A followup to Wednesday’s puzzle:
The assumptions are the same as on Wednesday:
I am basically a logic machine. There are certain axioms that I believe, and I never say anything out loud unless it can be deduced from those axioms via the rules of logic. (Fortunately, I can talk about many things, because my axioms include everything from the usual axioms for arithmetic to a rich set of beliefs about ontology, ethics, psychology, and everything else I care about.)
Now I’ve found myself in a whole new imaginary conversation with the same old imaginary Bob Murphy. This time I found myself saying out loud that “If I can prove there is no God, then surely there is no God.”
The Puzzle: Can I in fact prove there is no God?
Solution forthcoming on Monday.
Writing my answer in rot13 (https://www.rot13.com/) as it seems bad form to be the first commenter and have an easily viewable answer as it might spoil it for others.
Gur pbagencbfvgvir bs gur fgngrzrag lbh fnvq vf “Vs gurer vf n tbq gura V pnaabg cebir gurer vf ab Tbq”
Jr nyfb xabj sebz gur ynfg chmmyr gung “V pnaabg cebir K” vzcyvrf rirelguvat.
Gb cebir gurer vf ab Tbq Lbh pna pbafvqre gur gjb pnfrf “gurer vf n tbq” naq “gurer vf ab tbq”
Fhccbfr “gurer vf n tbq”, gura ol zbqhf cbaraf lbh jbhyq oryvrir “lbh pnaabg cebir gurer vf ab Tbq” naq gurersber lbh pna cebir rirelguvat, vapyhqvat “gurer vf ab tbq”.
Nygreangviryl vs “gurer vf ab tbq” gura “gurer vf ab tbq”.
Fb lbh pna cebir “gurer vf ab tbq”.
Leo, thank you for avoiding the spoiler!
“If I can prove there is no God, then surely there is no God.”
The Puzzle: Can I in fact prove there is no God?
You cannot say that statement unless it is true. If your antecedent is false, then your statement is true automatically. That is if you can’t prove there is a god your statement is true and you can say it out loud.
One way your statement is true is that you cannot prove there is no god.
If your antecedent is true, then the conclusion must be that surely there is no god.
Yesterday we discussed things that can be proven from within a formal system of logic and from outside that system. I think (but I am not sure) that you are making the claim that if you can prove there is no god from your axioms there is no god in the “outside world”. That sounds to be false on first look as you cannot say stuff about things outside your system. You as a logical system would know this.
(Thinking out loud, one of your axioms could be “there is no god”. Then within your system of logic you can prove there is no god, because axiomatic things are true. But that has no meaning outside your system of logic.)
So it seems to me that you cannot prove there is no god and therefore you can make your statement. I am looking forward to further elucidation.
A meta answer: The answer can’t be “no”, since nothing you say can exclude the possibility that you have “False” as an axiom and can therefore prove and say anything. So if there is an answer to this question, it must be “yes”.
Psouaggoy sup sx spply pcupoeens ply? Leyys!
Harold:
You cannot say that statement unless it is true.
Where did that come from? What’s given is that I cannot say that statement unless I can prove it. The mere fact that I can prove (from a set of not-necessarily-true axioms) does not make it true.
Leo:
Nice job (for the second time this week), and I second Harold’s thanks for avoiding the spoiler.
“The mere fact that I can prove (from a set of not-necessarily-true axioms) does not make it true.”
Indeed. I think I meant valid. I was thinking valid/true and ended up with true without thinking it through. But then again, maybe the concepts were mixed up in my mind as much as the words and I not sure I am using the term correctly.
I think I meant you can’t say that unless the argument is valid – that is can be derived from your axioms. If the antecedent “I can prove there is no god” is false, then your statement is true because any if/then is true if the antecedent is false. So if you know your initial premise is false, that is if you can prove that you cannot prove there is no god, then you are allowed to make that statement.
This would make you inconsistent and able to prove anything, as before.
This has treated the second part “surely there must be a god” as entirely incedental. In my analysis you could have said anything there. I therefore fear I am overlooking something as I have not used all the material. The word “surely” appears superfluous. I cannot see a difference between “surely there is no god” and “there is no god”. Maybe I am missing something.
I think I have established that if you are inconsistent you can make the satement and prove there is no god but I have not shown you are inconsitent. Lets say you are consistent, which means there is nothing you can prove you can’t prove. You cannot prove (know) your antecedent is false.
The if/then is only false if antecedent is true and consequent is false. If you are consistent you cannot know that antecedent is false. That you made the statement means you know the statement is true. Since you can’t know if A is true or not (as you are consistent) that implies you know the consequent is true.
I have rambled a lot, and I am not sure if I am getting anywhere or not. I will sleep on it.
Dan Christensen (#4): That’s great (though of course you get no official credit for it). Thanks for sharing it.
The following two problems seem equivalent to your problem.
1) You find yourself saying “If I can prove 1+1=2 then 1+1=2″. Is your formal system consistent?
2) You find yourself saying “P”, where P is a theorem in your formal system. Is your formal system consistent?
Especially in problem 2 there is no doubt that this system can be consistent. I was quite lost in Godel, Escher, Bach when Tarski got involved but that smells like the direction you’re going in. Perhaps the words “in fact” are actually essential. Something like; you can’t prove within a consistent formal system A that A describes some other system like reality. If that’s the case then something must be stopping you from taking “My formal system describes reality” as an axiom. Surely axiom 7 can consistently say “axioms 1-6 describe reality” but maybe it can’t say “axioms 1-7 describe reality”, either because that statement leads to contradiction or because it can’t be formalized at all.
Or maybe the key is that reality is consistent, so by asserting that your formal system corresponds to reality you are asserting it is consistent, which means you’ve proved it’s consistent which means it’s inconsistent.
This all would imply that even though you believe your own axioms, you can’t say out loud that they are “true” without proving yourself inconsistent. Here’s hoping I’m way off the mark.