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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: another solution | Comment 43 of 44 |
(In reply to another solution by Logan)

1.|S<=>(S=>G)
2.||S
3.|(S=>G)
4.||-S
5.|-(S=>G)
6.-(S<=>(S=>G)) from steps 3 and 5


I'm confused as to how you reached step 6. You can't form a proof by contradiction by supposing first an antecedent is true and then that it is false and then using both as if they were both true at the same time. This can be shown if we include premise dependencies along-side the statement dependencies:

1. P
2. P
3. From 2, from premises 1,2.
4. P
5. From 4, from premises 1,2.
6. From 3 and 5, from premises 1, 2 and 4.

Now instead of just saying "from steps 3 and 5", we see that it is from premises 2 and 4, respectively. Therefore, the conclusion is based on the premise "S and -S", which is a contradiction in the first place, and from a contradiction anything can be proven, including step 6.

But note that this isn't the same contradiction you claim to have shown - it's not a contradiction stemming from premise 1 (which you want to reject), but from premises 1, 2 and 4, which means you can't conclude simply by rejecting 1.

  Posted by Sam on 2005-03-09 10:30:43

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