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Gödel's ontological proof (wikipedia.org)
55 points by kdavis on Dec 28, 2014 | hide | past | favorite | 52 comments

The difference between Anselm's original ontological argument and modern ones like Goedel's or Plantinga's is that while Anselm's is full of terrible reasoning, Goedel's reasoning is (unsurprisingly) watertight and all the terribleness is concentrated in the axioms.

Specifically, I think the idea of classifying all properties as "good" or "not good" is hopeless: I do not believe there is any such classification that fits anyone's intuitive ideas about goodness well enough that calling something "godlike" if it has all "good" properties is credible. More specifically, I suspect it's consistent to suppose -- hence there are no humanly-comprehensible counterexamples -- that the only notions of goodness obeying all Goedel's axioms are ones that look like "P is good iff P(x)" for some fixed object x. (And, like Krishnaswami, I think the system of modal logic Goedel needs is awfully strong.)

In the spirit of Anselm's ontological argument, however, I offer the following proof of the nonexistence of God:

Consider a really bad argument for the existence of God. In fact, consider one so bad that no worse argument can be conceived.

Obviously a bad argument for something fails to prove what it purports to prove. But merely failing is a pretty mediocre kind of badness. The worst possible argument, surely, has to be much worse than that: it must conclusively prove the opposite of what it's meant to prove.

Now, the worst conceivable argument for theism clearly "exists in the understanding", as St Anselm put it. But it can't exist only there -- because a bad argument is more damaging to the premise it's meant to support if it's actually made.

Therefore, there is an argument for the existence of God which is actually a conclusive proof of the nonexistence of God.

And, of course, any proposition that can be conclusively disproved is false; therefore there is no God.

(This argument is in my opinion almost exactly as strong as the original ontological argument for the existence of God. Which is to say, it's absolutely hopeless. But I think it's fun.)

> The difference between Anselm's original ontological argument and modern ones like Goedel's or Plantinga's is that while Anselm's is full of terrible reasoning, Goedel's reasoning is (unsurprisingly) watertight and all the terribleness is concentrated in the axioms.

To me that makes it a success. Part of the entire game of philosophy is to use watertight reasoning to illustrate flaws in the original axioms.

I would think that any god that is proven to exist by simply defining it to be identical to something that exists is pretty worthless to begin with. You could always just short-circuit and just say "I am right by definition."

Which really gets us closer to the heart of what proof is: it is that which will compel a rational mind into agreement. It is not enough to follow formal rules. It must be measurably dangerous -- potentially a catastrophic failure -- for a rational mind to disagree with a proof. Someone should be able to build a dutch book on your expectations and obliterate you for failing to agree with that proof.

So until someone actually measures their god to be good, there are no consequences for a proof in either direction, and a rational mind should not be convinced.

Your objection to the notion of positive and negative properties is the first that stood out to me after reading the Wikipedia summary. If I'm understanding things correctly, Godel's proof seems committed to some objective notion of goodness--I'm not sure how such a thing could exist without some higher power, which makes the proof somewhat circular in my mind.

I could be missing something here as I only took my first philosophy class this semester and I'm not taking ethics until next, but is that basically what you're getting at?

What many people miss about Anselm's argument is that it is based on an ontology that essentially identifies existence with goodness. Arguments that attempt to derive existence from extreme badness are obviously wrong under that premise. On top of that, existence would not be considered an attribute of arguments themselves. You have trouble with this point yourself, where you quickly replace it with a notion of "being made". Speaking of which, existence would not be considered an attribute of acts, either.

I love it. I think Anselm's ontological argument is more a proof of confirmation bias than anything else. So as a corollary to his argument, the more beneficial something may be to us, the more biased we are to confirm its existence. Therefore, a maximally benevolent being incites maximal confirmation bias on our part, and any logical argument for its existence is therefore problematic.

Yet another interesting side of Anselm's and Gödel's proof is that they leave the possibility open for us to be gods.

wow! did you come up with this?

What, the anti-ontological argument? Yes.

There's another very-slightly-similar inverted ontological argument that may have been in the back of my head. It's due to a chap called Douglas Gasking, and it begins with a standard ontological argument -- which, as you know, delivers us a Maximally Excellent Being. Now, of course God's signature accomplishment is the creation of the universe; but any creative achievement is more impressive when it is accomplished under greater constraints (so, e.g., while it is impressive to come up with a proof of Fermat's Last Theorem it would be even more impressive to come up with a proof of Fermat's Last Theorem in the form of a perfect Shakespearean sonnet). Well, what would be the most impressive obstacle for God to have overcome when creating the world? His own total nonexistence, obviously. So our maximally impressive being must be a god who created the universe without existing; therefore God does not exist.

I have to confess that I like mine better.

When I think of Godel, the first thing I think of (based on accounts) is compulsion. He was really driven to explore the edges of reality using logic as a probe.

Here's an account of his discovery of a logical flaw in the US Constitution: http://morgenstern.jeffreykegler.com/

Not really an account of his discovery, just a story about how he claimed to have found one; no one knows what flaw he thinks he found (but one can guess).

Godel's proof is essentially Anselm's with some mathematical window dressing added. It is worth noting that Anselm's argument applies equally well to Satan as to God: The most evil thing we can conceive of would be more evil if it actually existed, therefore Satan must actually exist. If you're into this sort of mind game, it makes an interesting exercise to translate this version of Anselm's argument into formal modal logic.

To say that Godel's proof is essentially Anselm's is somewhat shortsighted. Godel used not only modern notation, but also modal logic (which would not exist until the mid-20th century) to prove the necessary existence of God.

My biggest point of contention with your claim is that Anselm didn't make a contingent/necessary distinction. Instead, he was caught up in a muddled "exists in understanding/exists in reality" distinction which is decidedly not an analogue of the former. To go a step further, Anselm merely assumed the possibility of God, whereas Godel actually proved it.

> Godel used not only modern notation, but also modal logic (which would not exist until the mid-20th century)

Formal modal logic wasn't invented until the 20th century, but Anselm is using modal logic nonetheless. A modal logic is simply a logic that allows operators to be applied to propositions. Necessary-truth and possible-truth are only two possible operators. "Knows" and "believes" are two other classic examples. Anselm's logic uses the operators "actual-truth" and "conceptual-truth". All these proofs have essentially the same structure whether the operators are actual/conceptual or necessary/possible. You could even render the same proof in the modal logic of knowledge and belief. In fact, modern evangelicals often do this. The result renders as something like, "Look at all the martyrs who died. No one makes that kind of sacrifice for false beliefs, therefore their beliefs must be true."

Even though you are correct and logicians like Avicenna and Peter of Spain wrote in gory detail about (what we take to be) Medieval Modality[1], this does not make Medieval Logic even remotely as rigorous as its modern incarnation. The issue here is about having a sound, consistent, and complete logic.

S5[2] (what Godel used) has those properties, whereas Anselm didn't even have the notion of soundness or completeness, let alone a logical system that would satisfy them.

[1] http://plato.stanford.edu/entries/modality-medieval/

[2] http://en.wikipedia.org/wiki/S5_%28modal_logic%29

Very well, I concede the point.

Would it really apply to Satan? Historically, most Christian thinkers consider him to be a creature and thus finite in every respect; likewise, evil is generally considered by the same thinkers to be insubstantial, i.e. it is the absence of good and not some kind of entity (or put another way, it's not a matter–anti-matter kind of thing). So it seems evil would have a kind of local maximum, or floor value, depending on your perspective, which would be "zero good". At the same time we could speak of evil being compounded over time, i.e. an accumulation of depraved acts and persons.

Of course, you may have other definitions of Satan and evil in mind, and I don't mean to imply that those concepts are solely the province of Christian minds.

One must not get too hung up on terminology here. The point is that the same argument that applies to "the greatest good that can be conceived" applies just as well to "the greatest evil that can be conceived." You can attach whatever labels to these things you like. The point is that if a great good becomes greater by actually existing, then so does a great evil, and so you can apply Anselm's rhetorical trick just as well to the latter as to the former.

But a "semantics of evil" was the point I was trying to bring to the fore. If the maximum local evil (e.g. in a person) is "zero good", i.e. the complete absence of good, then the greatest evil is a fixed value and doesn't seem subject to the same line of reasoning.

Nope, doesn't work. If you admit the existence of even a single affirmative evil act (torture, say) then one can always increase the amount of evil by doing more of that affirmative evil act.

Mathematically it's analogous to trying to reason from the premise that zero -- the "absence of anything" -- is the smallest possible number. You can do it, but it leads to incredibly messy math. As soon as you try to define subtraction, all hell (pun intended) breaks loose. And as soon as you admit -1 into your system you can no longer have a smallest number (if you have addition and induction).

Just a thought experiment:

What if we define The Good as unity, and evil as the ratio of The Good to relative attainment of the The Good. We might then define compounded evil as a ratio of The Good to the multiplication of the denominators of each ratio-evil consider on its own.

So an act of torture might be (/ 1 ½) and three acts of torture would be (/ 1 ⅛).

(/ 1 ½) == 2, and (/ 1 ⅛) == 8, so you're just mapping n onto 2^n. How is that supposed to change anything? You still have a total order and no upper or lower bound, which is all that matters.

The multiples of ½ were arbitrary, just an illustration re: denominators.

Also, your points above are well taken; I was simply trying to come at the matter (and my previous points) with a bit more precision than I had previously, but I cut my elaborations short as I needed to run out the door.

So with my idea above, you would basically have two notions for assigning magnitude to evil. Absolute evil would be defined as the complete absence of good in a person or other locus of circumstances and acts. So the greatest real evil would a fixed value, "zero good", and "negative amounts of good" would be a void construct.

The other notion of evil, "evil considered", would be a metaphor for understanding how evil gets compounded according to our usual perception that some lives and acts are more evil than others. It's basically a "score" which doesn't correspond directly to any substance.

> Absolute evil would be defined as the complete absence of good in a person or other locus of circumstances and acts.

But evil is more than the mere absence of good. There are affirmative evil acts (rape, murder, torture) so you can always become more evil by performing more evil acts.

Suppose more of a Manichaean devil rather than the Christian one.

Neel Krishnaswami wrote a quite readable description of the proof on his blog: http://semantic-domain.blogspot.com/2014/06/g-ontological-ar...

This proof is a mix between Anselm and a little bit of math.

Disclaimer; I've only read Anselm and Leibniz, and am just trying to piece together the discrepant sources around.

Premise 1: There are many worlds.

Definition 1: "x necessarily exists if and only if every essence of x is necessarily exemplified" (From the article)

Premise 2: Existence is good.

Definition 2: God is the being Good-er than whom cannot be conceived.

Sub-Conclusion 1: God if he exists, by definition is good, and by supposition necessarily exists because existence is good. He necessarily exists because if he did not, then a being good-er than him could be conceived (ie, one that existed), and that being would then not be god.

sub-conclusion 2: Because there are many worlds, there exists one with a being good-er than whom cannot be conceived. Therefore, this is a being whose existence is necessary.

Consider then, that God's existence would not be necessary, if the world that god governed was not necessary, because that world could, or could not exist, and so God could, or could not exist, and god's existence would not be necessarily be exemplified.

Therefore The world in which God exists must be necessary to our world, and by consequence, God's existence must be necessary to our world.

If someone who has actually read him could give me some feedback as to whether this is fairly close to what he means, I'd appreciate it.

>Premise 1: There are many worlds.

The premise is called the plenitude principle. "The principle of plenitude asserts that the universe contains all possible forms of existence."[1] You first see it in Plato's work like in the Cave Allegory. You can make a compelling argument that the plenitude principle was the precursor to the ideas of a multiverse and evolution.

1. http://en.wikipedia.org/wiki/Principle_of_plenitude

Readers might also enjoy the audio recording and transcript of a famous debate between Bertrand Russell and Fr. Frederick Copleston:

The Famous 1948 BBC Radio Debate on the Existence of God


"Your general point, then, Lord Russell, is that it's illegitimate even to ask the question of the cause of the world"

"Yes, that's my position"

"Well, if it's a question that for you has no meaning, it's of course very difficult to discuss it, isn't it?"

Russell, almost with a sigh: "Yes, it is very difficult." Then: "What do you say, shall we pass on to some other issue?"


What struck me most was the calmness and politesse they both maintained, and the steadfast articulateness at the speed of thought. There seemed to be an unspoken, shared assumption that you never, ever raise your voice or betray any hint of frustration, let alone insult your opponent in public. It's hard to imagine hearing something similar on the radio today.

I'm also curious how much of the decorum was a sign of the times, and how much was specifically British.

Russell and Copleston were titans of philosophy as well, and the philosophers I've seen tend to discuss philosophy in a calm, polite, and considered fashion.

Note that even if you accept this proof, it says nothing about a god that answers prayers or otherwise intervenes in daily life. It also says nothing about an afterlife. Unless you're a deist, ontological arguments are practically useless. They've been used in the defense of many gods now relegated to mythology.

I don't know about that. If you accept an ontological argument as valid, what it gives you is a god who's maximally perfect in every way. So then if you want to show (e.g.) that he answers prayers, all you have to do is to argue that answering prayers is more perfect than ignoring them.

Descartes gives a kinda-ontological argument for the existence of God, and then he does indeed very rapidly move to deducing useful properties of God from it -- e.g., that he wouldn't allow us to be systematically deceived about everything. It's the basis of his whole epistemology, at least ostensibly.

Whereas, say, a first-cause argument tells you there's something that's in some sense the cause or origin of the universe, but you've got a whole lot of work to do to get from there to anything relevant to actual religious belief. (Though, e.g., William Lane Craig does somehow manage to keep a straight face when transitioning at lightning speed from "something caused the universe to begin to exist" to "and that something must be personal, enormously powerful, etc.)

(Lest I be misunderstood, I'll add that I think both ontological and cosmological arguments fail badly, and that neither of them either manages to give much reason to believe in a maximally perfect being / first cause / whatever, or to believe that such a being if it existed would have anything much to do with the gods of traditional religions. But it's not my purpose to litigate any of those claims here -- Goedel's argument is much more interesting than any I'd make.)

I thought the same thing as you (answers prayers == more perfect), but figured proponents of the argument would ignore that. If a perfect being must answer prayers and prayers aren't answered (which seems to be the case), then it's kind of hard for a perfect being to exist. Either that or everyone is praying to the wrong gods.

> Goedel's argument is much more interesting than any I'd make.

Hardly, it's just obscurantist gobbledegook dressed up as modal logic. It uses symbols and math to make the same basic errors that all other proofs make.

It's disappointing that I'm being voted down. The argument only proves that perhaps some vague thing which Goedel slaps the 'god' label on might exist, or might have existed. The fancy looking methodology only serves to obscure this obvious limitation of logical argument proofs.

On the disagreement hierarchy[1], your downvoted comment is DH3 at best. Also, it's needlessly disparaging. I think you'd fare better if you avoided such digs at others' opinions in the future.

Also, don't worry about downvotes. HN isn't that big, and individual comments are quickly forgotten. Think of it as feedback, not condemnation of you personally.

1. http://www.paulgraham.com/disagree.html

I hadn't seen the disagreement hierarchy. Very cool.

I'm not that fussed about the downvote, I'm just innately curious as to people's reasons.

Or the FSM

and an implementation (I think unfinished) in Coq https://github.com/FormalTheology/GoedelGod

I only dabble in formal logic so I haven't come across this before, but it sort of feels like this rests on a circular argument. You know, the "can God make a rock so heavy that he can't lift it" or "does God have the power to make himself even more Good". I wonder if it's possible to prove anything if one of your axioms is actually a hidden circular argument, much like how one can prove anything from setting true to false.

All ontological arguments for God are of the form:

1. I carefully define God as having to exist.

2. Therefore, God exists.

All the work goes into the first part, however many volumes of obfuscation (hi, Dr Plantinga!) one hides it behind.

There are a number of problems with this approach.

What you're describing in the second part of your post is the Principle of Explosion[1]. It does not apply to Godel's argument as his axioms entail no contradictions.

I'm not sure what you mean by "circular" in this context. However, what people usually mean is an argument where the conclusion is contained in one of the axioms. Godel's proof doesn't explicitly suffer from this but I will say that his fourth axiom, i.e. Good(phi) -> [Necessarily]Good(phi) is cutting it pretty close.

[1] http://en.wikipedia.org/wiki/Principle_of_explosion

We see our world in terms of cause and effect, but these are temporal concepts that might have no meaning outside of linear time. It may therefore be nonsensical to ask what caused the universe.

We're still talking about God? There must be something to this "God" idea that is important / common to humanity.

The gullibility of children, usefully instilled into us by natural selection.

The fear of death, usefully instilled into us by natural selection.

Human minds have no experience of not existing, and tend to find the concept difficult to comprehend. We desire answers, but unfortunately the human condition is to prefer wrong answers over no answer.

Human brains also deceive us into believing we are distinct entities, whereas in reality the universe is probably a single entity.

Cute, and in some respects insightful, but as a contribution it's arguably an equivocation on the definition of entity. You could similarly argue that human brains deceive us into believing we are distinct entities, whereas in reality we are billions of entities experiencing the emergent properties of multi-cellular life.

> probably a single entity

"Probably"? How about "surely" or "surely not"? Is an entity comprised of many things or one thing? Does it matter? Are you merely pointing out that there is no true separation between anything and everything that exists?


It seems to me that he left out a couple of "if it is possible to exist" clauses here and there.

This is exactly the opposite of what is the case. Anselm implicitly assumed that it was possible for God to exist, but Gödel proves it as a theorem (theorem 2 on the Wikipedia page).

This assumes that "goodness" is an irreducible, ontologically basic property.

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