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.
(In reply to assumption
Federico, while it is certainly true that the two paradoxes are related, one difference is that "This statement if false" can not only not be false but also not be true, by it's very nature. This proof, however, doesn't contain within it any internal contradictions, the paradoxicalness must be seen from the outside.
Manuel, I'm not sure what you mean, but it is a common way of solving logical proofs to split an 'OR', then show that one side leads to a contradiction, and therefore prove the other side. We could take Descarte's proof (modified) as an example: Either I am or I am not. Let's suppose that I am not. If so, I can't be pondering this question. Since I am pondering it, we have a contradiction. Therefore I am.
In the proof above, we start with "either S is ture or it is false." It then goes on to show that you get a contradiction if S were false, and so proves that it must be true.
If it makes it any better for anyone, the proof can also be done in the reverse manner (but it can be a little more confusing):
S: If S is true then God exists.
1. Let us suppose that S is true (P1)
2. By modus ponens, God would have to exist. (by premise P1)
3. Therefore we can say that if S were true, God would have to exist. (with no premises)
4. This is exactly what S says, so S is True, with no premises (so therefore we have now proved S, instead of merely supposing it. Ie, nothing relies on our having said "let us suppose S").
5. S must be true so blah blah blah so God must exist.
That might be less intuitive to some, granted, but is actually equally valid in standard logic.
Edited on December 15, 2003, 4:49 am
Posted by Sam
on 2003-12-15 04:43:13