statements on that list whatsoever!

If you can honestly tell me, after all that, that you are utterly confident in the truth of every single belief on that list - well, I have to honestly tell you, that's not quite delusions of godhood, but it's pretty close. Even the Pope only claims infallibility in religious matters, you know.

Oh, oh, oh, but wait! If I've just persuaded you that one of your beliefs is false, then by statement #5 you must believe that one of your beliefs is false...

7. You do not believe what you claim to believe.

That list of all your beliefs has an error on it somewhere.

Nevertheless, you continue to believe them all.

You hypocrite.

Want to join the club? There's always room for one more. :)

* There is a mathematical field called intuitionistic logic designed to resolve complications like this, by only using operations that preserve justifiability, rather than truth as in classical logic. As a result (and despite the name), it's actually stricter about what conclusions you can make than classical logic is! Read up, 'tis fascinating.