• Philosophy and Spirituality
    Welcome Guest
    Posting Rules Bluelight Rules
    Threads of Note Socialize
  • P&S Moderators: Xorkoth | Madness

A Formal 'Proof' that God Exists

I believe a Christian God was first mentioned by swilow in post #8 of this thread, and he was actually making the same point you are. Even if the ontological argument proved that 'God' exists, it doesn't tell us which God it is; though, based on the argument I presented in the OP, it would have to be a God that was conceived of as being 'necessarily perfect'. (Strictly speaking, that is an overstatement; the argument does not directly claim that God is necessarily perfect, but it is this consideration that is generally used to motivate the first assumption.)
 
^ Even if that were true, it doesn't mean God would utilise such a power.

There is a paradox of omnipotence which many will be familiar with: 'Can God create a stone so heavy that He/She could not lift it?' - philosophy of religion is really not my thing - but there is some literature devoted to that question, with some theists saying it is basically unintelligible or not well-formed. Somewhat ironically, I believe at least one of the reasons that this has been said is that God can't alter or surpass the laws of logic, which are necessary. Proponents of this view think the question leads to a contradiction, which (at least according to classical logic), is impossible. (I can't recall the details exactly, and I may be a bit off the mark here; like I said, my interests don't overlap with religion at all, but if you are interested I could probably dig up some relevant literature for you.)
 
The proof in the OP reminds me of the various mathematical tricks one can use to hide illegal operations like division by zero and prove ridiculous falsities like 1 = 2.

I don't know anything about modal logic but my intuition would say that it's probably not as rigorously formalised as real mathematical algebra even if it imitates it somewhat through it's notation.

For that reason I don't believe it is possible to "prove" anything in this manner - unless the word "proof", in the context of modal logic, means something different to what it means in the context of mathematics or science. Proof requires that something be objectively testable and potentially falsifiable through scientific experiment - even if that experiment is only in the form of a series of mathematical operations.

But modal logic is not maths, perhaps it is an interesting conceptual tool to attempt to apply mathematical logic to concepts that are not usually quantifiable, and perhaps this kind of discussion is useful in refining the field of modal logic itself, and one's understanding of it to be applied in other ways and to other problems... but this is not proof of anything.
 
The proof in the OP reminds me of the various mathematical tricks one can use to hide illegal operations like division by zero and prove ridiculous falsities like 1 = 2.

There is no 'illegal' operation though, I pointed out which step of the proof doesn't work in weaker modal logics. The logic was not designed to prove the existence of God, it was formulated in a way that was thought to be theoretically well-motivated. It just so happens that the resulting logic validates the argument in the OP. You wouldn't be the first person to reject S5 as being too strong, and the fact that it validates the ontological argument may well provide some grounds for thinking such a rejection is well-motivated. Regardless of whether you accept or reject S5, it was not formulated in order to 'hide illegal operations' or to 'prove ridiculous falsities'.

I don't know anything about modal logic but my intuition would say that it's probably not as rigorously formalised as real mathematical algebra even if it imitates it somewhat through it's notation.

Modal logic has numerous mathematical applications. For example, one can do topology with S4, one can use modal logic to formulate a 'provability' logic for Peano arithmetic, and it has been used to advance the development of model theory - it also has numerous applications in linguistics and computer science. (Google it if you don't believe me.) Why would you suppose modal logic is less rigorous than algebra if you don't know anything about it? It seems like a fairly groundless assumption; prominent mathematicians like Tarski and Godel have done significant work in modal logic.

For that reason I don't believe it is possible to "prove" anything in this manner - unless the word "proof", in the context of modal logic, means something different to what it means in the context of mathematics or science. Proof requires that something be objectively testable and potentially falsifiable through scientific experiment - even if that experiment is only in the form of a series of mathematical operations.

Well, for the most part a 'proof' in logic is more certain than one in science, since logic uses deductive reasoning and science uses inductive reasoning.* (The same is true of mathematics since it is also a deductive enterprise; deductive reasoning can establish something more conclusively than inductive reasoning.)** The 'proof' in the OP demonstrates that the conclusion follows from two assumptions; if you reject one (or both) of the assumptions, or reject S5 as being too strong for the domain of discourse it is being applied to, then the proof doesn't follow. It is not a theorem of S5 that 'God exists', it is simply possible to use the logic to show that this statement is a logical consequence of a set of assumptions which some have found plausible.

The steps in the proof are a series of logical operations, it isn't clear why mathematical operations should have a privileged epistemological status over logical ones - especially when you consider that numerous prominent mathematicians and philosophers of mathematics have thought mathematics is reducible to logic. Not so many people think that now, but it is possible to axiomatise arithmetic with second-order logic. (Look up Frege's theorem if you are interested; note that I am not claiming all of mathematics can be derived from second-order logic.)

Ultimately maths and logic are reducible to various sets of axioms, and formulas which follow from said axioms - can you offer some reason for thinking that the mathematical axioms ought to enjoy a privileged epistemological status compared to the logical ones?***

But modal logic is not maths, perhaps it is an interesting conceptual tool to attempt to apply mathematical logic to concepts that are not usually quantifiable, and perhaps this kind of discussion is useful in refining the field of modal logic itself, and one's understanding of it to be applied in other ways and to other problems... but this is not proof of anything.

You have given no evidence for your claim that this is not proof of anything. I am sympathetic to the view that it is not a proof of the claim that 'God exists'; since I accept the logic S5, I do think it constitutes a proof that 'God exists' is a consequence of the two assumptions the conclusion of the proof rests on.****

Do you have anything other than a vague intuition that modal logic is less rigorous than mathematics to back up your assertion? You seem to be suggesting that logic should only be applied to mathematical arguments. This is not a completely untenable view, but it is an uphill battle; many valid forms of natural language argument can be adequately expressed in the language of formal logic, in the absence of some reason for thinking that modal discourse cannot in principle be formalised - or at least some reason for thinking Kripke semantics are not an adequate way to do so - the objection is fairly unpersuasive.


* Falsificationism was an attempt to conceptualise scientific reasoning as deductive; however, the Duhem-Quine problem raises some problems for this approach. Obviously, the strictly mathematical parts of scientific reasoning are deductive.

** See the problem of induction if you want to get a sense of why this is the case.

*** You might point out that there is less agreement between logicians on the correct way to axiomatise logic than there is among mathematicians on the correct way to axiomatise mathematics; this is true, but there is still plenty of dissent from the orthodoxy in mathematics. Various forms of constructivism and finitism, for example, reject certain parts of classical mathematical reasoning.

**** I have some reservations about the epistemological status of alethic modal claims; but, insofar as these modal claims are epistemologically justifiable, I think S5 is the most plausible candidate for a logic of necessity and possibility.
 
Last edited:
no offense, but I'd think you'd make a better case in this topic if you stated where you're coming from and that this is an attempt to /inspire/ people and not proselytize others. I mean posting a mathematical type formula can still be proselytizing :(

I presented something that I thought would be of interest in a forum about philosophy and spirituality. I am certainly not attempting to proselytize or convert anybody - not to belief in God, or to the dogmatic acceptance of the application of a particular modal logic to alethic modal discourse. As should be clear to anyone who read the OP, I am an atheist, I do not take the deduction I presented in the OP to show that God exists. Vastness expressed the belief that modal logic is 'probably' less rigorous than 'real mathematical algebra'* - this is simply false; in fact, modal logic can be treated semantically using algebra (and the more familiar Kripke semantics are expressed using first-order logic and set theory). I make no apologies for pointing out that this belief is false. Intuition is not enough to justifiably dismiss the rigour of decades of logical and mathematical work.


* One might be inclined to ask what makes a mathematical algebra 'real'; are there fake algebras?
 
Last edited:
IMO, a god who is all knowing, all powerful, who has always existed, is unknowable by the human mind. Believe what you want. That's called Faith. But until She lands her UFO in Central Park & gives an interview to MSNBC, I'll consider myself an agnostic.

ag?nos?tic
aɡˈn?stik/
noun
  • a person who believes that nothing is known or can be known of the existence or nature of God or of anything beyond material phenomena; a person who claims neither faith nor disbelief in God.

 
Clifford and Lie algebras, as opposed to the Cayley-Dickson Construction in my opinion

Edit: "Lie" algebra, too perfect if true

Clifford and Lie algebras seem like interesting, non-trivial algebras to me. I don't know why they would be regarded as 'fake'. The 'Lie' in Lie algebra is named after the mathematician Sophus Lie, the name is not intended to convey that there is anything fake or dishonest about the algebra.

The notation you mention is not what I am used to, but easy enough to understand. Actually, I was talking about logic with a computer programmer fairly recently and they told me that they write not-p as '!p' (it might actually have been 'p!', though the former makes more intuitive sense to me, who is used to seeing a negation sign appear before the formula being negated).

I don't know much about MIDI and am a bit short on time to look into what you are talking about, it sounds interesting though.

IMO, a god who is all knowing, all powerful, who has always existed, is unknowable by the human mind. Believe what you want. That's called Faith. But until She lands her UFO in Central Park & gives an interview to MSNBC, I'll consider myself an agnostic.

Did you read any of this thread? I have said over and over again that I don't believe in God.

Why do you think God is unknowable to the human mind? I don't believe in God, but if God created the Universe it isn't obvious why it couldn't endow some of its creations with some ability to know about God. In the context of someone presenting a supposed proof of God, it is fairly pointless to simply assert that you think God is fundamentally unknowable; obviously, the proof purports to show that we can know that God exists. In the absence of some reason to think that such knowledge is not available in principle, the assertion fails to engage with the argument at all.

I was really hoping to discuss the merits of the argument itself (not necessarily the formal presentation, even just the informal argument), that is the point of this thread. I don't accept the argument either, for reasons I have already mentioned. But, to just come in here and assert that God is fundamentally unknowable, or that a rigorous formal system which one has no familiarity with must be some kind of unrigorous trickery (I am not attributing this second claim to you, by the way), is not really engaging with the thread topic in any substantive way.
 
Last edited:
This is way off-topic, but, 1's and 0's simply represent truth values, they are not truth and falsity simpliciter. If one assumes there is such a thing as objective truth, it isn't clear how a computer which 'entangles' 1's and 0's could have any influence on the objective truths in the actual world. Of course, there is a sense in which truths about an existing computer system are truths in the actual world, so in the trivial sense that what is true of the computer system might be altered, objective truth in the actual world will correspondingly be altered. This is a long, long way from the reification of a deity, though.
 
I think if an intimate knowledge of a god(s) was intended it would be instinctual and not something we need to search/ repent/ pay donations or pray for. It would be like knowing for you air exists and is a fundamental and essential part of life.

If there is a god he clearly doesn't want to be found by us all since making himself known to the world for any omnipotent being is very doable.

/if every war, debate, argument and deep thoughts and study done into the existence of god was all funnelled into one mammoth project for the last 5000 years, I wonder what we could have done. Money and man hours considered here.
 
I think if an intimate knowledge of a god(s) was intended it would be instinctual and not something we need to search/ repent/ pay donations or pray for. It would be like knowing for you air exists and is a fundamental and essential part of life.

If there is a god he clearly doesn't want to be found by us all since making himself known to the world for any omnipotent being is very doable.

/if every war, debate, argument and deep thoughts and study done into the existence of god was all funnelled into one mammoth project for the last 5000 years, I wonder what we could have done. Money and man hours considered here.

I agree. And it could be done in a variety of covert ways, like eliminating cancer. Something like that would catch the attention of a lot of people.

The God of the OT did make himself known frequently but not so much in the NT.
 
Likely because in OT times the Jews had to justify a lot of warfare on other people and the wanton killing and enslavement and confiscation of goods. They needed a god to speak up and direct and condone it.
 
Well, thanks for such a considered response drug_mentor, I will do my best to respond where I can.


Regardless of whether you accept or reject S5, it was not formulated in order to 'hide illegal operations' or to 'prove ridiculous falsities'.
I don't doubt (or at least, don't doubt the possibility) that the purpose of it's formulation was an intellectually honest one, but intuitively I am not convinced that it cannot be used for the purpose of "proving" ridiculous falsities by an illegal operation which is not actually logical, but which is made to seem so by the current formulation. This is however a position which I accept I may struggle to defend.

Why would you suppose modal logic is less rigorous than algebra if you don't know anything about it? It seems like a fairly groundless assumption; prominent mathematicians like Tarski and Godel have done significant work in modal logic.
Admittedly, I suppose this because of an inherent distrust of the utility of certain areas of philosophical debate, although I concede that this is not likely to be a logically defensible position in any sense and that reference to utility is also problematic. So, I have read a little about modal logic, but it would seem to me still that algebraic mathematics is inherently more rigorous (or at least, more likely to be more rigorous) than modal logic, because algebraic mathematics (it seems to me) is essentially an application of classical logic, as it does not deal in possibilities (except in probabilities, of course). I think that the introduction of possibility into any argument necessarily makes it a less rigorous discussion - or, possibly, only that it is probably less rigorous - because it increases the complexity of that argument.

Well, for the most part a 'proof' in logic is more certain than one in science, since logic uses deductive reasoning and science uses inductive reasoning.*
I think that I accept the truth of this as a general statement, so I concede that my own statement may be at least partly incorrect or incomplete, but...

The steps in the proof are a series of logical operations, it isn't clear why mathematical operations should have a privileged epistemological status over logical ones - especially when you consider that numerous prominent mathematicians and philosophers of mathematics have thought mathematics is reducible to logic. Not so many people think that now, but it is possible to axiomatise arithmetic with second-order logic. (Look up Frege's theorem if you are interested; note that I am not claiming all of mathematics can be derived from second-order logic.)

Ultimately maths and logic are reducible to various sets of axioms, and formulas which follow from said axioms - can you offer some reason for thinking that the mathematical axioms ought to enjoy a privileged epistemological status compared to the logical ones?***
My reason for thinking this is the same as the point I made (or, tried to make) above regarding algebraic mathematics and/or classical logic versus modal logic. To be clear though again, I am not saying that mathematics is always going to be a more reliable measure of truth than modal logic, only that it seems to me more likely to be true - at least, in those areas to which it can be applied. I accept that classical logic and mathematics are far more equivalent in terms of their epistemological privilege, so to speak, even classical logic might be arguably slightly higher on the scale.

I did look up Frege's theorem and honestly I don't understand it but I think that is interesting, and may lend weight to the possibility that higher order logic may be closer to mathematics and classical order logic (in terms of epistemological privilege, likelihood of leading to a true statement, and/or perhaps the trustworthiness of the particular method of application of human reason that is in question... slightly losing track of what I am trying to argue here 8().

Do you have anything other than a vague intuition that modal logic is less rigorous than mathematics to back up your assertion?
To be blunt - at the moment, no.

You seem to be suggesting that logic should only be applied to mathematical arguments.
I don't believe that logic should only be applied to mathematical arguments - in fact quite the opposite. I would like to believe that logic can be applied to practically everything in the vast sphere of human experience, even if in some areas it is not currently clear how best to do this.

I am feeling like ultimately I have used a lot of words to say very little in this post, and concede that you have rightfully called me out on trying to argue something about which I know little to nothing. Admittedly, my original stance is based on a bias towards science and maths over practically every other method of seeking truth - I have generally found this to be a fairly reliable position to hold but obviously acknowledged bias is not in itself a defense of anything. I have since tried to get at least a basic understanding of the differences between different order types of logic and other terms you have used in order that I might construct a somewhat more coherent response, worthy of your own fairly thorough critique of my original argument, and 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.

When I have more time I will look into modal logic further and have another look at your original post when I can (hopefully) understand the notation. It is interesting that modal logic has been used by mathematicians and may be applicable to arithmetic, this does change my view of it somewhat.
 
Last edited:
You can argue all day long about the existence of a god but there is no way to prove its existence.
 
Right you are it seems. It's moot but seems to be a captivating topic just like the free will debate. Much ado about nothing.
 
(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

The final result (11) is a contradiction with the assumptions (1) and (3), so I think that all this chain of reasoning actually proves is that some of the assumptions made are false. In this argument, it's kind of first said that it is not necessary that God exists, but in the end it actually is necessary, like something previously false has suddenly become true during the course of this reasoning. If this is possible in some versions of modal logic, then excuse my ignorance.

When some logical reasoning leads to a contradiction, it's not necessarily obvious which one of the assumptions made in the beginning has been wrong. To me it seems that in the application of reductio ad absurdum at line (9) it is arbitrarily decided that the false assumption was (3) - "it is not necessary that God exists".
 
Last edited:
Vastness, thanks for the reply. I am a little short on time right now and want to address some of Polymath's concerns with the formal argument, I will come back and address what you have said. :)

The final result (11) is a contradiction with the assumptions (1) and (3), so I think that all this chain of reasoning actually proves is that some of the assumptions made are false. In this argument, it's kind of first said that it is not necessary that God exists, but in the end it actually is necessary, like something previously false has suddenly become true during the course of this reasoning. If this is possible in some versions of modal logic, then excuse my ignorance.

Nice to see you round the forum Polymath! I hoped you might reply to this thread, I remember discussing modal logic briefly with you in another thread quite some time ago.

The assumption on (11) is derived from assumptions (1) and (2). A contradiction was derived on line (8 ) from the set of assumptions (1), (2) and (3). It is fairly uncontroversial to say that there are no true contradictions,* so if a contradiction is derived from (1), (2) and (3) then those assumptions cannot all be true. The argument takes it for granted that assumptions (1) and (2) are true, it therefore follows by reductio ad absurdum that the assumption on line (3) is false. Hence, line (9) says, the falsity of (3) follows from (1) and (2).

'It is not necessary that God exists' is the assumption on line (3). The falsity of this assumption follows from the assumptions on (1) and (2). Of course, one can take issue with those assumptions; personally, I reject (1), the point of the proof is to show that the conclusion 'God exists' follows from (1) and (2) - at least, it does in S5.

When some logical reasoning leads to a contradiction, it's not necessarily obvious which one of the assumptions made in the beginning has been wrong. To me it seems that in the application of reductio ad absurdum at line (9) it is arbitrarily decided that the false assumption was (3) - "it is not necessary that God exists".

It is certainly true that, from the derivation of a contradiction on line (8 ), one could just as well have chosen to retain (3) and one of the other assumptions to derive the negation of the unretained assumption. However, the decision to retain (1) and (2) is not arbitrary, those are the premises of the informal ontological argument which this is supposed to be a formal proof of. It is important to realise that the conclusion only follows if one accepts both assumptions; but, if one does accept those assumptions then the application of reductio on (9) is completely natural.

In natural language, the argument does not seem valid to me at all. It is interesting (to me, anyway) that this argument can be proved using a strong modal logic. One might well think that the validation of a prima facie invalid argument provides some reason for rejecting S5 as a logic of necessity and possibility, though, one would usually want to produce more than a single suspicious argument in order to justify the rejection of an otherwise plausible and useful logic.

* Though, a dialetheist would deny this; but, they would generally confine their claims of true contradictions to things like the Liar Paradox and Russell's Paradox, not the contradiction on line (8 ).
 
Nice to see you round the forum Polymath! I hoped you might reply to this thread, I remember discussing modal logic briefly with you in another thread quite some time ago.

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"?
 
drug_mentor, I had another look at your original post earlier on a low dose of acid and I found that actually you had already explained a lot, or at least pointed in the right direction to look for further explanations. I have to confess I really did skim over everything on my first read, pretty much dismissing it as not worth even bothering to try to understand, but 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. =D Anyway, with a slightly clearer head I have now had another look at your original argument.



drug_mentor said:
(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
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"?

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...



drug_mentor said:
(6) []~p {1, 3} - From line 5, by []-introduction***
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.

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.



drug_mentor said:
(7) ~[]~p {2} - From line 2, by the definition of <>
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).



drug_mentor said:
(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
However, if we accept that both (6) and (7) hold, then of course everything that follows is obvious. Personally however I do not see how this can be the case.
 
Right you are it seems. It's moot but seems to be a captivating topic just like the free will debate. Much ado about nothing.

Yes, the free will vs determinism debate is very interesting, as is the topic of consciousness & self-awareness. I also enjoy discussing evolution vs creationism & intelligent design.
 
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.
 
Top