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Proof of Anything (Posted on 2003-12-13) Difficulty: 4 of 5
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.

See The Solution Submitted by Sam    
Rating: 3.6250 (8 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: statement 2 | Comment 22 of 44 |
(In reply to statement 2 by donnmike)

Hey Donmike,

this is a sometimes contriversial but quite standard rule in logic. Specifically, if you have an If...Then statement, the only way that it can be false is if the first part, the antecedent, is true and the concequence is false. Thsi means that the statement
"If it is sunny tomorrow I will play soccer"
will only be false if it IS sunny tomorrow yet I DON'T play soccer. Any other combination of events will result in the statement being called true (it is sunny and I play, it isn't sunny and I don't play, or it isn't sunny and I do play).

A consequence of this is that if the first part of the conditional is false, the statement will always be true.
Therefore, in the argument above, we said "supposer S is false." This means that any conditional with "S is true" in the beginning will always be true (still asumming that S is false). We then find that we get a contradiction, so S must be true.
  Posted by Sam on 2003-12-15 20:03:01

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