Sunday, March 09, 2008

Fun With Absurdist Logic

What can you prove starting from the assumption that 1 = 2? Technically speaking, anything. One of the basic rules of logic is that if you include a false statement in your assumptions, you can prove any statement from those assumptions, regardless of whether the resulting statement is true or false. (P ⇒ Q follows from ¬P, regardless of Q.) This is often used for proofs "by contradiction" where you assume the opposite of what you are trying to prove, show that it results in a false statement, and thus conclude that your original assumption was false.

Though we know that this is true, rarely do we exercise it, and it can be a lot of fun to do so. Bertrand Russell once remarked "Give me any false statement and any other statement to prove and I will prove it," and I'll be cribbing from him here to use "1 = 2" to prove that I am a walrus. (He proved that he was God, but I'm no Bertrand Russell.)

Assume 1 = 2.
Consider the following set of two elements: {me, a walrus}.
This set has size 2, but because 2 = 1 it must also have size 1.
Therefore, me and the walrus must be the same element, and thus, I am a walrus.

There's great potential for a game here: One player chooses a false statement to start from, and the challenger responds with a fantastically outlandish statement to prove. Failure to prove it in 1 minute results in consequences appropriate for your age group. (If you are in college, you know what to do.) Astute readers will note that a similar argument to the above shows that I am also the eggman. Proving "goo goo g'joob" is left as an exercise to the reader.

1 comment:

Michael Ryan said...

This reminds me of that constructive logic class and specifically tutch, that proof checking program. Such fond memories... ;-)