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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: Poor Sam...Trying to be a philosopher while ignoring the Ancient Greeks !!!! | Comment 11 of 44 |
(In reply to Poor Sam...Trying to be a philosopher while ignoring the Ancient Greeks !!!! by Penny)

Penny, I'm not quite sure that you've understood the point of these puzzles on Flooble. The point of my posting this is certainly not to say "my paradox is right, I'm so smart" for two reasons: 1) I cannot lay claim to the paradox, much as I would like to, as it is a simple extension of Curry's paradox and Lob's theorem, and the liar's paradox before that, and 2) at the bottom of the post I say "What, if anything, is wrong with this proof?" Thus statements such as "Poor Sam...Trying to be a philosopher while ignoring the Ancient Greeks !!!!" really mean nothing.

That said, this discussion is becoming interesting, even though you as usual resort to your "proof by cut-and-paste" style of argument that we have all grown to love.

Gamer's comment about infinite loops is actually very interesting. Consider the statement

S: Statement S has five words

This looks ok until we try to replace S with it's referent:

"Statement S has five words" has five words

looks correct, while

S: "Statement S has five words" has five words

doesn't look quite as nice, and the problem only magnifies when we start unpacking the statement further.

[Edit: actually, it only starts to look really bad once we unpack it twice:
S: ""Statement S has five words" has five words" has five words
is quite definately false]

(On a side note, Hofstadter had an interesting statement uttered by a djin who asked a favor from "GOD," which stands for "GOD Over Djin," and thus asked the favor from an infinite number of djins above him.
Also related, the statement might be considered analogous to the formal language <S: aSb>, which is of course non-terminating, and so not adequately described.)

On the other hand, are
S: Statement S has five words, and
S: "Statement S has five words" has five words

truely equivalent? Clearly they are refering to different *sentences*, if not to different statements. We know from Quine (or is it Kripke? I forget) that sometimes referents cannot be simply inserted, even in a normal statement. Consider:

The president might not have become the president


George Bush might not have become the president.

The second statement looks unabiguously true in modal logic, while the first less so.

(or more formally in intensional logic, the distinction between 1) In all worlds it is necessary that the president is the president vs. 2) In all worlds it is necessary that George Bush is the president).

Therefore, if it might be the case that we cannot equate "Statement S has five words" with ""Statement S has five words" has five words", must we assume that we can equate S: S=>T with S:(S=>T)=>T?
But even if we can, is it such a bad thing? Godel's proof is no less valid just becuase

S: S cannot be proven under F

is equivalent to

S: "S cannot be proven under F" cannot be proven under F

indeed, it is quite necessary!

And speaking of which (to bring it back to the puzzle at hand), one can generalize the exact sentence above to get to Godel's second incompleteness theorem.

Lob's theorem states that:

D: If D is provable, then D

is always provable (and you can see that his statement has much the same structure as my original statement S above).

From here we can reach Godel's second proof that, in any consistant system F, F's own consistancy cannot be proven from within that system (don't ask me to try to prove this right here). Godel's second theorem is usually simply presented as a corollary to his first, but Lob's theorem can be used to springboard right into it.


None of this is supposed to be a final answer to anything people have written, it's all just food for thought, but I hope that it goes further than a mis-guided attempt at a competition. I for one am extremely interested in all the directions one can go from such paradoxes.
Edited on December 14, 2003, 12:17 am
  Posted by Sam on 2003-12-13 23:01:44

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