A Formal 'Proof' that God Exists

[drug_mentor]

Nice to see you again, too, d_m!

So, if instead of the sentence "God exists", I try to prove the opposite: "God does not exist", then doesn't the same line of reasoning work equally for that one, too? Or is the statement about the existence of God somehow special in the property (1) that it being "true" will also lead to it being "necessarily true"? Also, what about other statements that aren't either necessarily true or impossible, like "There are alien fish living under the surface of Jupiter's moon Europa"?

The existence of God is thought to be special with respect to assumption (1). The argument goes something like this: God is a necessarily perfect being, every property of a necessarily perfect being must contribute to its perfection, therefore, every property which God has, God has necessarily. If existence is taken to be a property,* then (1) is supposed to follow from this line of reasoning. It is only because God is described as 'necessarily perfect' that (1) is thought to be plausible; there are a good many things that it would be ridiculous to assert (1) of, for example, I exist, but it does not follow from my existing that my existence is necessary.

More or less the same line of reasoning can be used to prove the non-existence of God. If one assumes 'necessarily, if it is not the case that God exists, then it is necessarily not the case that God exists' and 'it is possible that it is not the case that God exists' then one can prove 'it is not the case that God exists'. I will post a proof up in the post following this one, in case you are interested. (I owe Vastness a reply and am trying to avoid making this post unreasonably long.) (EDIT: polymath, I didn't see your most recent post until I had posted this, I will reply in my subsequent post.)

I think that it is likely that I would consider S5 to be too strong to be applied to this particular argument, as you mentioned. Although, to be honest I am not 100% clear what the word "strong" means in this context either.

In this case 'strong' means that it validates more inferences than a 'weaker' logic. Every argument that is valid in a weaker modal logic like S4 is valid in S5, but some arguments, like the one in the OP, are invalid with respect to S4 and valid with respect to S5. S5 is 'stronger' than S4 because it contains S4 and it also validates some inferences which are invalid in S4.

I decided to focus most of this reply to your most recent post, if you feel there is anything from your previous post which I ought to have addressed and failed to, please draw my attention to it and I will do so when I get an opportunity.

since I have started reading about modal logic I am now finding it a little addictive. There is also an element of quite delicious absurdity about a lot of expressions of quite logical axioms when expressed in natural language. Anyway, with a slightly clearer head I have now had another look at your original argument.

Logic is a lot of fun if you like argument and/or mathematics. I am glad that this thread topic inspired you to look into modal logic, and that you are enjoying it!

I am having some difficulty with line (5) although it is quite possibly down to my limited understanding of modal logic rather than a problem with the statement itself. I accept that modus tollens is a logically sound path of reasoning, but I think in this case that it may serve to highlight some problems with "God" as a subject of logical discussion...

Presumably ~p should be interpreted as "Not god exists", or maybe "No god exists"... right? But if this is the case this statement in itself seems quite ambiguous and to lend itself to several possible meanings. Which interpretation is most accurate here? "Things that are not god, exist" or "The idea that there is not a god exists"?

So, in this case, the best way to read '~p' is: 'it is not the case that God exists'. It is simply the negation of 'p' ('God exists') - which in classical logic means it is the expression of the falsity of p. 'It is not the case that God exists' is a bit long-winded, but I think it is the best expression of the meaning of the formula in natural language, and it removes any ambiguity around its interpretation. When you consider that '~p' is true if and only if 'p' is false, I hope it will be fairly obvious that I am not suggesting an artificial interpretation of the formula in order to make the argument seem more plausible.

I hope it is fairly intuitive why, on the above interpretation of the formula, the step of modus tollens on line (5) is valid.

I think either is possibly correct but they mean quite different things, which makes me think that "God" itself is a problematic concept and may be the hidden illegal division-by-zero operation in the vast majority of logical arguments involving it, because it is an inherently quite illogical concept in which contradictory statements can be simultaneously true, which leads quite neatly onto my next points...

I hope the foregoing explanation of how to interpret '~p' has gone some way to mitigating your suspicion here. But I will try to say a little bit more on the matter. If you take the interpretation of '~p' to more ambiguous than my explanation suggests, then the problem must surely lie with the logical language, rather than with the concept of God. After all, p stands for 'God exists' by construction; I could just as easily have used q, r, etc. to stand for this sentence. Suppose q stands for the sentence 'Vastness is a Bluelighter', one could raise the same difficulties of interpreting '~q' that you have raised for '~p': does it say 'Not-Vastness is a Bluelighter', 'No Vastness is a Bluelighter', etc. The most reasonable interpretation for '~q' is surely 'it is not the case that Vastness is a Bluelighter'.**

I am struggling with the concept of []-introduction here too. I can accept that as a general rule, if x then []x but I am not sure that this can be universally said to be the case, and especially in this case absent context. Necessarily for what? For the statement to be true in the first place? If so, this strikes me as an obvious tautology but in general an unnecessary one, which adds an unnecessary layer of complexity to obfuscate what may be a logical fault line, so to speak.

It is not generally valid to infer '[]A' from 'A'. If it were, then I could assume anything and infer that it was necessary; from 'I am sitting in my lounge room' it would follow that 'necessarily, I am sitting in my lounge room' - any plausible logic of necessity and possibility is not going to allow such a trivial inference. The application of []-introduction on line (6) is justified by the fact that the occurrence of '~p' on (5) rests entirely on fully-modalized (f.m.) assumptions. A f.m. formula is one in which every occurrence of a sentence letter (p, q, r, etc.) occurs within the scope of a modal operator ([] or <>). The weakened rule for []-introduction in S4 says one can infer '[]A' from 'A' only if A rests on fully necessitated assumptions, i.e. assumptions in which every sentence letter occurs within the scope of [].

Since the '~p' on (5) rests only on assumptions (1) and (3) (i.e. [](p-->[]p) and ~[]p), it may superficially seem like the step of []-introduction on (6) is S4-valid. To see why this is not so, consider the following: []p is equivalent to ~<>~p, therefore '~[]p' is equivalent to '~~<>~p, by double negation this is equivalent to '<>~p', so the sentence letter 'p' in (3) actually occurs within the scope of '<>', thus (3) is not a fully necessitated assumption.

Interestingly, I think that []-introduction works for the second statement I proposed above as a possible interpretation of (5), but not for the first, which to me just serves to strengthen the case for []-introduction being, potentially, an illegal operation here.

I hope I have sufficiently explained why (5) is not really ambiguous, as well as the justification for the step of []-introduction on (6). You are certainly right that the most plausible step of the proof to regard as faulty is the one on (6); though, I expect you will likely agree that if the inference is faulty, it is not due to any ambiguity in the formula on (5).

It strikes me that both (6) and (7) cannot be true. I can accept that (7) quite obviously follows from (2) so, for me, Occam's Razor dictates that the problem lies somewhere along the more complex path of reasoning that takes us through steps (1) to (6).

(6) and (7) certainly cannot both be true, I conjoined them on (8 ) in order to derive an explicit contradiction, thus justifying the step of reductio on (9). If one only wants to reject the conclusion, one need not think a faulty step has been made, rejecting one or both of the assumptions would be sufficient (though denying (2) is arguably question begging, I am strongly inclined to deny (1)). Are you convinced there must be a faulty step of inference because the argument in natural language strikes you as obviously being invalid?


As an aside, I just wanted to say that I appreciate both the intellectual honesty you have displayed, and the effort you have put into addressing the arguments I put to you.


_{* Immanuel Kant rejected the argument because he held that existence is not a property.
** Negation doesn't behave uniformly in natural language. The negation of 'it will be sunny tomorrow' is 'it will not be sunny tomorrow', the negation of 'no-one is here' is 'Someone is here'. '~' expresses the contradictory of the formula that occurs within its scope (in the case of '~p', 'p' occurs within the scope of '~').}

 

[drug_mentor]

As promised to polymath in my previous post (#47), here is a 'proof' that 'it is not the case that God exists'. So that this argument can be more easily compared with the argument in the OP, I mark each numbered assumption with a ', so that they take the form (n'). Once again, let p be 'God exists'.

I am proving the following informal argument:
(1') Necessarily, if it is not the case that God exists, then it is necessarily not the case that God exists ([](~p-->[]~p))
(2') It is possible that it is not the case that God exists (<>~p)
Therefore,
(C') It is not the case that God exists (~p)

Before proceeding with the proof, I want to point out that I can not think of an intuitively plausible informal argument to motivate (1') in the way that the proponent of the ontological argument has an at least semi-plausible argument to motivate her (1). (1') can be obtained from (1) by Uniform Substitution (replace every occurrence of p with ~p); this is not the same as (1') being entailed by (1), but perhaps one might appeal to this fact in order to motivate the plausibility of (1') - I am not so sure how plausible this strategy is, and I do not presently have time to pursue the thought further, but thought it was worth highlighting nonetheless. Now, on with the proof!

(1') [](~p-->[]~p) _{1'} - Assumption
(2') <>~p _{2'} - Assumption
(3') ~[]~p _{3'} - Assumption*
(4') ~p-->[]~p _{1'} - From line 1', by []-elimination
(5') ~~p _{{1', 3'}} - From lines 3' and 4', by modus tollens
(6') []~~p _{{1', 3'}} - From line 5', by []-introduction**
(7') ~[]~~p _{2'} - From line 2', by the definition of <>
(8') []~~p & ~[]~~p _{{1', 2', 3'}} - From lines 6' and 7', by and-introduction
(9') ~~[]~p _{{1', 2'}} - From lines 3' and 8', by reductio ad absurdum
(10') []~p _{{1', 2'}} - From line 9', by double negation elimination
(11') ~p _{{1', 2'}} - From line 10', by []-elimination
QED

Ok, I read some material about modal logic, and it seems that the statement "if p is true, then p is necessarily true" doesn't hold for all statements, and the assumption (1) here is a special property we assign to sentence p = "God exists". An example where the same logic is not true is the sentence "Dogs are kept as pets", which is true in the world we inhabit but wouldn't necessarily have to be true if no one had ever gotten the idea of taming dogs (or cats or some other animal). The "possible worlds" here are similar to the elementary events of probability.

However, the "proof" doesn't tell anything about the properties (which religion?) of the God it proves to exist.

Modal logicians most commonly refer to 'possible worlds' as maximal consistent sets of propositions; there is some controversy in philosophy as to what the metaphysical status of 'possible worlds' could be. David Lewis famously maintained that they were concrete entities which are neither spatially nor temporally linked with the actual world (this committed him to the view that, in the context of referring to worlds, 'actual' is an indexical term), a more moderate form of modal realism says that possible worlds are abstract objects. On the anti-realist side, some maintain that possible worlds are mental constructions, others that they are sets of sentences which can all simultaneously be true.*** Personally, I find the debate a bit tedious. I don't see how we could have epistemic access to the kind of possible worlds that the various types of realism posit, and taking possible worlds as mental constructions seems to have the consequence that modality is more of an epistemological notion than a metaphysical one - a most unwelcome consequence to modal metaphysicians (the argument in the OP is an example of 'modal metaphysics', broadly construed). Prima facie, the most plausible interpretation is as sets of sentences, which is closest to what the logician says; though, it seems to me that such a definition may implicitly rely on one of the modal notions which possible worlds were being invoked to explain.****

It is certainly true that, at best, this argument establishes the existence of God - it is silent on the kind of God that exists. If one is inclined to accept (1) on the basis of the supporting argument I outlined in my previous post, then one may also conclude that God is 'necessarily perfect' - though, this is quite vague and doesn't pin down many (if any) other precise properties.



_{* Note that the formula on this line is equivalent to <>p, so, like the argument in the OP, this argument is not valid in S4.
** This is justified by the fact that the assumptions that the formula on this line have been derived from are fully modalized; note that one of these assumptions is (3'), so it does not rest on fully necessitated assumptions, thus the step of []-introduction on (6') is not valid in S4.
*** I don't mean to suggest that I have given an exhaustive survey of the views on possible worlds here, these are just the ones with which I am familiar.
**** Doesn't 'this set of sentences can all be true simultaneously' amount to 'it is possible for all of the members of this set of sentences to be true simultaneously'?}

 

[polymath]

Thanks d_m for going through the effort of writing this reply. Is the argument "God is a necessarily perfect being, every property of a necessarily perfect being must contribute to its perfection, therefore, every property which God has, God has necessarily." from Thomas Aquinas or someone else?

 

[drug_mentor]

Thanks d_m for going through the effort of writing this reply. Is the argument "God is a necessarily perfect being, every property of a necessarily perfect being must contribute to its perfection, therefore, every property which God has, God has necessarily." from Thomas Aquinas or someone else?

No worries!

Philosophy of religion and the history of philosophy are far from my areas of focus, so unfortunately I am not able to give you as good an answer as I would like to. I am actually not sure who to attribute that argument to, it was given to me second-hand by a logician in a seminar about 18 months ago. The ontological argument in the OP was presented to me as an Anselm-inspired argument, which is to say that it owes a lot to Anselm's ontological argument, but differs from it in certain respects. I am not really the best person to get into the finer points of difference, though.

I looked up Anselm's argument briefly, it seems quite distinct from the argument I presented in the OP. It seems there is no real consensus on how to interpret Anselm's original argument, the SEP article on ontological arguments presents five different formulations of this argument. The fourth formulation is definitely the most similar to the one I presented, though it is still quite different. (As an aside, the fourth formulation/interpretation seems to be attributed to David Lewis, who I mentioned briefly in my last post in connection with a strange view on the metaphysical status of possible worlds.)

 

[belligerent drunk]

I've been meaning to reply to this for a month, but haven't got around to it due to not having time and forgetting.

The way I see it, this sort of proof (or any purely logical proof) is flawed for the following reason. There are 2 possible scenarios, which both already assume that the logical argument is sound (which it seems to be):

1) The entity you're trying to prove is not special, as in it has no property that makes its existence more (or less) motivated under pure logic than any other entity, such as a tennis ball, Zeus, a virus, a planet - anything. In this scenario, if you can prove "god", you can also prove anything you like - such as a tennis ball swimming around in the oceans of Europa. So even if the logic is sound, it's a proof of nothing, because most of the things you can "prove" with it don't make any sense.

2) The entity you're trying to prove is somehow special. In this thread you mentioned many times how a god is assumed to be "perfect", which somehow makes the proof work for it, but not for any random entity. In this case rather than addressing the proof itself, the hardest part is justifying special treatment for god but not for Ceres on vacation in Neptune's blue methane. The question is something along the lines of "what property of god makes this proof work for it, but not for other gods or random objects? And how do we know that god can have that property? Why can't other entities have it?". And the way I see it, trying to answer the latter is going to boil down to opinion vs opinion until something empirical can shift it towards one or the other. And currently there is nothing of the like.

Anyway, that's my take on it. It is most likely biased, because I'm a sucker for empirical evidence. IMO, while logic is great and indispensable for natural sciences, doing abstract logic for the fuck of it can lead to extremely absurd results as illustrated in the first scenario, or in the end require empirical evidence to boot in order to avoid the first scenario. In other words, no matter how sound the logical argument is, I personally will not believe in a statement if it has no empirical motivation (clear distinction from observation, as you need not directly observe something in order to have motivation to believe in it - cf. Big Bang, which is strongly motivated by indirect observations and physics models that work for most of observed phenomena, but obviously has never been directly observed).

 

[JessFR]

It's interesting how many assumptions people make, and don't even realize they're making about the nature of God.

The Christian God is said to be the one and only true God, to he perfect and all knowing and all powerful. But these aren't inherent qualities of a God. Many cultures have believes in various gods that haven't been all knowing or all powerful or invincible. The idea that a God has to possess those qualities is already making an assumption born from your cultures assumptions about deities to start with.

All things considered, this idea of there only being one all powerful all encompassing God is actually fairly new. For a long time before then human societies generally believes in many gods that didn't have those attributes.

Even the origins of the Christian/Jewish God "יהוה‬" is likely the evolution of beliefs that started with the Christian God as just one of many until people came to see him as the only legitimate God. Assuming you believe in and study history rather than get everything you know from biblical literalism that is.

 

[drug_mentor]

Hey b_d, nice to see you posting in P&S! Thanks for your considered reply.

I've been meaning to reply to this for a month, but haven't got around to it due to not having time and forgetting.

The way I see it, this sort of proof (or any purely logical proof) is flawed for the following reason. There are 2 possible scenarios, which both already assume that the logical argument is sound (which it seems to be):

1) The entity you're trying to prove is not special, as in it has no property that makes its existence more (or less) motivated under pure logic than any other entity, such as a tennis ball, Zeus, a virus, a planet - anything. In this scenario, if you can prove "god", you can also prove anything you like - such as a tennis ball swimming around in the oceans of Europa. So even if the logic is sound, it's a proof of nothing, because most of the things you can "prove" with it don't make any sense.

2) The entity you're trying to prove is somehow special. In this thread you mentioned many times how a god is assumed to be "perfect", which somehow makes the proof work for it, but not for any random entity. In this case rather than addressing the proof itself, the hardest part is justifying special treatment for god but not for Ceres on vacation in Neptune's blue methane. The question is something along the lines of "what property of god makes this proof work for it, but not for other gods or random objects? And how do we know that god can have that property? Why can't other entities have it?". And the way I see it, trying to answer the latter is going to boil down to opinion vs opinion until something empirical can shift it towards one or the other. And currently there is nothing of the like.

This argument definitely opts for 2), the first assumption is a special assumption about God. I think you are right to reject that assumption. There are some reasons to think it is more plausible that something like 'God' or 'divine creator' is necessarily perfect (i.e. could not have failed to be perfect) than it is to think ordinary contingent beings are necessarily perfect. But, I nonetheless agree that it is a contentious assumption that is very difficult to justify. I said a little bit elsewhere in the thread about some arguments I have heard to motivate it, but I don't think they are persuasive.

Anyway, that's my take on it. It is most likely biased, because I'm a sucker for empirical evidence. IMO, while logic is great and indispensable for natural sciences, doing abstract logic for the fuck of it can lead to extremely absurd results as illustrated in the first scenario, or in the end require empirical evidence to boot in order to avoid the first scenario. In other words, no matter how sound the logical argument is, I personally will not believe in a statement if it has no empirical motivation (clear distinction from observation, as you need not directly observe something in order to have motivation to believe in it - cf. Big Bang, which is strongly motivated by indirect observations and physics models that work for most of observed phenomena, but obviously has never been directly observed).

(My emphasis.) One might ask, if doing logic in the abstract can lead to absurd results, why should we be convinced it is a good tool for reasoning scientifically? It seems to me that logic either does what it says on the tin - i.e. preserves truth from premises to conclusion - or it doesn't. If it does do this, then surely abstract reasoning is not the problem, false premises are. On the other hand, if it doesn't do this, then it seems it can still quite easily lead us to error when we use empirical premises. You might agree with all that, but suggest the premises used when reasoning scientifically are likely to be true; I am not sure that the history of science would vindicate such a claim. We might say that we should only reason logically with premises we know to be true, but that condition would seem to be strict enough to rule out lots of worthwhile scientific reasoning. I have fairly strong empiricist leanings, but it seems to me that a lot of highly abstract conceptual reasoning with logic is respectable; for example, in metamathematics. Intuitively, it doesn't seem entirely unreasonable to think that mathematical concepts enjoy a privileged status over other concepts we might reason with. However, I am not so sure that this intuition can be justified; surely, for the justification to be non-circular we would need to employ non-mathematical concepts, and employing non-mathematical concepts in order to justify the privileged status of mathematical concepts seems self-refuting to me.

 

[belligerent drunk]

Nice to see you too, and thanks for creating this interesting thread. It gave me a lot to think about when I first read it.

(My emphasis.) One might ask, if doing logic in the abstract can lead to absurd results, why should we be convinced it is a good tool for reasoning scientifically? It seems to me that logic either does what it says on the tin - i.e. preserves truth from premises to conclusion - or it doesn't. If it does do this, then surely abstract reasoning is not the problem, false premises are. On the other hand, if it doesn't do this, then it seems it can still quite easily lead us to error when we use empirical premises. You might agree with all that, but suggest the premises used when reasoning scientifically are likely to be true; I am not sure that the history of science would vindicate such a claim. We might say that we should only reason logically with premises we know to be true, but that condition would seem to be strict enough to rule out lots of worthwhile scientific reasoning. I have fairly strong empiricist leanings, but it seems to me that a lot of highly abstract conceptual reasoning with logic is respectable; for example, in metamathematics. Intuitively, it doesn't seem entirely unreasonable to think that mathematical concepts enjoy a privileged status over other concepts we might reason with. However, I am not so sure that this intuition can be justified; surely, for the justification to be non-circular we would need to employ non-mathematical concepts, and employing non-mathematical concepts in order to justify the privileged status of mathematical concepts seems self-refuting to me.

I now realize that you're right. The problem with absurd results is not the fault of logic itself, but is because of faulty assumptions/premises. I guess getting premises right to the absolute degree is actually the trickiest part, at least IMO again. The main point of my reply was to show the importance of justifying your assumptions, and how hard it can be to do so objectively.

It's also quite noteworthy that you being an atheist participate in such exercises.

 

[drug_mentor]

Nice to see you too, and thanks for creating this interesting thread. It gave me a lot to think about when I first read it.

I am glad you got something out of it!

I now realize that you're right. The problem with absurd results is not the fault of logic itself, but is because of faulty assumptions/premises. I guess getting premises right to the absolute degree is actually the trickiest part, at least IMO again. The main point of my reply was to show the importance of justifying your assumptions, and how hard it can be to do so objectively.

That was certainly the main point I took away from your initial post. I picked up on the sentence I emboldened only because it was an opportunity to raise some interesting questions and challenge your strongly empiricist viewpoint somewhat. Apologies if I came across as nit-picky or failed to adequately acknowledge the central point of your post. For the most part, I agree with your diagnosis of where the argument goes wrong, and I appreciate your substantive engagement with the initial argument.

It's also quite noteworthy that you being an atheist participate in such exercises.

If I come across something that relates to philosophical logic I will almost certainly pay attention to it. I only know about this because it came up in a logic seminar a while back, I can't say I actively pursue this kind of thing. These days my formal interests are generally centered around the 'quest' for a good conditional and attempts to resolve paradoxes. I have very little interest in God or religion, the argument is only really of interest to me because some might think it raises the issue of whether S5 is too strong. It is also kind of interesting that an argument which strikes me as clearly invalid in natural language can be shown to be deductively valid. It's a fun exercise to think about this kind of stuff and try to figure out where things are going wrong.

 

[mp44god]

Can someone explain this theory in layman's terms for the people on the lower end of the IQ spectrum? Jesus christ. All that math makes my head hurt.

 

[Grace&Rest]

This is such a funny thread ... It is actually what made me join BL
As a Christian Acts 2 has the 'proof' that God Exists ... it is the proof of the indwelling of the Holy Spirit - speaking in tongues.

When I approached Father God in the way He defined in the Bible - He responded with the Proof that He said He would.

Since then - I have raised my son from being dead through Prayer (documented - it happened in a hospital)
that happened 24 years ago ... and the miracles have not stopped!

God Bless you all ... keep the fun happening!