An awesome logical paradox!

Read about this the other day. Intriguingly, it relates to what I was talking about in the last entry on choice.

Here goes. Consider this statement:
If this statement is true, then giant parakeets rule the known universe.


Well, if that statement is true, then it's true, and thus giant parakeets rule the known universe.
But that's what the statement says, and that's true. So it must be true.
Therefore, giant parakeets rule the known universe.
Logic says so, so it must be true.

On the other hand, we could also say:
If this statement is true, then giant parakeets do not rule the known universe.

Then we come to the conclusion that giant parakeets do not rule the known universe.

Isn't that awesome? You can prove anything!

Here's the argument formally:
Suppose X = (X --> Y).
Obviously, X --> X.
Substituting, X --> (X --> Y).
Therefore X --> ((X --> Y) and X).
We know that ((X --> Y) and X) --> Y.
Therefore X --> Y is true.
By definition X = X --> Y.
Therefore X is true.
Therefore Y is true.

And it doesn't matter what Y is!

Where's the problem? Right at the start, the statement X = (X --> Y).

This very definition confuses an empirical X with a hypothetical X --> Y.
In setting these two (essentially incomparable) entities equal, we can create a paradox so deep that it destroys all logic completely.

So I guess we better not do that, eh?