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.

So has anyone actually solved the problem yet? I think we've been too preoccupied with "If ~A, then ..." is true when A is true and other stuff. Self-referencing statements doesn't seem to be what Sam had in mind. So what is the problem really?

Speaking of the matter of antecedents, what if the antecedent is true only part of the time? For example, "If my computer is on, then I'm on Flooble" is not true (though comes close). Even if my computer is off, the statement is still false, isn't it? Following this same logic, statement S might have only been true part of the time. The other times, it still has the opportunity to be false. Of course, the paradox is still there since you can prove anything was true at least at one time. That's why I call this a partial disproof, because it only replaces the paradox with another.

Posted by Tristan on 2003-12-17 19:27:29