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.

Hi, I've just joined and ran across this puzzle. (Title caught my eye.)

Seems to me that the mistake is that calling S and (S --> G) both "Statement S" results in an equivocation.

So, step 8 is a no-go. The label "Statement S" can't refer to both S and (S --> G).

Using material implication, you could build (S --> G) out of S (by adding ~G, then perfoming implication). But just because this operation is permissible doesn't seem to license the claim that S 'contains' the conditional (S --> G), and thus that the statements are one and the same (allowing this reductio to 'work'.)

Posted by Matthew Pianalto on 2004-03-09 20:29:07