A Formal 'Proof' that God Exists

[drug_mentor]

I first encountered this formal argument well over a year ago and have been meaning to post it here since. I think it crosses the two forum topics of philosophy and spirituality rather nicely. I take it that many here will be unfamiliar with modal logic, but, the proof is quite easy to follow; and, even if some struggle with the proof itself, there can still be fruitful discussion concerning whether the conclusion follows deductively from the adduced premises.

Logical Preliminaries: Read []p as 'it is necessarily the case that p' and <>p as 'it is possibly the case that p'. Note that one can define possibility in terms of necessity as follows - <>p =_def ~[]~p, I hope that the interdefinability of possibility and necessity is fairly intuitive, but I can elaborate on this if required. The subscripted numbers in {}-brackets signify the set of assumptions which a particular line of proof rests on; so, for example, a line of proof which contains _{{1, 2}} has been derived from the assumptions made on lines (1) and (2).

I am going to present a supposed proof of Anselm's Ontological Argument resting on the following two assumptions:
(1) Necessarily, if God exists, then it is necessary that God exists (let p be 'God exists', then we can express the assumption as follows: [](p-->[]p))
(2) It is possible that God exists (<>p)
Therefore,
(C) God exists (p)

The first assumption is motivated by claiming that it is part of the concept of God that, if God exists, then it is necessarily the case that God exists. This assumption seems questionable to me, but let us take it for granted at present. As for the second assumption, I am an atheist myself, but it does seem overly strong to assert that it is impossible that God exists, even if one is quite confident that, as a matter of fact, God does not exist. Certainly, it seems to me that denial of the second assumption would require considerable argument.

I should note that I will begin the proof by making three assumptions, however, one of these assumptions will be discharged using a rule known as reductio ad absurdum; so, the conclusion really does rest on just two assumptions.

Before proceeding, I want to note that I am proving this using a particularly strong modal logic known as S5, the proof does not work if one is using the slightly weaker modal logic known as S4.* So, one might well accept the premises of this argument and insist that the problem is the adoption of a logic which is too permissive in the inferences it allows one to make. This feature of the proof makes it of particular interest to me.

(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

What do people think? Have I just proven that God exists? If not, which assumption(s) would you challenge and why? Or, is the issue here that S5 is too strong of a logic to adequately handle inferences concerning necessity and possibility?

_{* Specifically, the inference on line (6) is not valid in S4.
** What is necessarily the case must actually be the case - this fact justifies the inference from [](p-->[]p) to (p-->[]p) by []-elimination (as well as the inference from []p to p using the same rule).
*** This inference is justified by the fact that the line of proof rests solely on assumptions in which every occurrence of a sentence letter is contained within the scope of a modal operator (modal operators in this case being [] and <>). In S4 []-introduction is more restrictive, blocking the inference on line (6).}

 

[Xorkoth]

Alright I'm gonna make an attempt to answer this. Thanks for contributing by the way, drug mentor. I hope some other people do too. I'll start by saying that I have never studied academic philosophy/logic. And to be honest, I find it a bit absurd sometimes, such as in this example. I also think people who haven't studied this stuff are unlikely to be able to grasp everything happening here, because to some extent you have to be familiar with the operators used and how it works, which I am not (and I'm a computer programmer so I have a firm grasp of logic and basic logical operators).

The first assumption I think is just obviously valid although it would also be true if "god" was a variable that could be absolutely anything. Obviously, if something exists, then it's necessary for it to exist. Isn't that no different from saying that if x is true, then x is true? This seems extremely trivial and I don't understand how it helps to prove any sort of question.

The second assumption is valid, in my opinion. When dealing with something intangible that cannot be proven one way or the other, I think the only logical conclusion to make is that it could be true, no matter how certain you personally feel one way or the other.

I'm not clear what the third assumption that you say is irrelevant is, so I can't comment on that or whether I agree it is irrelevant.

See, the problem I have with this is that it feels like academic card tricks; I mean, honestly, different, more and less strict forms of modal logic making an 11-step exercise in logical algebra either valid or invalid as a proof for a question like "does god exist" strikes me as totally absurd and like people have overthought themselves into an imaginary realm of significance - no offense meant. I mean looking at the two assumptions, you're saying:

"If god exists, then god must exist, and it's possible he exists, therefore god definitely exists"

You can't convince me that this is a sound argument regardless of how complex you make your answer seem.

I realize this isn't your proof so I'm not directing my probably obvious scoffing at you personally.

 

[alasdairm]

isn't the entire thing built on the premise "if god exists..."? well, if he doesn't exist then it is not necessary that God exists, right?

alasdair

 

[Kittycat5]

I dont care what logic you use, it is not factual proof. You are making it so your statement is valid but in the real world, there is no rigorous evidence for the existence of god.

 

[drug_mentor]

Thanks for the replies Xorkoth, alasdair and Kittycat!

The first assumption I think is just obviously valid although it would also be true if "god" was a variable that could be absolutely anything. Obviously, if something exists, then it's necessary for it to exist. Isn't that no different from saying that if x is true, then x is true? This seems extremely trivial and I don't understand how it helps to prove any sort of question.

Generally, it isn't thought that if something exists, then it is necessary for it to exist. Think of it this way, my parents met at a bar, it seems hard to deny that it is possible that one or both of them could have decided to stay home that night, if this had eventuated then my parents would never have got married and procreated, so I would never have existed. In this sense, even though I do exist, it is possible for me to have failed to exist. In philosophical jargon, we would say that my existence is a contingent fact, as opposed to a necessary one.

Temporal considerations might also motivate one to reject the view 'if something exists, then it is necessary for it to exist'. To say something exists necessarily is equivalent to saying it is impossible for it to not exist. Mortal creatures have finite lives, so it is implausible to suggest that it is impossible for us to not exist; it is unlikely you or I will be around in 100 years time. God is supposed to be infinite, so this objection doesn't apply to the claim that it is necessary that God exists.

The view that, 'necessarily, if God exists then it is necessary that God exists' is not taken to be trivially true in the sense you are interpreting it. People generally argue for the premise by appealing to the claim that God is defined as a necessarily perfect being, from this it is supposed to follow that whatever properties God may have, God has those properties necessarily. If 'existence' is thought of as a property, then the foregoing considerations suggest that if God has the property of existence, then God has this property necessarily (i.e. God exists necessarily).

Following Leibniz (and subsequently Kripke) necessity is generally defined as 'truth in all possible worlds'. However, there is more than one sense of necessity, for example, one can talk of logical necessity, physical necessity, etc. For the purposes of this argument, necessity should be thought of as metaphysical necessity. Metaphysical necessity is a fairly vague notion, generally all logical and mathematical truths, conceptual truths, and some empirical truths (such as true identity statements) are thought to be metaphysically necessary. Personally, I am a bit suspicious of most necessity claims, they generally seem to be underpinned by quite a lot of vagueness, and I am not sure that it makes sense to think that people could ever know what is necessarily true, as opposed to just true.

If you are interested I can briefly explain the semantics of normal modal logics, this might make it easier for you to see why there are stronger and weaker modal logics. Modal logics are really versatile, useful, and both philosophically and mathematically interesting. Actually, you can do topology with the modal logic S4 which I mentioned. A certain feature of modal logic allows one to model various logics on different kinds of discourse which can't be treated adequately by plain classical logic, like temporal, deontic, and epistemic discourse.

I'm not clear what the third assumption that you say is irrelevant is, so I can't comment on that or whether I agree it is irrelevant.

That assumption is 'it is not necessary that God exists'. Strictly speaking, this assumption is not irrelevant, as the proof would not go through without it. The reason that the conclusion does not rest on this assumption is that this assumption is inconsistent with the other two assumptions. I assumed it so that I could derive a contradiction, and thus derive the negation of the third assumption, resting on the initial two assumptions.

See, the problem I have with this is that it feels like academic card tricks; I mean, honestly, different, more and less strict forms of modal logic making an 11-step exercise in logical algebra either valid or invalid as a proof for a question like "does god exist" strikes me as totally absurd and like people have overthought themselves into an imaginary realm of significance - no offense meant.

I doubt if many logicians or philosophers would see this as a genuine proof of God's existence. The reason it is interesting is because the argument is deductively valid according to a logic (S5) which many have thought to be the most plausible candidate for an alethic modal logic. For me, the interest is mostly confined to philosophical issues about logic; I am very interested in logic and not at all interested in religion. This is about the best I can do for a 'spiritual' topic, and so far it seems to be generating some interest.


Alasdair and Kittycat, something has come up and I have to run out the door. I will be back to reply to your posts later.

 

[spacejunk]

D_m, when i read posts like this, i'm reminded why i never majored in philosophy. It ties my mind in knots.
I enjoy reading it though, even if i find it hard to follow (let alone reply in any meaningful way)

Sorry, this is a bit of a nonsense post, but nice to see such intellectual posting here

 

[drug_mentor]

isn't the entire thing built on the premise "if god exists..."? well, if he doesn't exist then it is not necessary that God exists, right?

The argument does rely on a hypothetical/conditional premise ('necessarily, if God exists then it is necessary that God exists'), but this assumption does not say anything about whether God exists. The idea is that, if one reflects on the concept of God as a necessarily perfect being, one will accept the conditional premise regardless of whether one thinks God exists or not. By way of analogy, one might accept the truth of the conditional 'if it is raining, then it is wet outside' irrespective of whether they think it is raining or not.

For the purposes of proving the existence of God, we wouldn't want to make an assumption either way as to the existence of God. As soon as one takes the existence (or non-existence) of God for granted, you no longer have any sort of 'proof' regarding God's existence, just a circular argument. The idea behind this argument is that one can accept the conditional assumption without making any assumption as to whether God exists or not. If one also accepts the premise 'it is possible that God exists', then we can derive the conclusion 'God exists' using the logic S5.

The argument is supposed to demonstrate that we can derive the proposition 'God exists' from some (allegedly) plausible assumptions, without making any assumption as to whether God exists. In order to reject the conclusion we need to reject one (or both) of the premises, or reject S5 as an adequate logic of necessity and possibility. I like S5 more than S4 for philosophical reasons that don't bear directly on this particular argument, so I am inclined to reject the conditional assumption (1).

I dont care what logic you use, it is not factual proof. You are making it so your statement is valid but in the real world, there is no rigorous evidence for the existence of god.

Well, the proof isn't supposed to appeal to empirical evidence. The argument is supposed to be an a priori proof that God exists, which follows from nothing more than a couple of conceptual truths about 'God' and deductively valid principles of inference. For this reason, the objection that there is no empirical evidence which suggests that God exists kind of misses the mark.

The most charitable way I can interpret your objection is as follows: Logic is a useful tool, but it has limitations, we can't derive substantive conclusions about the world from purely conceptual assumptions. I am quite sympathetic to this point, but I am not sure how tenable this view really is. Let us consider the following argument:
P1. All husbands are married
P2. No bachelor is married
Therefore,
C. No husband is a bachelor
(I trust that the validity of this argument is rather obvious, it seemed gratuitously unnecessary for me to prove this formally using first-order logic, but if anyone questions the validity of the argument I can happily supply such a proof.)

'No husband is a bachelor' seems to me to be a fact about the world, yet, the premises which I used to derive it are conceptually true. You might concede this point, but object that the conclusion is not just a fact about the world, it is also a conceptual truth, so it is no surprise that we can derive it from related conceptual truths. The situation with God existing is not analogous, for 'God exists' is not a conceptual truth. I am sympathetic to this intuition, but it seems to me that is plainly fails as an argument, for the proponent of the ontological argument maintains that 'God exists' is a conceptual truth, as demonstrated by its logical derivation from other conceptual truths. Denying that 'God exists' could be a conceptual truth, in the absence of supporting argument, commits the fallacy of begging the question against the proponent of the ontological argument.


I appreciate the compliment, spacejunk. I find your posts very thoughtful and informative. I am quite certain that if you are finding my posts hard to follow the fault lies with my exposition, not with your philosophical dispositions.

 

[swilow]

God, as a being, lacks nothing. Therefore, God cannot lack the quality of existence. Therefore, God must exist.

That is how I had this argument explained to me, but I still believe the whole argument is predicated on the assumption that God exists. It is highly circular logic. Its kind of like an optical illusion almost, it makes sense at certain points before devolving into gibberish. Therein lies my problem with formal philosophy. At its heart, it is simply semantic and may or may not have any relation to the real world.

The ontological argument also doesn't tell us anything much about the nature of god and its manifestation in the universe. Anselm may have found it compelling, but it bears no actual relationship with Christianity or the like. None of it tells us we should go to church on Sunday or not eat our children. The god presented in this argument is more like a functionality of the universe, like gravity or nuclear force, rather than a powerful, interventionist supernatural and benevolent organism sending children to a speck of a dust in an eternal expanding void.

(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

The above is one reason I have no interest in philosophy. That makes no sense to me, whatsoever. My brain literally does not want to understand that.

I've always had more fun dismissing god based on evil humans and dying children and stuff.

 

[drug_mentor]

God, as a being, lacks nothing. Therefore, God cannot lack the quality of existence. Therefore, God must exist.

That is how I had this argument explained to me, but I still believe the whole argument is predicated on the assumption that God exists. It is highly circular logic. Its kind of like an optical illusion almost, it makes sense at certain points before devolving into gibberish.

That's an interesting paraphrase of the argument. Phrased in that way, the argument is viciously circular; when you say 'God has property X' you presuppose that God exists, saying 'if God exists, then God has property X' doesn't presuppose that God exists in the same way, arguably, it only presupposes that the concept of God exists (which seems uncontroversial).

Therein lies my problem with formal philosophy. At its heart, it is simply semantic and may or may not have any relation to the real world.

A lot of philosophical problems arise out of semantic confusion. A big part of analytic philosophy is an attempt to achieve conceptual precision in order to try and solve or dissolve these problems. There is a sense in which any form of human enquiry that is couched in language is 'semantic', and even the supposition that science has 'any relation to the real world' depends on a number of assumptions that are rather difficult to justify.

I do agree that the primary aim of philosophy is not to discover facts about the world in a way analogous to the physical and social sciences. In terms of fields of enquiry, I think philosophy is probably most continuous with law. It is highly normative, and more about argument and interpretation than hard facts - as a general rule, once a question can be rigorously studied in an empirical fashion then the question ceases to be philosophical.

The ontological argument also doesn't tell us anything much about the nature of god and its manifestation in the universe. Anselm may have found it compelling, but it bears no actual relationship with Christianity or the like. None of it tells us we should go to church on Sunday or not eat our children. The god presented in this argument is more like a functionality of the universe, like gravity or nuclear force, rather than a powerful, interventionist supernatural and benevolent organism sending children to a speck of a dust in an eternal expanding void.

That's true, but the argument is appealing to a conception of God as a necessarily perfect being. Christians and the like might be able to argue that this entails God having some of the properties they say God has, like omniscience and benevolence. Certainly, nothing in this argument suggests that we should go to church, or that there are objective moral facts. However, once the theist has 'proved' God exists, they may be able to use this as a premise with other reasonable assumptions in order to try and derive the existence of moral facts. (I don't know if anyone has actually done this, it would be interesting if so!)

The above is one reason I have no interest in philosophy. That makes no sense to me, whatsoever. My brain literally does not want to understand that.

I only did basic maths in high school, and I was very reluctant to take 'intro to symbolic logic' out of fear that it would be very dry and difficult. I did enrol, because several working philosophers told me that if I hoped to pursue philosophy further then logic was not really negotiable, particularly given that my interests tend to lie in the more abstract and theoretical side of analytic philosophy. I couldn't be more glad I took the class, to my surprise I fell in love with logic very quickly and have since taken every undergraduate class in philosophical logic that my University offers.

It isn't as difficult as it seems. That derivation is about as difficult as a moderately hard problem in arithmetic, that might even be overstating the matter. Once you learn the symbols and the inference rules then doing logic by natural deduction is not overly difficult. The meta-theory and philosophy of logic can be very difficult and complex, but natural deduction is relatively easy and (in my opinion) well worth learning.

 

[drug_mentor]

isn't the entire thing built on the premise "if god exists..."? well, if he doesn't exist then it is not necessary that God exists, right?

alasdair

I was thinking about this post a bit more, I was worried I had misinterpreted it in my initial response to you. On my initial reading, I took you to be suggesting that the argument might be assuming the conclusion somewhere. Now I am wondering if you are suggesting that, taking a premise that follows from (1), we might be able to prove that God's existence is not necessary. This thought inspired me to do a little bit of work.

It turns out that, taking the initial assumption as 'necessarily, if God doesn't exist, then it is not necessary that God exists' ([](~p-->~[]p)) then it appears one needs to assume 'necessarily, God exists' in order to prove 'God exists', so trying to proceed in this way would be circular:

Here is the proof:
(1) [](~p-->~[]p) _{1} - Assumption
(2) []p _{2} - Assumption
(3) ~p-->~[]p _{1} - From 1, by []-elimination
(4) ~~[]p _{2} - From 2, by double negation introduction
(5) ~~p _{{1, 2}} - From 3 and 4, by modus tollens
(6) p _{{1, 2}} - From 5, by double negation elimination
QED

Note the assumption on line (2), this is clearly a circular argument, and the assumption on line (1) (as well as many lines of the proof) are totally redundant, since the following would suffice:
(1) []p _{1} - Assumption
(2) p _{1} - From 1, []-elimination
QED

One can prove 'God exists' from 'necessarily, if God doesn't exist then it is necessary that God doesn't exist' and 'it is possible that God exists':
(1) [](~p-->[]~p) _{1} - Assumption
(2) <>p _{2} - Assumption
(3) ~p-->[]~p _{1} - From 1, by []-elimination
(4) ~[]~p _{2} - From 2, by definition of <>
(5) ~~p _{{1, 2}} - From 3 and 4, by modus tollens
(6) p _{{1, 2}} - From 5, by double negation elimination
QED

This proof is better than the proof in the OP, it is shorter and there is no need to make a third assumption only to discharge it later. I suppose that it might be hard to see the first assumption as being equally plausible as the first assumption of the proof in the OP. In order to motivate the plausibility of this view, I have established that 'necessarily, if God doesn't exist then it is necessary that God doesn't exist' is a logical consequence of 'necessarily, if God exists, then it is necessary that God exists'.

Proof of the foregoing claim:
(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
(12) ~p _{12} - Assumption
(13) p & ~p _{{1, 2, 12}} - From 11 and 12, by and-introduction
(14) ~<>p _{{1, 12}} - From 2 and 13, by reductio ad absurdum
(15) ~[]~p _{15} - Assumption
(16) <>p _{15} - From 15, by the definition of <>
(17) <>p & ~<>p _{{1, 12, 15}} - From 14 and 16, by and-introduction
(18 ) ~~[]~p _{{1, 12}} - From 15 and 17, by reductio ad absurdum
(19) []~p _{{1, 12}} - From 18, by double negation elimination
(20) ~p-->[]~p _{1} - From 12 and 19, by conditional proof
(21) [](~p-->[]~p) _{1} - From 20, by []-introduction
QED

Apologies for the lengthy and somewhat messy proof. I tried to establish whether the formulas on lines (1) and (21) were equivalent. As the above proof shows, the formula on (1) implies the formula on (21); however, I am having a bit of trouble determining whether (21) implies (1). I suspect that it does not, but I smoked a little weed earlier and might postpone the investigation until a later date.

 

[Xorkoth]

Generally, it isn't thought that if something exists, then it is necessary for it to exist. Think of it this way, my parents met at a bar, it seems hard to deny that it is possible that one or both of them could have decided to stay home that night, if this had eventuated then my parents would never have got married and procreated, so I would never have existed. In this sense, even though I do exist, it is possible for me to have failed to exist. In philosophical jargon, we would say that my existence is a contingent fact, as opposed to a necessary one.

Alright, I see, this makes sense. I do exist, but it is not necessary that I exist, I could have just as easily not existed. Or I guess that depends on your view of things anyway (deterministic vs nondeterministic universe), but it certainly isn't a given in any case.

I'd like to address some other points but I'll do so later because I've spent too much time on Bluelight this morning and I have to catch up on some work.

 

[RDP89]

I don't see how you can possibly prove anything based entirely on an assumption.

 

[drug_mentor]

Xorkoth, I look forward to your extended reply. I think even a determinist can make sense of necessity and possibility to some extent, they can say that every event is necessitated by antecedent events and the laws of nature, but conjecture that the initial state of affairs could have been different. If the initial state of affairs was different, then the causally determined events would also have been different. As a determinist myself, I have more trouble with the notion of probability than possibility, and have wondered if the former might be analysed in terms of the latter, though, that is probably a topic for a different thread.

I don't see how you can possibly prove anything based entirely on an assumption.

Well, to be fair, it's two assumptions. More seriously, in logic, any conclusion that is not a logical truth (essentially conclusions that rest on no explicit assumptions (though, clearly, they do rest on whatever assumptions the logic itself rests on; one could argue that, for example, proving the Law of Excluded Middle with classical logic is circular), things like the Law of non-Contradiction) will rest on one or more assumptions. (I favour Lemmon style natural deduction systems, if one is more influenced by Fitch then one would be more likely to refer to these assumptions as hypotheses, but this is a purely terminological point.)

This is more an epistemological point than a logical one, but I expect your (and everyone else's) basic beliefs are all assumptions themselves or derived purely from assumptions. If you can't get from assumptions to substantive truths, it is difficult to see where you could possibly begin. Take this for example, you can't prove that there are physical objects; when you observe objects that seem exterior to you, you posit that these objects are mind-independent and would exist even if you were not there to observe them. How could you prove this? Your observations are perfectly consistent with the hypothesis that they are generated by your mind - yet, this hypothesis is generally regarded as absurd! But what tells you that it is absurd? Your experience? How could it, when your experience is consistent with a hypothesis which denies that there are mind-independent objects? You might say, well it is more parsimonious to think that the uniform nature of my perception is explained by sensory apparatus engaging with relatively uniform objects. But, doesn't this reply rely on at least three assumptions? Namely, that mind-dependent entities will be less uniform than mind-independent ones, that explanatory parsimony should be favoured over ontological parsimony, and that it is a virtue of an 'explanation' that it be parsimonious?

 

[Godel]

This is logic at its weakest. Really, all logic can do is dive deep, deep into grammar. This is a consequence of the fact that any formalization is, at bottom, a translation of an interpreted common language. It can't exceed the features of language in any way, unless you credit it with clarification in very specific cases. But even there, to clarify is to already have an interpretation in mind. For what you are trying to do, this "proof" is a contrivance for you to show off you ability to perform rudimentary formal proofs of the sort learned in any introduction to logic course. I am dubious of your bringing in modal logic together with your reference to kripke; its smacks of a google search of: "is anything important still happening in analytical philosophy?" Why frame a simple argument in an alienating academic technician's language, except to provide yourself an opportunity to make stock comments about logic to people who don't have the training to counter argue?

And:

Assume the argument is valid- [](p--->[]p) is a still huge, huge metaphysical assumption. So the arguement is not sound, because the truth value of one of your premises is indeterminate, at best (which is more problematic than its being false, really.) Without soundness all you really prove is the course of a few very basic operations of formal logic; there is no content to your proof, just form. It is not a mirror of Anselm's argument.

Your generalization of Anselm's (1) to your (1) is inappropriate. They do not have the same valency. Your (1) is just your (1), while his is packed with ontological assumptions. If God exists, then it is necessary that god exists by no means stands on its own: To say, the Egyptians, the existence of gods could be considered contingent, they may die or be isolated or change status in the world. The necessity of god is a deeply cultural idea- god as the pillar of the earth, the entelechy of creation - and all these, (1.1) (1.2) (1.3) as it were, elements intrinsic to Anselm's argument are lost in yours, which is problematic because you treat them interchangeably. As opposed to clarity, this only buries the real, labyrinthine difficulties deeply under the surface, leaving the appearance of a level plane.

Lastly, and my logic is rusty, but you seem to be confusing premises and assumptions. Your assumptions are actually premises. You only show that your premises contradict one another, and so your argument is fallacious, and if that isn't true, then it is still fellatious for the reasons above.

 

[drug_mentor]

Lastly, and my logic is rusty, but you seem to be confusing premises and assumptions. Your assumptions are actually premises. You only show that your premises contradict one another, and so your argument is fallacious, and if that isn't true, then it is still fellatious for the reasons above.

A proposition can be both a premise and an assumption in an argument. In a formal proof, every line except the conclusion contains a premise of the argument.* But, only the lines that introduce formulas which are not derived from any other line(s) in the proof are assumptions. If you are most familiar with Fitch-style natural deduction, you might be more used to seeing what I am calling 'assumptions' referred to as 'hypotheses'.

Your logic is quite rusty, the assumptions which the conclusion rests on don't contradict one another. I introduced a third assumption, which is inconsistent with the initial two assumptions, in order to derive the negation of the third assumption by reductio, thus 'discharging' the third assumption. Do you deny that reductio ad absurdum is a valid form of inference?

If you believe my argument is invalid please indicate which line(s) of the proof you think is faulty and why.

But even there, to clarify is to already have an interpretation in mind. For what you are trying to do, this "proof" is a contrivance for you to show off you ability to perform rudimentary formal proofs of the sort learned in any introduction to logic course. I am dubious of your bringing in modal logic together with your reference to kripke; its smacks of a google search of: "is anything important still happening in analytical philosophy?" Why frame a simple argument in an alienating academic technician's language, except to provide yourself an opportunity to make stock comments about logic to people who don't have the training to counter argue?

This is a simple proof and I said as much in one of my responses to swilow. It's disappointing to be accused of showing off for sharing something that I think is interesting and trying to make it accessible. It is difficult for me to see how Kripke semantics for modal logic relate to what is happening in analytic philosophy now in any important sense, considering they were developed in the late 1950's...

I made a thread about logic about 18 months ago and several people expressed in that thread that they were familiar with formal logic, including modal logic. Moreover, a few people who work in IT are on here and presumably have some familiarity with logic. Obviously, most people don't know modal logic, but I had some reason to expect that some posters might be familiar with it.

Personally, I don't find the ontological argument very interesting in itself. I do find the fact that it can be 'proved' using a well-known logic interesting, and I thought framing the thread around the claim that there is a formal 'proof' of God's existence would be more likely to generate some interest.

This is logic at its weakest. Really, all logic can do is dive deep, deep into grammar. This is a consequence of the fact that any formalization is, at bottom, a translation of an interpreted common language. It can't exceed the features of language in any way, unless you credit it with clarification in very specific cases. But even there, to clarify is to already have an interpretation in mind.

It seems obvious that a highly restricted formal language will not be able to exceed natural language in terms of expressive power, quite the contrary. However, as you note, logic might exceed natural language insofar as it makes certain important features of arguments more perspicuous. It's true that any logic needs to stipulate how the logical operators behave, but that doesn't mean clarification through logic is a circular exercise. Stipulating the behaviour of various operators does ultimately determine how any complex argument will be interpreted, but that doesn't mean that this interpretation of an argument will be perspicuous before one translates the argument into the logical language. In this sense, logic can show us the deductive consequences of conceiving of logical operators (as well as particular expressions in natural language) in particular ways.

This argument is formally interesting for two reasons: (i) I don't think it is at all obvious that 'God exists' follows logically from assumptions (1) and (2), but translating these statements into a logical language which I find very plausible suggests that it does indeed follow; (ii) The argument being S5 valid and invalid in S4 casts light on the inference rule that is responsible for the proof going through in S5 - this provides an opportunity to reflect on whether a particular interpretation of how modal operators behave is the correct one. So, while I am somewhat sympathetic to your point, you seem to be missing the point that the issue you mention is actually one of the reasons that this is a philosophically interesting formal argument.

Assume the argument is valid- [](p--->[]p) is a still huge, huge metaphysical assumption. So the arguement is not sound, because the truth value of one of your premises is indeterminate, at best (which is more problematic than its being false, really.) Without soundness all you really prove is the course of a few very basic operations of formal logic; there is no content to your proof, just form. It is not a mirror of Anselm's argument.

Yes, the argument is (almost certainly) not sound. I explicitly said, more than once, that I am inclined to dispute the first assumption. I never claimed the argument was sound; I have been clear about the fact I am an atheist (so obviously I don't regard the proof as sound), that I think the second assumption is fairly hard to dispute, and the first rather questionable.

The claim that a premise being indeterminate is "more problematic than it's being false" is a strange one. In general, it seems as though there is good reason to want to know what follows logically from one's beliefs, even if the truth-value of some of those beliefs is indeterminate. I can't think of many cases where it would be very interesting to know what follows from propositions that are known to be false. (Though, the discussion surrounding 'counterpossibles' in the literature on counterfactual logic, as well as debate surrounding 'the principle of explosion', are two interesting examples of this.)

The idea behind formal logic is that arguments are valid or invalid due solely to their form and not their content. In light of this, it is difficult to see how the claim that 'there is no content to the proof, just form' is supposed to function as a substantive objection.

Your generalization of Anselm's (1) to your (1) is inappropriate. They do not have the same valency. Your (1) is just your (1), while his is packed with ontological assumptions. If God exists, then it is necessary that god exists by no means stands on its own: To say, the Egyptians, the existence of gods could be considered contingent, they may die or be isolated or change status in the world. The necessity of god is a deeply cultural idea- god as the pillar of the earth, the entelechy of creation - and all these, (1.1) (1.2) (1.3) as it were, elements intrinsic to Anselm's argument are lost in yours, which is problematic because you treat them interchangeably.

I called this argument Anselm's Ontological Argument, and it is true that the argument as I have presented it is probably not 100% the same as Anselm's - it would probably have been more appropriate to call it an Anselm Inspired Ontological Argument, but such pedantry seemed a bit superfluous in the context. To whatever extent my argument differs from Anselm's, I don't believe this makes the argument any less interesting.

In post #5 I briefly discussed why some have found the first assumption of the argument plausible. I didn't claim that these reasons exactly mirror Anselm's.


_{* The exception to this would be if one makes a superfluous assumption which is not actually used to derive the conclusion.}

 

[Godel]

Ugh I'm so sorry. I got some bad shit last night and behaved belligerently and my entire post was un-necessary. I also enjoy logic, but only took half a class before I had to drop out to work; I only got through FOL and the most basic propositional calculus. I read alot of Wittgenstein and it made me very critical of many application of logic, although I enjoyed it... it never consummated the excitement I felt reading Leibniz's sketch ideas on pure or unmediated symbols.

By more problematic I was thinking of non-sense (very denotative non-sense.) If premises are false, then you are still talking about something. It isn't that we don't know now, but might at some future point, arrive at a truth value for (1), but instead that the meaning of that. I wrote a bunch more but I lost it all. I settled on saying the act of translation introduces artifacts, that you started by convertigna compound proposition written in a sort of shorthand (anselms (1)) into a elementary one and hence the divergence between his and yours... I wrote more elegantly but I posted it an it never popped... maybe it is being reviewed by admins for the naughty language I sometimes use in the course of speaking of logic. Sorry to act the fool I know it is rare to be able to communicate one's interests to others openly and with reciprocated interest.

 

[drug_mentor]

Ugh I'm so sorry. I got some bad shit last night and behaved belligerently and my entire post was un-necessary.

That's okay, I hope that you are having a better day today.

I also enjoy logic, but only took half a class before I had to drop out to work; I only got through FOL and the most basic propositional calculus. I read alot of Wittgenstein and it made me very critical of many application of logic, although I enjoyed it... it never consummated the excitement I felt reading Leibniz's sketch ideas on pure or unmediated symbols.

If you are interested in pursuing logic in your own time I can recommend some good books.

I have read quite a lot of later Wittgenstein and interpretations of his work in the last six months, his work is very compelling and interesting. I actually just read Wittgenstein on Rules and Private Language by Saul Kripke last week, which altered my perspective on Wittgenstein's Philosophical Invesitgations a little bit. Kripke makes an interesting connection between Wittgenstein's concerns in philosophy of mind and philosophy of mathematics. It's worth checking out if you haven't read it.

By more problematic I was thinking of non-sense (very denotative non-sense.) If premises are false, then you are still talking about something. It isn't that we don't know now, but might at some future point, arrive at a truth value for (1), but instead that the meaning of that. I wrote a bunch more but I lost it all. I settled on saying the act of translation introduces artifacts, that you started by convertigna compound proposition written in a sort of shorthand (anselms (1)) into a elementary one and hence the divergence between his and yours...

I don't really see how false premises refer to something in a way that premises with an indefinite truth value do not. Without getting sidetracked by theories of meaning, I can't think of any that would obviously have this consequence. It certainly isn't a consequence of a (later) Wittgensteinian conception of meaning, which shifted the focus from truth conditions to assertability conditions.

I know it is rare to be able to communicate one's interests to others openly and with reciprocated interest.

It can be difficult to do so successfully, especially with things like formal logic and analytic philosophy. I hoped that the idea that there was a proof of God was contentious enough to bait some people into seeing there is some interest here, and thought it might also attract some believers who might want to defend the first assumption.

 

[Godel]

I had been until that point, but I always felt like I was stuck between learning logic/proofs, and trying to figure out what logic is representing or doing... relationships between language, thought, meaning. I wish I finished propositional calculus, but I didn't ultimately feel like I was gaining clarity... even my teacher ultimately seemed to appeal to vauge and metaphorical notions to describe things like logical form. I was always waiting for the curtain to be pulled back to concretely show the foundations were as rigorous as the activity itself; but then again I suppose generations have struggled for and against such foundations. I've read Quines little book, which was surprisingly candid and interesting, but I was more amazed than anything to see even he was vexed by them still, having done so much work in philosophy. I thought of Wiley Coyote continuing to run even though the cliff ended a ways back... Besides that, really only the Tractatus and webpages on techniques and ideas in logic.

Speaking of the Tractatus, it was more in that vein I was referring to nonsense. I think there is a tractarian sense in which non-sense is outside of the world, and there is no outside the world. Which the tractatus sets out to prove, and then finally reveals itself to be a book of nonsense even trying to talk about this. God I loved that book, and even if he revised his ideas it stands with some compelling ideas (to say nothing of its achievements as a work of art; poetry, maybe.) I did read Rules and Private Language, and actually enjoyed it very much. Kripke was described by a friend of mine as the sort of godfather of classical, conservative analytical philosophy, so I expected a text that aimed at negating Wittgenstein. I was quite wrong. He takes Witty absolutely seriously, and he isolated/expanded the rule following paradox excellently. I don't think Wittgenstein would have thought of it as a "paradox", but Kripke explicitly states that this is PI "as it struck him" which I respect enormously. He had dialog with him instead of cheaply seeking exegesis. I might also recommend Murray Shanahan's Embodiment and Inner Life. Its a book by an embodied AI researcher that uses Wittgenstein to escape the philosophical morass that is Phil. of Mind and clear the terrain for designing a theoretical framework for describing a wide area of the "space of possible minds". The movement from philosophy to research program is astounding, and he is very lucid on Wittgenstein. Paul Feyerabend buries it deeper, but is very much a Wittgensteinian.

His work has congealed into so many different areas outside of philosophy; it is interesting to see how and where, for me at any rate. Wittgenstein and Spinoza belong to a class of their own for single-minded rigor and fertility.

IF oyu have any good reccomendations, I'd like to hear them. I came to philosophy piecemeal from chemistry and virology, but now I work so, so far away from everything in direct service public health- less time for mental beach combing. These is still alot to think of, but most of it is pretty practical or metaphysical... trying to work out little theodices. It must be a gnostic, implacable god, I think. Some shitbird demiurge in between us making all the myriad troubles of the earth.

In a sense I am a believer, but it is in no sense clear what people are talking about when we talk about god. It is easier if you bring in plato and his basterdizations to talk about spiritual worlds or planes, and their imbred offspring, afterlives. But if there is only the world, and nothing else, God is a strange mystery.

 

[drug_mentor]

Speaking of the Tractatus, it was more in that vein I was referring to nonsense. I think there is a tractarian sense in which non-sense is outside of the world, and there is no outside the world.

I don't see how a proposition with an unknown truth-value is 'outside of the world' - but it's a good while since I read the Tractatus. (I agree that it's a great book, but Wittgenstein's later work has had considerably more influence on me.)

Kripke was described by a friend of mine as the sort of godfather of classical, conservative analytical philosophy, so I expected a text that aimed at negating Wittgenstein. I was quite wrong. He takes Witty absolutely seriously, and he isolated/expanded the rule following paradox excellently. I don't think Wittgenstein would have thought of it as a "paradox", but Kripke explicitly states that this is PI "as it struck him" which I respect enormously. He had dialog with him instead of cheaply seeking exegesis. I might also recommend Murray Shanahan's Embodiment and Inner Life. Its a book by an embodied AI researcher that uses Wittgenstein to escape the philosophical morass that is Phil. of Mind and clear the terrain for designing a theoretical framework for describing a wide area of the "space of possible minds". The movement from philosophy to research program is astounding, and he is very lucid on Wittgenstein. Paul Feyerabend buries it deeper, but is very much a Wittgensteinian.

I am fairly certain Kripke is an externalist about meaning himself, and it wouldn't surprise me if he first got the idea from Wittgenstein.

Thanks for the recommendations, I might check out the Murray text when I have more time. I am vaguely familiar with Feyerabend, I'm not overly sympathetic to his views, but if I read him perhaps that would change.

IF oyu have any good reccomendations, I'd like to hear them.

What interests you? It's a lot easier to give recommendations if I have a better sense of what areas of philosophy you're most interested in.

So far as logic: start with Beginning Logic by E. J. Lemmon to get the basics of classical propositional and predicate logic; Modal Logic: An Introduction by Brian Chellas is a solid introductions to modal logic; If you are interested in non-classical logic then Graham Priest's An Introduction to Non-Classical Logic has a really nice overview of lots of logics and their respective philosophical motivations (I personally don't like his 'tableau' presentation of proofs), An Introduction to Substructural Logics by Greg Restall is a really good text on a variety of non-classical logics and presents an interesting way to conceptualise different logical systems; If you are interested in relevant/relevance logic specifically then Relevant Logic: A Philosophical Interpretation by Edwin Mares is a good introduction to the relevant logic R but is more philosophical than formal/technical, Anderson & Belnap's Entailment (I can only speak for Volume I) is an entertaining book with a wealth of technical material on various relevant logics - be warned though, Entailment is a technically demanding text.

 

[LopLover]

You guys are so far above my thinking that my head is now spinning! One point--in someone's post--I can't find it now--they said something about Christianity. But who is positing a Christian God? As opposed to a Jewish, Muslim, or some other kind of God, that is?