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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)

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re: Welcome to Logic 101 | Comment 8 of 44 |
(In reply to re(2): Welcome to Logic 101 by Penny)

"Self-referencing statements in logic are exactly like dividing by zero in mathematics. They are totally bogus. They lead to all kinds of paradoxes, and are disallowed. That is what is wrong with Sam's proof."

I'm confused as to where you came up with this idea. Did you invent it yourself? Can you show me where they are "disallowed"? I'm actually quite interested, because I must have completely forgotten to teach my students this when I was a TA for Logic 101 (a true, but completely coincedental reference to the title above), so I guess it's lucky none of them failed!

Which of these statements are "bogus":

-This statement has five words
-This statment has four words
-This statement starts with "this"
-S: Statement S starts with "statment"
-This statement would be included in the set of all statments that refer to themselves
This statement would not be included in the set of all statments that refer to themselves
-S: S is not a letter
-S: "S" is not a letter
-The referent of this statement is not what it appears to be

Now some of those statements are true, and some are false (and that last one looks a little confusing), but I don't remember being able to assign "bogus" to a statement. My professor must not have gotten around to that.

Self-reference is actually quite vital. Godel's proof, which many would consider to be the most important mathematical proof ever written, relies on it most explicitly:
S: S cannot be proven true under system F

Possibly some people wrote him letters where they said "come now, statement S is neither provable not disprobvable, it's just bogus" but these people were neither mathematicians nor logicians, who realized the importance of the proof.

So do tell me, where are these logical laws that you refer to?
Edited on December 13, 2003, 6:14 pm
  Posted by Sam on 2003-12-13 18:09:09

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