### Wednesday Puzzle

Here’s what you should know about me: 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.)

Here’s what else you should know: Last night, in the course of an imaginary chat with an imaginary Bob Murphy, I found myself admitting out loud that “I cannot prove that there is no God.”

First Puzzle: Can I in fact prove that there is no God?

Second Puzzle: Can I prove that there is a God?

Third Puzzle: Based on the information given, can you determine whether there is a God?

I’ll answer tomorrow, or in a few days depending on how the comments play out.

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#### 21 Responses to “Wednesday Puzzle”

1. 1 1 Bennett Haselton

Having met you and read your books, on my first reading of this post I took your opening paragraph at face value, before realizing it was the premise of the puzzle.

As for my attempt at the answers:

1) If you allow the possibility that your axioms are inconsistent, then possibly yes, possibly no. Perhaps your two axioms are
(i) I cannot prove there is no God
(ii) There is no God
and made the statement because it followed from (i), but even so, you can prove there is no God from your axiom (ii). On the other hand, perhaps you’re missing axiom (ii), and you really can’t prove there is no God.

On the other hand, if you require your axioms to be consistent, then you cannot prove that there is no God as a consequence of any of your axioms, because that would contradict your other statement that you cannot prove it.

2) Maybe, maybe not (independent of any requirement for your axiom set to be consistent). Perhaps your axiom set is “There is a God”, in which case Yes. Or perhaps your axiom set is empty, in which case No. (Either way, you could still say “I cannot prove that there is no God.”)

3) No because there is no guarantee that your axioms are true.

But even if your axioms are assumed to be true, there is no guarantee that the your axioms and logic are complete (capable of proving all true statements), so even if you can’t prove there is a God, there might still be a God.

2. 2 2 Sub Specie Æternitatis

This looks like fun, so let me try to take a cut at this before everybody else posts their, indubitably superior, answers:

1. At first glance this question would appear not to be answerable from the premises. All possible relations between what Logic Machine Steve Landsburg (LMSL) merely “says” and what LMSL actually knows are logically consistent.

However, one notes that the problem uses the word “admit” rather than merely “say” or “claim.” Generally one cannot “admit” to anything which one does not believe to be true and LMSL’s believes about internal facts (such as what LMSL can or cannot prove) would tend to be accurate, absent some severe mental disorder not in evidence.

So further assuming that the Meta-Steve Landsburg (MSL) who posted the puzzles can be relied upon (and that is a premise which nearly every puzzle must imply to be meaningful or entertaining), LMSL really cannot disprove the existence of God.

So, subject to the above qualifcations, I conclude that LMSL cannot prove that there is no God.

2. This one, at least on first reflection while struck by insomnia at 3 AM, I cannot answer.

On the one hand, being able to prove the existence of God is entirely consistent with being unable to prove the nonexistence of God, so it is possible that LMSL can prove the existence of God.

On the other, LMSL from the description appears to be sufficiently logically powerful that Goedel’s famed theorems (and that is what these puzzles are aiming at, right?) apply to him. Hence there must be undecidable propositions for LMSL. The existence of God might be one of these. So being unable to prove the existence of God is also entirely consistent with being unable to prove the non-existence of God.

3. Again, my deduction fails me. Based on the above it is possible that LMSL can prove the existence of God and hence that God exists.

However, if the question is undecidable for LMSL, he could add either the existence or the non-existence of God as additional axiom to his given set and would not encounter a contradiction either way. So in this case, the existence or non-existence of God is really LMSL’s choice.

3. 3 3 Jonathan Kariv

I get no,maybe,no.

I’m assuming a consistent set of axioms here.

It might be that the “god question” is undecidable from your axioms (fifth postulate style). Or that you can proof god exits within your framework (maybe “god exists” is an axiom).

This seems a bit too straight forward for this blog, so interested to see where you’re going with it.

4. 4 4 Harold

First Puzzle: Can I in fact prove that there is no God?

Second Puzzle: Can I prove that there is a God?

Third Puzzle: Based on the information given, can you determine whether there is a God?

Sometimes I find it helpful to use an analogy or two to avoid the baggage that comes along with concepts like God. To this end I consider two things about which you may make the claim you did. Steve Landsburg and do I have coins in my pocket?

1) You only say out loud what can be deduced logically from the axioms you believe. You said out loud you could not prove there was no god. Therefore that you cannot prove there is no god CAN be logically deduced from the axioms and therefore you cannot prove there is not God.

Godels incompleteness theorem roughly states that there are true statements within any set of axioms that cannot be proved using those axioms. Therefore the fact that you cannot prove something using the axioms does not make it not true. It may be that there is no god, but you cannot prove this without stepping outside your system.

You could equally make the same claim about SL and the coins in my pocket. You cannot prove the non-existence of either.

2) Using only the information given it seems we cannot determine this. The information given is consistent with there being a proof for God. For all we know God existing is one of your axioms.

For example, Steve Landsburg existing (or the I to whom you refer) is a logical necessity or one of your axioms. You therefore cannot prove SL does not exist. However you can prove there IS a SL. However, you cannot prove I have no coins in my pocket. Neither can you prove I do have coins in my pocket. It is simply not within your system of axioms to be able to derive this information. From the information given we do not know why you cannot derive the non existence of God, so it may be possible to do so.

3) No, of course not.

5. 5 5 Ricardo Cruz

I think it is important to define God first. If you make historical claims, it might be possible to disprove it, yes. If you make it too abstract, e.g. “God is energy”, then you might be able to prove it. If you make it too fuzzy, then no, you will not be able to prove it or disprove it.

6. 6 6 Ally

1) No.
2) Impossible to say based on the information given.
3) No I cannot, as this is also impossible to say based on the information given.

Explanations:
1) No. Given that you are basically a logic machine and you never say anything out loud unless it can be logically deduced from certain axioms that you believe, we can conclude that your statement “I cannot prove there is no God.” is a logical necessity given whatever axioms you believe.

2) Impossible to say based on the information given. It is possible that you have a proof for the existence of God – and this would be perfectly consistent with your previous statements (about being a logic machine and being unable to prove there is no God).
However, if you were unable to prove the existence of God, this would also be perfectly consistent with your previous statements.

3) No I cannot, as this is also impossible to say based on the information given. The information given is consistent with there being a God and with you having a proof for God. It is also consistent with there being a God, but you being unable to prove it. It is also consistent with there not being a God, but you being unable to prove this. The only state of the universe that is ruled out by the information provided is that there is no God and you being able to prove it.

Of course, one or more of your axioms may be wrong. But since we aren’t told what your axioms are, that’s impossible to tell.

7. 7 7 Alan Wexelblat

This sort of reasoning some years ago was what caused me to stop saying I was an atheist and begin admitting that I am an agnostic. I just don’t know.

8. 8 8 Sub Specie Æternitatis

@Alan Wexelblat:

I think I understand and appreciate your argument. You cannot conclusively disprove the existence of God, so you claim not to know, a-gnosis literally.

But let me offer a response offered by Russell, IRC: What is your view on the claim that there is a perfectly formed 1:1 scale Victorian teapot in orbit around Proxima Centauri?

With current technology you cannot neither prove nor disprove that claim, much as the claim about the existence of God. Yet very few would have any difficulty denying it.

I’d say that both claims fall to Occam’s razor. Now the razor is admittedly merely a heuristic, not a proof. Yet I’d consider it sufficiently strong a heuristic that I’d make the claim that there is no God with the same degree of certainty I’d make empirical claims about any fact on which I place the highest degree of probability.

9. 9 9 Denver

1. Yes
2. Yes
3. No

Belief in an axiom does not mean that axiom is a true statement about the world, nor does it mean that I share said belief. Putting it a little more technically, you can make valid proofs for the existence or nonexistence of God, but that doesn’t imply such proofs are sound, nor does it imply you can convince me of said proofs.

10. 10 10 dan

There are three possibilities. Your axioms lead you to the conclusion that (or the axioms themselves state):

a) there is no god
b) there is the possibility of a god
c) there is a god

The statement “I cannot prove that there is no God.” is consistent with b an c.

If b is true, then the answer is 1)No, 2)No, 3)No
If c is true, then the answer is 1)No, 2)Yes,3)No

11. 11 11 dan

Actually, b) above should be: it’s not impossible for god to exist, as opposed to, it is possible for god to exist.

12. 12 12 nobody.really

Now it is such a bizarrely improbable coincidence that anything so mind-bogglingly useful [as the Babel Fish, which translates all languages] could have evolved purely by chance that some thinkers have chosen to see it as the final and clinching proof of the non-existence of God.

The argument goes something like this: “I refuse to prove that I exist,” says God, “for proof denies faith, and without faith I am nothing.”

“But,” says Man, “The Babel fish is a dead giveaway, isn’t it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don’t. QED.”

“Oh dear,” says God, “I hadn’t thought of that,” and promptly vanishes in a puff of logic.

“Oh, that was easy,” says Man, and for an encore goes on to prove that black is white and gets himself killed on the next zebra crossing.

― Douglas Adams, The Hitchhiker’s Guide to the Galaxy

13. 13 13 James Kirby

Using God, rather than god, implies the god is (one of the three) Christian god. Proving that God doesn’t exist leaves all those other gods, like Allah or the Spaghetti Monster. Since the general rule in English is that capitalization indicates that the subject is a specific god, it’s confusing that the commentators here use both “god” and “God” as if they were interchangeable.

14. 14 14 nobody.really

Can’t improve upon Ally’s comment @6. But Sub Specie Æternitatis’s remarks about the “say”/”admit” distinction may warrant some attention. More generally, it strikes me as odd that Landsburg would specify “say … out loud.” This strikes me as a needless detail—which makes me wonder if it isn’t needless at all.

Of course, I’m fishing for the idea that the puzzle turns on some apparently incidental feature, rather than on some fundamental insight of logic. Landsburg hasn’t generally gone that direction, so I may be misleading myself.

15. 15 15 Jimbino

James Kirby
July 25, 2018 at 2:09 pm
Using God, rather than god, implies the god is (one of the three) Christian gods. Proving that God doesn’t exist leaves all those other gods, like Allah or the Spaghetti Monster. Since the general rule in English is that capitalization indicates that the subject is a specific god, it’s confusing that the commentators here use both “god” and “God” as if they were interchangeable.

16. 16 16 Jonathan Kariv

@Sub #8: Beautifully put

17. 17 17 neal

If gods are of a range, then perhaps any structures would be contrary and subject to contest.

Of course, any proof of that would just seem to be history and math and war mostly all over.

One would have to exist in a world without faith to see that as anything other. That is disconcerting.

18. 18 18 Leo

Many formal logical systems have a result where if you can prove X and Not X, i.e. that the system is inconsistent then you can derive any statement. Often called the principle of explosion.

Godel’s second incompleteness theorem says that within a suitable logical system that is consistent then you cannot prove that it is consistent.

The contrapositive of the principle of explosion is that if you cannot prove a statement then the logical system must be consistent.

So your statement about the provability of God implies the consistency of your system, but that means you could prove that, but that’s a violation of the second incompleteness theorem. So your system is inconsistent so you can prove anything so 1) and 2) are true and 3) is false

There are a few concerns about the reasoning above, firstly the principle of explosion is obviously unintuitive so maybe a formal system that didn’t imply it might be better, and more in keeping with having a rich set of beliefs

Secondly the statement about being able to prove something might be a statement about whether you can physically prove it, however you might believe you have a finite lifetime or you only have finite space in your brain for logical computation, I think something will go wrong with the godelian arguments if that’s what you meant.

19. 19 19 David Pinto

I was hoping you were talking to the former Mets announcer.

20. 20 20 Bob Murphy

I want to see more posts involving imaginary chats with an imaginary Bob Murphy.

(Like other readers, I’m assuming this is just a fun way to motivate a discussion of Godel stuff, and has nothing to do with God. But I’m curious to see where this goes because my answers to your puzzles are pretty trivial.)

21. 21 21 nobody.really

Whoa.

We’re already doing heavy lifting speculating about the existence of God. Simultaneously speculating about the existence of Bob Murphy will completely overload the circuits.

One thing at a time: Let’s stick with imaginary Bob Murphys for now. If our theory works in that special case, then we can develop a more general theory encompassing both contingencies.