Schnee (schnee) wrote,

Santa Claus

Consider the following sentence (not created by me; I merely found it in my notes again yesterday, having picked it up years ago from goodness knows where):

If this sentence is true, then Santa Claus exists.

At first glance this does not seem to tell us much. But let's see what knowledge we can glean from it.

Suppose, just for the sake of the argument, that the sentence is true. Then it is true that, if the sentence is true, Santa Claus exists. But the sentence is true, by assumption, so Santa Claus really does exist.

Let's step back for a moment and consider what we have shown: starting from the assumption that the sentence is true, we have concluded that, under that assumption, Santa Claus exists.

But that is exactly what the sentence says! So we have shown that the sentence is in fact unconditionally true. So it is unconditionally true that, if the sentence is true, Santa Claus exists. But the sentence is true, as we've just shown, so Santa Claus really does exist.


Can you find the flaw in this argument? It's the self-reference, of course: the fact that the sentence refers to itself. To see why this is a problem, let's call the sentence A, and the proposition that Santa Claus exists B. Then the sentence can be rendered as follows:

A → B

What happens if you try to remove the self-reference from A? If you replace A with itself, you'll end up with this sentence (A', say):

(A → B) → B

That still contains A, so we're not done. But it's obvious that if we continue replacing A with its statement that A → B, we'll never finish. In other words, the "final form" would be this:

( … ( … → B) … → B) → B

With an infinite number of implications.

And this, in turn, would mean that even if we accept that an infinite chain of implications can, in principle, have well-defined semantics, the final statement in this chain, B (which, remember, asserts that Santa Claus exists) does not ever come to rest on any definite statement. An implication is a statement of the form "if this, then that", but here there is never any "this"; hence we cannot conclude anything about "that".

So the statement A is not well-founded: ill-founded, as it were.

The problem with the above approach, then, lies in the initial working assumption that the sentence itself is true. Implicit in this is the assertion that the sentence can be true, i.e. have a logical truth value assigned to it in a consistent manner. Considering that the sentence is really an ill-founded statement, as we've just seen, this isn't the case. And if you cannot possibly assign a truth value to it, then in particular, the assumption that it is true does not make sense, and anything derived on the basis of that assumption will be invalid as well.

Shame though. I was just about to state that "if this sentence is true, I'm a millionaire". :P

Tags: logic, mathematics

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