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.

It seems to me the problem with this logic is the same as the problem with statement

S: This statement is false.

Of course, if the truth value of the above statement is true it contradicts itself. The same can be said if the truth value is false. Similiary, this problem arises when dealing with a set of inner-referring statements such as

S: Statement T is true.
T: Statement S is false.

Classically speaking, there has been no definitive answer to this type of paradox.

--Graatz

Posted by Jeff on 2003-12-23 19:46:02