## Wednesday, October 24, 2007

### If-then statements

Difficulty: 1 of 10

There is a classic puzzle that involves four cards on a table. On the cards are written 4, W, 6, and R. You are told that each card has a letter on one side and a number on the other. You are asked to consider the statement, "If there is an R on one side, then there is a 4 on the other." Which cards must you flip if you want to find out if the statement is true or false for the four cards? Continue only after you've made your guess.

Ok, so it's not really much of a puzzle. It's really easy, but still seems to catch a lot of people. If you answered the cards with an R and 4, you've got it wrong.

The reason you don't flip over the 4 is that if there is an R on the other side, the statement is true, and if there isn't, it's still true. Unless the first part of the if-then statement (called the antecedent) is true, it doesn't matter whether the last part of the if-then statement (called the consequent) is true or false. For example, it doesn't matter what number you find behind the W, since there is no R and the antecedent is false. You must flip over the 6 because if there is an R on the other side, the statement has been confirmed false. So the correct answer is the R and the 6.

If you really want to be tricky, I guess you could write an R next to the 4, thus assuring the statement is false without any flips.

To introduce some symbolic logic (if you already know about this, skip ahead), all statements in logic can be represented by a single letter, just like in algebra. So the above if-then statement can be shortened to "if X, then Y" where X is the antecedent and Y the consequent. The symbolic way of writing this is "X ⇒ Y". By definition, "X ⇒ Y" is false if and only if X is false and Y is true. There's a related logical statement that uses "if and only if" (often shortened to the not-a-typo "iff") in which it does matter if Y is true when X is false. But that's not the case here.

Ok, I'm sure some of my readers are bored by now, but there's an interesting paradox to conclude this post. Consider the statement, "If this statement is true, then God exists." If the statement is false, then the antecedent is false, and therefore the statement is true. Since being false leads to a contradiction, the statement must instead be true. Therefore God exists. Now that that's settled, I guess there's no longer any need to discuss atheism.

Anonymous said...

Anonymous said...

I was only able to read the first half before I got lost...
but If there were *four* cards, wouldn't there be two of each "R" "W" "6" and "4"? or Is there supposed to be two cards on the table, or is there supposed to be a card underneath another card.
I don't know if you were able to understand that or not.

miller said...

d-bunny, there are four cards, but you can only see one side of each.

Anony, I thought you'd never ask!

If we put the statement--let's call it statement S into symbolic logic, it would say "S⇒G".

If we expand that, we get "(S⇒G)⇒G". If we expand that, we get "((S⇒G)⇒G)⇒G" and so on. Since the statement is self-referential, it is impossible to ever fully expand it, and thus the statement is not well-defined. A statement that isn't well-defined is neither true nor false. Of course, there also exist self-referential statements that don't result in paradoxes, like "This statement is true", but that doesn't mean it is well-defined.

At least, that's my interpretation of the matter. If you want to look up what real logicians think, it's called Curry's Paradox or Lob's Paradox. My google search turned up very little I could understand, but maybe you'll have better luck.