Here is a nice little paradox:

Statement S: If S is true then God exists
Logically, statement S must be either true or false.

1. Suppose S is false.

2. If S if false, then any statement that starts with "If S is true..." is true *(see note)

3. Specifically, the statement "If S is true then God exists" would be true

4. This is exactly what S says, so S would have to be true

5. This is in contradiction with 1., so S cannot be false.

6. Therefore S is true.

7. So the statement "If S is true then God exists" is true.

8. By modus ponens, since S is indeed true, then the second half of that statement is true.

9. God exists.

Note of course that you can make the same argument to prove that God doesn't exist, or anything else.
What, if anything, is wrong with this proof?

*Note: This is the part that I expect most people will comment on. It is one of the standard logical rules that if something, A, is true, you can say "If (~A) then..." and that will always be true. For instance, I could say "If George Washington is alive then the moon is made of cheese" and that would be considered true in natural logic.

The problem is that we are assuming that "statement" S is, indeed, a statement (in fact it is, but only because the antecedent of the statement is already true; "God exists" was a poor choice here).

On the other hand:
   T: If statement T is true, then 1 + 1 = 3.
Is not a statement, because it can be neither true nor false.
The next line, "Logically, S must be true or false," is not necessarily true for every statement of this form.

A more appropriate title for this riddle may be "Proof of Anything That is Already True."

Finally, a statement is not an arrangment of words, but a logical idea. So, something like:
   U: Statement U contains five words.
Is meaningless, because
   U: There are five words in statement U.
Is the same statement, although it is worded differently (they are the same statement, but not the same sentence).

Posted by DJ on 2003-12-18 22:43:01