In 1970, Gödel — amidst the throes of his worsening hypochondria and paranoia —entrusted to his colleague Dana Scott[1] a 12-line proof that he had kept mostly secret since the early 1940s. He had only ever discussed the proof informally in hushed tones among the corridors of Princeton’s Institute for Advanced Study, and only ever with close friends: Morgenstern, and likely Einstein[2].
This proof purported to demonstrate that there exists an entirely good God. This proof went unpublished for 30 years due to Gödel’s fears of being seen as a crank by his mathematical colleagues — a reasonable fear during the anti-metaphysical atmosphere that was pervading mathematics and analytic philosophy at the time, in the wake of the positivists.
Oskar Morgenstern remarked:
Über sein ontologischen Beweis — er hatte das Resultat vor einigen Jahren, ist jetzt zufrieden damit aber zögert mit der Publikation. Es würde ihm zugeschrieben werden daß er wirkl[ich] an Gott glaubt, wo er doch nur eine logische Untersuchung mache (d.h. zeigt, daß ein solcher Beweis mit klassischen Annahmen (Vollkommenheit usw.), entsprechend axiomatisiert, möglich sei)’
Oskar Morgenstern
About his ontological proof — he had the result a few years ago, is now happy with it but hesitates with its publication. It would be ascribed to him that he really believes in God, where rather he is only making a logical investigation (i.e. showing such a proof is possible with classical assumptions (perfection, etc.) appropriately axiomatized.
Oskar Morgenstern, translation (mine)
Gödel ultimately died in 1978, and the proof continued to circulate informally for about a decade. It was not until 1987, when the collected works of Gödel were released, that the proof was published openly. Logicians perked up at this, as when Gödel — who, it’s fair to say, is one of the all-time logic GOATs[3]— says that he has found a logical proof of the existence of God you would do well to consider it seriously indeed.
The Proof
The Logic
Godel’s proof takes place in a canonical version of modal logic. This form of modal logic is usually called S5. It contains all the standard rules of propositional logic: modus ponens, conjunction; all of the standard tautologies. As well as four rules for the so-called “modal operators.” The modal operators are represented by a “Box” and a “Diamond”. Whenever you see a box, you should think “it is necessary that…”, or “in all possible worlds” and whenever you see a diamond, you should read “it is possible that…”, or “in at least one possible world.”
Rule 1
Brief justification. This one is pretty easy. It says only that if it is necessary that A is true (i.e. there is no world where A is false), then A must also just actually be true. We won’t really need to use this rule.
Rule 2
Brief justification. This rule states that if it is necessary that “A implies B,” then if it is necessary that A is true, it is also necessary that B is true. You can imagine this as saying the following:
Suppose that there are no worlds where A is true and B is not true (so that it is necessary that A implies B).
Then also, if in every world A is true.
Then in every world B is also true.
Rule 3
Brief justification. This rule says that if it is possible for A to be true, then it is necessary that A is possibly true. That is, if in some world it is possible for A to have been true (so that A is contingent), then it is also necessary that A could have been true, so that A could have been true in any world.
This is the most contentious rule of S5, but it is equivalent to some other rules that are less contentious — I think it’s not so bad. But you already know the final stop on this train — you may choose to get off here, though I would recommend a later station.
Rule 4 - Necessitation rule
Brief Justification. This rule just states that if a logical statement follows from nothing, i.e. it is a logical tautology, then it is necessarily true in all possible worlds. This makes sense, since we suppose that the rules of logic apply in all possible worlds (the possible worlds we’re considering here are supposed to represent all the logically possible worlds). There is no logically possible world where a tautology is not true, since it is a tautology (a note that this statement is itself basically a tautology).
This rule is usually not stated explicitly as one of the axioms of S5, as it is so widely accepted that all of the modal logics basically take it as given. The modal logics that satisfy this rule are called the normal modal logics.
The Proof—Axioms and Definitions
I will walk through the proof that Scott transcribed[4], which is in the usual logical notation that modern philosophers and mathematicians are familiar with. It is equivalent to Gödel’s original proof, but Gödel wrote his version in traditional logic notation — which is in my opinion much harder to read. The proof takes place in second-order logic, so we will have “properties of properties” (i.e. a property can be “perfect[5]” or a property can be “imperfect”). This is not particularly controversial.
Axiom 1 - Monotonicity of perfection
I know it’s starting to look scary — but I promise it’s not as bad as it looks. The P that appears here is one of those “properties of properties” I mentioned just before. It says only that “this property is perfect”.
The sentence above says, therefore, that “If property 1 is perfect, and it it is necessarily true that whenever an object has property 1 it must also have property 2, then property 2 must also be perfect.”
This seems justifiable enough — if property 1 is perfect, and there is no world in which you have property 1 and don’t also have property 2, then property 2 should also be perfect, otherwise how could we have said that property 1 was truly perfect in the first place?
Axiom 2 - Polarity of perfection
This says that if a property is perfect, then not having that property is not perfect. This just means that it can’t be the case that having a certain property is perfect and also not having that property is perfect. This seems sensible enough — we cannot say it is perfect to be all-knowing and also perfect to not be all-knowing, this would be nonsensical.
Notably, this is not saying that every property is either perfect or not perfect — it is just saying that if a property is perfect, then the negation of that property can’t also be perfect.
Axiom 3 - Possibility of perfection
This axiom says that if a property is perfect, then it must be possible for there to exist something with that property.
It would seem unreasonable to say that a property is perfect and also that there are no worlds where anything can actually instantiate that property. Surely it must at least be possible for something to have the property if we’re calling it perfect, even if nothing in our world actually has that property. Otherwise we can just resign this property to the collection of neither-perfect-nor-imperfect properties, or else simply call it imperfect, since nothing can ever have it.
Definition 1 - Definition of God
This introduces the definition of God that we’ll be working with in this proof. It should be relatively familiar — something has the property of “Being God” if it possesses every perfect property. Also, if something possesses every perfect property, then we can call that thing God. It is a being which possesses every perfect property, what word would you like to use for it?
It does not say that God only possesses perfect properties, God can also possess properties that are neither perfect nor imperfect — however God cannot have any imperfect properties, as that would lead to a contradiction by Axiom 2.
Axiom 4 - God is Perfect
This axiom just says that the property of being God — you know, the property of having every perfect property — is itself perfect.
This seems reasonable, it almost follows from axiom 1, however since there is no one perfect property that implies you are God, God is not perfect as a result of axiom 1, so we need to introduce a special axiom to say that God is perfect.
Again, how would you describe a being that has every perfect property — it seems ridiculous to say that such a being is not itself perfect.
Definition 2 - Perfect-Generating Property
This definition introduces the notion of an “Perfect-Generating” property. This basically says that a perfect-generating property of a thing is a property that captures everything that makes that thing “perfect.”[6]
For example, suppose that the perfect properties of a triangle are things like:
having all sides equal
having all angles equal
being maximally symmetric
having maximal geometric regularity
Then the perfect-generating property of a triangle would be the property of “being an equilateral triangle,” as that implies all of the other perfect properties.
To describe the logic of the statement more explicitly, it says:
We say property 1 is a perfect-generating property of X iff X has property 1, and for any other perfect property 2, if X has property 2, then anything else that has property 1 must necessarily (i.e. in all possible worlds) also have property 2.
So it cannot be the case that there is some Y that has the perfect-generating property of X, and yet X has some perfect property that Y does not have. In fact, there is no world where that happens — it is necessary that anything with this perfect-generating property also has all the other perfect properties of anything else with that same perfect-generating property.
Note that this doesn’t mean that the “perfect-generating” property is itself a perfect property (although it will be whenever we use it in the proof).
Definition 3 - Perfect-Essential Necessity
This is just another definition.
We say that something has the property of “perfect-essential necessity” if (and only if) whenever there is a perfect-generating property for that thing, it implies that there necessarily must exist something (i.e. in every world) which instantiates that perfect-generating property.
This would also mean that there must necessarily exist a thing that has all the perfect properties that follow from that perfect-generating property.
This is just a definition of what it would mean for something to have “perfect-essential necessity.” We have not claimed that any such thing exists, we are just saying that if a thing has a perfect-generating property that must necessarily exist, then it also has the property of “perfect-essential necessity.”
It is just a definition — it cannot hurt you.
Axiom 5 - Necessary Perfection
This axiom states that whenever we say that property is perfect, it must be perfect in all possible worlds. It must be necessary that the property is perfect.
If there is some world in which a property is not perfect, this axiom claims, then it was never truly perfect to begin with. We are not talking about your “garden-variety” perfection — we’re only talking about the real cream of the crop perfect properties.
Axiom 6 - Perfect-Essential Necessity is Perfect
This claims that the property of having “perfect-essential necessity” — whereby all your perfect properties must necessarily be instantiated in the world — is itself a perfect property. This seems sensible to me! If it were the case that all of the perfect properties of a thing had to be instantiated in the world, it seems like having that property would also be perfect.
Is it not itself a perfect property for all of your perfect properties to be instantiated in the world? Is it somehow imperfect for all of your perfect properties to be required to exist — well then clearly they were not perfect to begin with!
The Proof—Derivation
Well, if you haven’t gotten off the logic-train at any of the steps above, then you know what’s coming now. Let’s show that these axioms entail the existence of God.
Theorem 1
This says “it is possible for there to exist an X, such that X is God.”
We prove this using axiom 3, which stated that any perfect property must have the property that it is possible that something instantiates that perfect property.
Then we apply this to axiom 4, which said that “Being God” is a perfect property.
So, just by replacing the φ in axiom 3 with G, it follows that it must be possible for there to exist a God.
Theorem 2
This says that “If there is an X with the property of Being God, then Being God is a perfect-generating property of that X,” i.e. it captures all the other perfect properties that such an X would have.”
This is a more involved derivation, so let’s go step by step. What we need to show is that if there is something with the property of “Being God”, then it satisfies the definition of a perfect-generating property. So let’s look at definition 2 more closely.
So we need to show the conjunction on the right holds when φ is replaced with G. It is clear that the first clause of the conjunction holds, i.e. if something has the property of “Being God,” then it has the property of “Being God.” That was easy!
The trickier part is showing the second clause. This says that for any property ψ, if God has that property and that property is positive, then it is necessarily the case that whenever anything has the property of “Being God,” it must also have that positive property ψ.
Let’s begin by again noting the definition of God
And since this is a definition, we can apply the rule of universal generalization from logic[7]. We’ll also drop the reverse implication, since it isn’t necessary for the result.
This is basically just a restatement of our definition. It’s saying for anything which has the property of “Being God,” it also satisfies our definition of “Being God.” We’ve also replaced φ with ψ, just so it eventually lines up with our definition of a perfect-generating property — but we could’ve replaced it with anything.
However, since this is a theorem that we can derive from just our definition of God and the rules of logic alone, we can apply rule 4 from our modal logic system, the necessitation rule. This must apply in any world (since we assume that it is necessarily true that all the laws of logic are true). Therefore, we can just write that the above statement is necessary.
Now we have that in every possible world, “Being God” implies that if a property is positive, you have that property.
From this point on, it’s going to get a little rocky if you aren’t familiar with basic logic. Explaining the rules of basic logic would unfortunately take too long, and this post is long enough already, but feel free to skip to Theorem 3. I promise nothing here is a trick, or at all contentious.
To reach the conclusion for the definition of a perfect-generating property, we can begin by assuming that the ψ we’re working with is perfect. Otherwise there’s nothing we need to show about it (since the first clause of the implication we’re trying to show would be false, so the implication would be vacuously true). So if we assume this, axiom 5 lets us say:
And therefore we can also assume that it is necessary that ψ is a perfect property. Then we can apply a theorem of S5’s modal logic (a basic corollary of proposition 3.17 here, but don’t worry about it too much), to Lemma 1 and get:
Then we can generalize back again, going back to thinking about this being true for anything we include in the formula — since we had a general formula originally in Lemma 1 — which gives us:
Which is the conclusion that we wanted to reach on the right hand side of the definition of a perfect-generating property. So we’ve shown Theorem 2 must hold.
Theorem 3
This is the final theorem we’ll need to prove. That it is necessary for there to exist a being with the property of “Being God.”
For this, we’ll need to prove a small lemma, which is:
To prove this lemma, we begin by assuming there exists something that has the property of “Being God;” from this we then want to show the right-hand side, that there must then necessarily exist something with this property. Let’s pick an arbitrary thing that has this property of “Being God” to work with[8], so that we have:
for some a. Then by axiom 6 — that perfect-essential necessity is itself a perfect property, and so it is a property that anything with the property of “Being God” must have, were such a thing to exist — and the definition of God, we have:
Then, by restating theorem 2 for this generic a that we’ve chosen:
Then we use the definition of perfect-essential necessity, and plug in “Being God” as the perfect-generating property (we only need to use the forward implication from the definition):
So, finally, let’s combine our assumption that there exists something with the property of being God, with statements (1), (2), and (3). Then following the chain of implications, we arrive at:
So it is necessary that there exists something with the property of “Being God.” But remember, we have to include the initial assumption we used to prove this, so ultimately all we have shown is:
So the lemma has been proven.
Then we need one last step from our modal logic, we’ll start with theorem 1, which stated:
Then we’ll apply our Lemma 2, to change existence into necessary existence, to get:
So we have that it is possible for it to be necessary for there to exists something that has the property of “Being God.” Then finally, we’ll apply Rule 3[9] from our modal logic to get:
And so it is necessarily true that there exists something with the property of “Being God.” That is, there is something that is “Being God” in every possible world.
Checkmate, atheists.
Conclusion
Well, that was a pretty intense modal-logic session we just went through. I hope that I managed to walk you through it effectively, and perhaps hope that I managed to convince you that there is a God — though I must say, I have my doubts.
Now, I departed from the proof in one place, which was my statement of Axiom 2. Originally, Gödel stated it so that every property is either perfect or it’s negation is perfect, and I have stated it to leave open the possibility that some properties can be neither perfect nor imperfect. This also meant that I had to modify the definition of a perfect-generating property, which can be phrased in a slightly cleaner way if you accept that every property is perfect or imperfect — but I prefer my phrasing[10]. The proof goes through the same way. It is worth noting, though, that in Gödel’s original proof, the bipartite nature of perfect properties (i.e. that every property is perfect or imperfect) ends up leading to something called “modal collapse,” in which everything that is true is also necessarily true — which seems like an unfortunate consequence. I am not certain that my formulation totally avoids modal collapse — though it certainly prevents the most obvious way to get there.
However, I remain uncertain about the existence of God — why is that? After the proof I just gave. Well, it is because I am not sure that any property is perfect, aside from perhaps the property of “Being God”, and the property of “perfect-essential necessity.” I think my arguments for those being perfect properties are good enough, though is it the correct turn-of-phrase to say something has the property of “Being God” if it only has that property, and every other property is simply contingent — I am not so sure. There also remain ontological uncertainties about whether S5 is a legitimate modal logic to consider for our reality. Rule 3 is fairly strong, and I am not sure if I want to accept it, and as you saw in Theorem 3, that was a fairly essential part of the proof — the whole edifice would collapse if we rejected that. But it is implied by some less-controversial axioms, as I mentioned, so perhaps we ought to believe it.
One other comment on the proof — as with so many ontological arguments, we could go through the proof and entirely modify the language, so that instead we are talking about “Perfectly Evil” properties, or indeed any other properties[11]. Perhaps, what this proof has really done is shown us the truth of Manichaeism.
A note that this proof has actually been formalized (in Gödel’s original formulation, not my variant) in a computer-assisted theorem prover! I’m sure the Germans who did that realized the magnitude of the headlines they could generate — “Proof of God Verified by Computers!” Perhaps those of us engaged in similarly wacky disputes could formalize an argument (that works, as all arguments do, only by accepting certain axioms) and then generate a headline that says what you actually believed all along has now been demonstrated beyond any doubt to be certain. This may be a fruitful generalizable PR stratagem.
- ^
Unfortunately, I’m talking about Dana Scott the logician — of Scott’s Trick fame — not the incredibly attractive lawyer from world-renowned TV show Suits.
- ^
There is no evidence for Gödel sharing it with Einstein, though it is well-known that Gödel and Einstein were incredibly close during their time at Princeton, so it is hard to imagine Gödel never brought it up with Einstein.
- ^
I know that means all-time logic greatests of all time, there’s two all-times. I think Gödel probably deserves two all-time awards.
- ^
Okay, I’ve made a minor modification to make it more satisfying to me personally. I’ll discuss this in the conclusion.
- ^
I will use perfect to mean the classical English definition of perfect, not the technical Leibnizian meaning. The argument feels more justifiable to me in this vocabulary than the standard vocabulary of “positive” or “good.”
- ^
This is where I’ve made a slight departure from Gödel’s original proof — I think the proof I’m going to provide here is just a bit more convincing, since otherwise you need to have a biconditional on perfect properties (i.e. every property is perfect or its negation is perfect), and the essential property needs to capture everything about something, not just everything that is perfect about it. There are reasons to prefer Gödel’s original formulation, but I prefer mine (of course). Don’t worry, I’ll talk about the original and my twist on it in the conclusion.
- ^
- ^
- ^
This isn’t quite rule 3, but it’s a corollary of Rule 3. I’ll write a quick proof here. What we want is:
and what we have is:
We get this through a process called “taking the dual.” Begin by replacing p with ~p:
Then contrapose to get:
Then we can use the fact that “it is not possible in any world for not p” is equivalent to “it is necessary that p,”
Then we use the fact that there is an equivalence between “it is not necessary that p” and “in some possible world not p” to get:
Then finally, if it “is not possible that not p” then it must be “necessary that p,” and so we get:
Just by rearranging, we get our result:
- ^
This is not a claim to originality, I’m sure some other logicians have arrived at the same conclusion — it is not a particularly difficult adjustment to notice.
- ^
Although for certain other properties I think you would need to be more careful about the arguments for axioms surrounding necessary existence.
I think if you replace the word “God” with “top” and “perfect” with “highest”, it would be much more clear what the proof actually implies about the real world: Very little.
Definition 0: Say that ψ is higher than φ if □ ∀x φ(x) → ψ(x).
Axiom 1: A property higher than a highest property is also highest.
Axiom 2: The negation of a highest property is not highest.
Axiom 3: If a property is highest, then in some world there exists an object with that property.
Definition 1: An object is top if it is every highest property.
Axiom 4: The property “top” is highest. Note: This looks like a a type error to me.
Definition 2: A property φ is highest-generating for an object x if it is true of x, and every highest property of x follows from φ in every possible world.
Definition 3: Superior object. (I changed your name for this as well.) An object x is “superior” if for every highest-generating property of x, every world has an object with that property (not necessarily x).
Axiom 5: A highest property is highest in every world.
Axiom 6: The property “superior” is highest. Probably also a type error.
Sheesh, there sure are a lot of axioms. At this point, I’m not even sure we have a
coherent logicsane logic anymore! Especially with those potential type errors.The end result we prove is that in every world, there exists a top object.
When you divorce the “God” and “perfect” language from the axioms, we don’t really get anything that implies much about the real world or Christianity, do we?
It’s definitely not a coherent logic as those are defined to be first-order, while this is explicitly a second-order logic.
I didn’t know that coherent logic was actually a term logicians used! I’m not a logician myself—I’m a programmer. Thanks for letting me know!
Actually it’s been formalized, as I mention in the conclusion, so indeed we know that there are no type errors, and that the logic is coherent.
The reason I was saying it looked like a type error was because of the self reference. I’m extremely wary of self-referential definitions because you can quickly run into problems like Russell’s paradox. It seems like sometimes it’s okay to have self-referential definitions (like the greatest lower bound), but I’m not confident that Axiom 4 actually avoids those problems.
What’s the self-reference?
An object is defined to be G if it has every perfect property, and then G is assumed (by axiom) to be a perfect property, hence being G requires being G. Now that I think about it a bit more, though, this seems more like a greatest-lower-bound situation than a Russell’s paradox situation.
I don’t know how there’s any self-reference at all? Having any property X requires having that property X? I mean, you can say a lot of things about Gödel, but the man understood self-reference ;)
If I were to try to translate this into classes instead of properties, it would look like, “The class of perfect properties contains the property of being every perfect property in this class”. That seems self-referential to me.
Why do that translation?
To try to clarify why it felt self-referential. I think there’s a self-reference regardless of whether you talk about classes or not, but it’s more obvious if you talk about classes.
I think the correct mathematical term is “Impredicativity”, not “self-referential”, but I’m no logician.
To say that an object y has property X only if the object y has property X is self-referential? I think it’s more like a tautology than self-reference.
Ohhhh, after reading the wikipedia article on impredicativity I think I understand the confusion better now! Yes. It is kind of strange, in a way, but it basically collapses to something that’s okay. Let’s take an example.
Suppose I say that “A bachelor is a married man,” this is a fine definition. I could then also say “a man is a bachelor iff he is a married man and he is a bachelor.”
This is indeed a super weird way of defining stuff, and you’d never normally do it (unless you have to because of weird second-order logic things, this is what’s going on in the proof above). But you can recollapse the definition for bachelor to be about everything that doesn’t require you to be a bachelor. So a bachelor is just a married man.
So God is something that has every perfect property except that it doesn’t have to have the property of being God to actually Be God. Once it has all the other perfect properties I say that it also has the property of being God, and that itself is also perfect. And it’s okay. If you like we could define two properties “Being God 1” and “Being God 2“ and then define two properties “perfect 1” and “perfect 2” and separate it all out. But it would be kind of annoying.
EDIT: I’ll leave it as an exercise to the reader to separate it all out ;)
Nit: a bachelor is an unmarried man.
My argument has been destroyed, and I am ashamed! (Seriously though, thank you for the correction)
Could you please link me to that formalization?
https://www.isa-afp.org/entries/Types_Tableaus_and_Goedels_God.html
Also, I’ll just note for completeness that the justification for the axioms is going to be much more difficult with these “meanings” (or lack thereof). I did try to provide some justification for the axioms. You don’t have to agree with all those justifications. I certainly don’t, as I mention in the conclusion. But one can still make reasonable arguments here.
Dana Scott is famous for many things, but, first of all, for “Scottery”, the breathtakingly beautiful theory of domains for denotational semantics, see e.g. https://en.wikipedia.org/wiki/Scott_continuity.
:-) And he looked approximately like this when he created that theory: https://logic-forall.blogspot.com/2015/06/advice-on-constructive-modal-logic.html :-)
Now, speaking about what I should do to try to “grok” this proof...
And considering that I don’t usually go by “syntax” in formal logic, and that I tend to somewhat distrust purely syntax-based transformations...
For me, the way to try to understand this would be to try to understand what this means in terms of “topological sheaf-based semantics for modal logic” in the style of, let’s say, Steve Awodey and Kohei Kishida, https://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf 2007 paper (journal publication in 2008: https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/abs/topology-and-modality-the-topological-interpretation-of-firstorder-modal-logic/03DE9E8150EE26B26D794B857FF44647).
The informal idea is that a model is a sheaf over topological space X, the “possible worlds” are stalks growing from points x of X, and statement P is necessarily true about the world growing from a base point x if and only if there is an open set U containing x, such that for every point u from U, P is true about the world growing from u.
So the statement is necessarily true about a world if and only if this statement is true about all worlds sufficiently close to the world in question.
This kind of model is a nice mathematical “multiverse”, and one can try to ponder what the statement and the steps of the proof mean in that “multiverse”.
We’ll see if I can follow through and actually understand this proof :-)
I tend to find Kripke semantics easier fwiw. But if you’ve done a bunch of sheaf theory then this perspective also seems reasonable (though I will say it seems somewhat overkill to me, but what can I say, I’m no sheaf theorist).
Also I somewhat prefer to have the image of Dana Scott from Suits in mind when I think about Scott! It’s a lie I tell myself that makes me happy. But thanks for showing the man himself—a truly brilliant logician!
Yeah, I just have an entirely unreasonable love for continuity :-)
These days, of course, we are not surprised seeing maps from spaces of programs to continuous spaces (with all these Turing complete neural machines around us). But back then what Scott did was a revelation, the “semantic mapping” from lambda terms of lambda calculus to a topological space homeomorphic to the space of its own continuous transformations.
I voted this down. Why? Because if you’re going to post an old argument which has been disputed many times, you should acknowledge the common responses to it and be able to articulate some reason as to why you don’t accept them. This post has utterly failed at doing that.
I would really like to know what criticism I did not include that I should have included, if you are going to make a comment such as this?
Did you read the conclusion?
EDIT: If you did, I am curious to know what you think I didn’t mention that I should mention re: criticisms. I also could see no other great post on Gödel’s ontological argument (which is, in meaningful ways, different from others such as Anselm’s). I thought it would make for interesting discussion!
It reads like an essay arguing X with a tiny disclaimer at the end to cover yourself, but fair enough. I’ve retracted the downvote.
It is about a proof which argues X. Sorry, I’m quite a high decoupler. For what it’s worth, the proof truly does not convince me
EDIT: but thanks for letting me know how some may see it! :)
For context, I think this is a powerful, often-overlooked argument for the existence of God, which I find compelling, and potentially completely convincing that God exists. I do also have other arguments that convince me that God exists, so hard to say whether this alone would be sufficient, but it might.
However it is worth discussing some objections, some of which are worth taking seriously and some of which are not.
First, I think we should dismiss any objections which turn on the colloquial meaning of “perfect”. I think this is a mistake that philosophical readers might make, but mathematical readers might not. One way to read the proof is that it is saying “There are such things as perfect properties, here are some claims about them”. Another reading, the correct reading, is “Here is a category of properties, defined by these axioms, which I am going to label “perfect”″. Mathematicians are more likely to read the axioms as a definition, rather than as premises.
I occasionally see objections like “The property of being omniscient or a murderer is not perfect, but is entailed by omniscience, which is perfect, so the entailment axiom fails”. That is just a misunderstanding, in this proof, we’d be happy to concede that “being omniscient or being a murderer” is a perfect property.
Another commonly cited misguided objection is one Graham Oppy has made, where the proof can be used to show the existence of multiple Gods. Oppy’s claim is that if we take our set of perfect properties, there’s another set with all those properties but one, and another almost-God who has all of those properties.
However if we have “constructed” our set of perfect properties by taking all the properties entailed by necessary existence and God-like-ness, then we can’t take any perfect properties out of our set without violating the entailment rule. And for people who want to draw on more literature, the classical theist tradition is full of arguments that the properties of God all entail each other, and are in reality only a single property.
I think the questions about S5 are more serious ones, I have a few thoughts on these.
The argument about S5 can be thought of as an accessibility relation question—are all possible worlds which “exist” (in whatever way we think they might) S5 accessible? For S5, the accessibility relation needs to be reflexive, symmetric, and transitive.
We could do a bit of a Proofs and Refutations move here—rather than arguing about whether all worlds are S5-accessible, let’s just talk about the subset of worlds which are S5 accessible. We can just restrict ourselves to the S5-accessible worlds! There’s at least one of those—this one.
So we could instead just restrict our thinking to all the S5-accessible worlds from this world. Maybe there are some possible worlds which are not S5-accessible, but that doesn’t need to bother us. Just specify that when we use [] and <>, we are talking about only S5 accessible worlds.
What tangibly does this mean? Well, it makes the argument a bit weaker—it opens up the possibility that God does exist in *some* possible world, but that world is not S5-acessible to us, so then God doesn’t necessarily exist here. Or said differently, “God S5-possibly exists” is a harder claim to establish than “God possibly exists”.
But we aren’t coming at it directly like that, as you would with Plantinga’s argument. Instead we are trying to show a kind of compatibility between necessary existence and “perfect” properties. But if S5-necessary-existence is compatible with the “perfect” properties, then the argument still succeeds, and there’s no reason to think that S5-necessary-existence is any more incompatible than just bare necessary existence, whether you take that to be S4 or something else.
I have some other intuition pumps for why S5 is appropriate here, but these are intuition pumps rather than rigorous arguments.
The objects we are dealing with are properties, and God who might or might not exist with some of those properties.
It seems obvious that God is the type of thing which, if possible, is possible in every possible world. If God exists, then God is the ultimate source of all worlds, and so God must at least be possible (and indeed necessary) in all worlds. S5 is the right system to use for discussing God.
What about properties of God? Again, if God possibly has some property, then could there be a world in which God does not possibly have that property? It seems not, as there’s nothing in any world which can constrain God from having a particular property, other than Himself. So S5 seems reasonable to use when discussing potential properties of God.
I think this exhausts my thoughts on potential objections to the argument as given, looking forward to hearing what you have to say!
I don’t have too much to add to this, honestly. But this is a super high-effort comment which was a joy to read, so I’ll give some commentary (mostly not arguments though).
I think the metaphysics of S5 are weird indeed, but I probably end up thinking it’s okay. But I could be convinced either way on this, very weakly-held. I think your points here are good considerations about this. I think my comment thread with @jessicata raised interesting questions about metaphysics of S5, and whether we should believe that perfect-essential necessity is a property realizable at all. Though I do disagree with her, but her arguments were fantastic.
I guess my main disagreement is that, well, I would say I am a more “mathematical reader” of this argument (I also authored this post, but in how I evaluate it). However, I do think it is sensible to think about what else we could interpret the second-order predicate P to mean, and so whether this argument “proves too much.” Especially if one doesn’t have a huge amount of logic-background to understand it, this can give you a first-order reason to argue against it. Although, again I probably agree with you that most such objections tend to be sort of subtly misguided, probably due to lacking mathematical background.
About Oppy’s objections I’m not as familiar. I think I agree with you that the entailment thing ends up being pretty strong here. Maybe strong enough to be a reason to reject that perfect properties exist (since they have to satisfy the implication rule), but in so-believing I also end up rejecting Oppy’s argument.
Modal logic seems interesting. Where can I quickly learn more about it?
It is quite interesting!
Philosophical perspectives: https://plato.stanford.edu/entries/logic-modal/, https://plato.stanford.edu/entries/phil-multimodallogic/
Mathematical perspectives: https://bd.openlogicproject.org/bd-screen.pdf, https://ncatlab.org/nlab/show/modal+logic
Not sure if there’s another perspective really? I hope this argument works somewhat as an introduction, although including second-order logic and all this “God” stuff might obscure the modal logic somewhat. But at least it shows how an argument using modal logic can work (especially in the second half of the post) :)
Well, it surprisingly did. Until the moment where a predicate symbol that had only previously been applied to variables representing objects in the universe of discourse (i.e. a normal variable) was applied to a variable representing a predicate. It was at that moment that I understood that I don’t understand this stuff.
This was fun to read through!
I would rewrite the conclusion of the proof as follows, curious to see if you would endorse this:
Either there are no perfect properties, or God exists.
To me, this makes it much more clear that this is a pretty continent claim on there being perfect properties. I realize it was just one of a list of axioms, but it seems like a natural one to be skeptical of if you were trying to translate this into the real world.
Yeah seems like the most doubtful one to me too.
Roughly, axiom 3 requires perfect properties to be realistic. Then axiom 6 is roughly assuming, as an easy implication, that perfect-essential necessity is realistic. But why would perfect-essential necessity be realistic?
Okay, I think I understand the objection. Let me expand it out:
perfect property → it must be possible for that property to be realized
perfect essential necessity means that if X has a perfect property that captures everything else that is perfect about X, then something is realized which has that property
perfect-essential necessity is perfect → it must be possible for something to have the property of perfect-essential necessity.
I don’t know… can I like appeal to the triangle example here? A triangle exists. Depends on what you allow as a perfect property really.
We already have reasons to think triangles are logically possible. Based on axiomatic systems like Euclidean geometry. We don’t have a similar mathematical model of perfect-essential necessity, that shows it to be realistic/possible.
Also we do have a model of these axioms, to be clear, since it was formalized in Isabelle, routing through checking for the existence of a model. Here if you want to read it yourself :) https://www.isa-afp.org/entries/Types_Tableaus_and_Goedels_God.html
EDIT: I think the version I gave is kind of like a mix of Anderson’s or Fitting’s, not sure, I made the change myself. But they’re pretty similar anyway.
No I’m saying the if “being regular is a perfect-generating property of a triangle” then “it is necessary that there are regular triangles” means that “there is a thing that exhibits the property of perfect-essential necessity.” If you think it is logically necessary that triangles exist (or something that has the property of being a regular 3-sided shape), then perfect-essential necessity can be called perfect without the contradiction you point out.
I think if you think anything has a perfect property, then you believe that perfect-essential necessity is realistic, basically. If you disagree that anything has a perfect property, then yeah, I think that’s pretty reasonable.
Couldn’t I think the property of “being logically consistent” is perfect, and possible, without thinking “perfect-essential necessity” is perfect (and therefore possible)?
No, because then there exhibits something with perfect-essential necessity, right? So it’s coherent to call it perfect, because it is possibly realized. As long as being logically consistent is a perfect-generating property of… something. An axiomatic system perhaps?
In this case I haven’t accepted that perfect-essential necessity is a perfect property. Peano arithmetic, for example, is (I claim) logically consistent, but does not have the property of perfect-essential necessity. And suppose I claim logical consistency is a perfect property. This seems ok so far.
Ah sure, so do I have you correct in saying that you think there is no essence that generates all the perfect properties of Peano Arithmetic? (Could I get around this by just taking the conjunction of all of its perfect properties, and saying that this is the perfect-generating property? It would satisfy the definition of a perfect-generating property, assuming you think that the conjunction necessarily implies each of the individual perfect properties in every world, which I think is not too controversial of a claim).
Idk. But suppose there is an essence that generates all perfect properties of PA. How would “perfect-essential necessity is possible” follow? (And if it does follow, why do you need axiom 6?)
Peano Arithmetic is (EDIT: necessarily) possible.
Peano Arithmetic has a perfect-generating property.
It is possible for that perfect-generating property to be (EDIT: necessarily) instantiated (it is instantiated in Peano Arithmetic).
Therefore it is possible for something to have the property of perfect-essential necessity.
To your other point, we are making the claim that perfect-essential necessity is perfect. I think “perfect-essential necessity is possible” is a bit weaker, but axiom 6 does also claim that. I thought we were discussing whether axiom 6 is strange because you think nothing has the property of perfect-essential necessity, which would be required to be able to call it perfect.
EDIT: No hostility intended, by the way, I’m just wanting to be clear that I’m addressing the correct point.
“PA is possible” sure ok.
“PA has a perfect generating property” let is suppose this. for example, maybe there is some list of perfect properties of PA, and they all follow from some property of it.
“It is possible for that perfect-generating property to be instantiated (it is instantiated in Peano Arithmetic)” sure, ok
“Therefore it is possible for something to have the property of perfect-essential necessity.” absolutely wild claim.
“We say that something has the property of “perfect-essential necessity” if (and only if) whenever there is a perfect-generating property for that thing, it implies that there necessarily must exist something (i.e. in every world) which instantiates that perfect-generating property.”
Seems like a wild claim. For PA to have this property would mean that its perfect-generating property (perhaps logical consistency, or soundness?) has a corresponding object having that property (e.g. soundness) in every possible world. But that’s a wild claim. Maybe there’s a possible universe with so little information that no sound logical system exists there.
You think there are logically possible worlds without sound logical systems? I guess it depends on your interpretation of modal logic, but I interpret it as “logical necessity” or “logical contingency,” so we aren’t like breaking the rules of logic or anything...
EDIT: I will probably also defend this interpretation reasonably strongly.
Imagine a universe that consists of a single particle. The law is that the particle stays in the same state always. Actually, the particle only has 1 possible state in this universe. So it’s just always in the same state as a matter of math not just physics.
There is no way for PA to exist in this universe.
Okay, noting that you made a stronger claim, that there is a universe where there is no sound logical system (i.e. P and not P are true in some possible universe), but I’m happy to move to this, sure.
What do you think it is about a Universe that is “making PA exist.” If tomorrow physicists found out there are <=10^googol particles in the universe, and 2^(10^googol) possible arrangements of states of the universe, or some other laughably huge number, would you then say “PA doesn’t exist in this universe?” Because PA proves 2^(10^googol) + 1 exists? I’m unclear about the ontology you’re assuming?
Let us say that PA consists of a formal system with exactly the standard axioms of PA. There is some quibbling about which “paper copies” work or which “translations” work. But PA only exists in a universe if it is instantiated as physical information. Instantiating large numbers does not instantiate PA.
Not what I meant. Two interpretations (a) in the single-particle universe, no logical system, sound or unsound, is instantiated, because no non-trivial physical info is instantiated. (b) Perhaps we could say that PA’s prefect-generating property is stronger than soundness. It is soundness plus “having provably distinct terms that number in the countable infinity”.
I feel like the single particle universe clearly instantiates the logical system: ∃x (particle(x)).
Let me move to a different example, why do you think our universe instantiates PA? Or do you not? Why do you think it has to be instantiated physically to exist? E.g. It can be proven that Con(ZFC) is true iff a certain turing machine with 745 states halts. This can be done in the metatheory of PA, I’m relatively confident (though not certain) Do you think that our universe then either instantiates Con(ZFC) or doesn’t? Since PA exists? Seems like a weird ontology for maths to me, but I guess all ontologies for maths are weird.
Nope, because they didn’t write down the system. (This might be quibbling. I mean it doesn’t write down that formal system anywhere.)
Because mathematicians have written down the rules of PA. And they have programmed computers to use PA.
Our universe instantiates formal system such as ZFC + Con(ZFC), because mathematicians can write them down. (It also instantiates inconsistent formal systems such as “ZFC + axiom of choice is false”.)
Backtracking a bit. I reject axiom 6 (which is not to say I actively disbelieve it, but that I don’t actively believe it) and I’m giving an idea of why. If you could logically prove axiom 6, you wouldn’t need it to be an axiom. So you’re going to have some trouble convincing me of it on logical grounds. Rather it’s more metaphysical and so on.
Yes, yes yes, this is of course fine to reject the axiom I agree. But the metaphysics is where it actually gets good (I only did a maths degree and am leeching the metaphysics of it out of my system with posts like these lol).
I mean I think this is a plausibly reasonable account of when mathematical objects exist I guess, kind of a “structuralism” flavour to it, I’m actually somewhat sympathetic. I didn’t think of that objection!
Though I will note this is only one example of a potential object with perfect-essential necessity. I’ve linked the formalization which comes up with a logical model of these axioms (including all the necessity, perfect-essential necessity, etc) in a different thread on your original comment, if you’re curious!
If you’d like to stop back-and-forthing about metaphysics, seems reasonable. I’m sure we’ll make lots of progress if we keep going debating our priors about this! \s
Yeah I think it might be more productive, if I wanted to make progress on this, to look at the math rather than the metaphysical back-and-forth.
Indeed!
Axiom 3 is wrong. If there are facts about what’s possible or not, then these facts must be proved; pleading that “surely it must at least be possible” doesn’t cut it. Surely the chicken must at least be white, but he ain’t. This flaw has been known for centuries, I tried to write a short explanation sometime ago too.
It’s an axiom in a modal logic setup. Axioms can’t be “wrong.” I could indeed instead have included it as part of the definition of what I mean by a perfect property. Would you then say the definition is “wrong”? It is a definition.
If it is not possible for a property to be instantiated, then you simply say that it is not a perfect property. So even under the charitable reading of “by wrong I mean that there are no properties like this,” I still disagree.
EDIT: I actually spoke too quickly here. It might be the case that if you take it as part of the definition, then there are no properties like this. This seems plausible.
If the axiom refers to a notion of “possibility” purely within the misty abstract world of modal logic, then sure, I agree. But then the “God” whose existence is thus proved also resides in that misty world, not in ours. For the proof to pertain to our world, the notion of “possibility” in the axiom must correspond to the notion of possibility that we humans have. And understood that way, the axiom can be wrong, and is wrong.
Ah sure, you disagree that modal logic is an apt way of describing the world. I think that’s very plausible indeed! I mention it in the conclusion.
Yeah. Or rather, I guess modal logic can describe the world—but only if you meet its very strict demands. For example, to say something is “possible”, one must prove the impossibility of finding a contradiction between the thing and all evidence known so far, to either the speaker or the listener. If that requirement is met, then modal logic will give the right answers, at least until new evidence comes along :-)
I’m not super confident in this conception… I’d have to think about it. If I’m honest your argument doesn’t totally convince me (because logically possible worlds do seem like a thing I should be able to talk about). Unless the “evidence” you’re talking about is what we know to be logically possible? Anyway, seems like this is probably the crux. Think it is quite reasonable to say modal logic is too strong (especially S5, which is on the stronger end).
I guess this time I spoke too soon! Indeed if we talk about logical possibility, then we “only” need to prove that the imagined world isn’t contradictory in itself. Which is also hard, but easier than what I said.
Yeah but who knows if that’s really “correct” in that the world-we-actually-exist-in logically behaves like this. Not to mention the precise details of the rules for necessity and possibility. I fear we’re in danger of switching positions.
It’s nice that we got to the notion of logical possibility though. It’s familiar ground to me.
Let’s talk for example about mathematical properties of musical intervals. When a major scale C D E F G A B is played on a just-intonation instrument, all pairwise ratios of frequencies are fairly simple: 2⁄3, 15⁄16, all that. All except the interval from D to F, which is an uglier 27⁄32, unpleasant both numerically and to the ear. This raises the tantalizing possibility of a perfect tuning: adjusting the frequencies a little bit so that all pairwise ratios are nice, not all except one. The property of a tuning being perfect can be described mathematically.
Unfortunately, it can also be shown mathematically that a perfect tuning can’t exist. What does that mean in light of your Axiom 3? Must there be a “possible world”, or “logically possible world”, where mathematics is different and a perfect tuning exists? Or is this property unworthy of being called perfect? But what if we weren’t as good at math, and hadn’t yet proved that perfect tuning is inachievable: would we call the property perfect then? What does your framework say about this example?
Well, I don’t want to take too much credit—it’s Gödel’s framework, not mine. I guess it would be the case that then it would not be a perfect property, since it is logically impossible to achieve. We just wouldn’t know that without doing a bunch of mathematics first, and might be confused to begin with. There’s many similar examples in this vein, for sure. But just because you don’t know something, doesn’t mean it’s not true or false. There either is a perfect property, or there is not.
I’m now going to address something that’s sort of irrelevant to your main point, but perhaps interesting:
It would seem strange to say that frequencies of music are inherently perfect. This seems to rely on a lot of contingent facts about how human ears work, why we like certain sounds and not other sounds. What is pleasant to us. So I wouldn’t call any property like this “perfect” in this sense. But I know it’s meant as an illustrative example, so this is not really to your point.
Wait, if we can be confused whether a property is perfect or imperfect—then why do we assert (in axioms 4 and 6) that some specific properties are perfect? What if they’re also impossible, like the perfect tuning?
For those ones we just say that they are, since we need those for the argument to hold logically. So we stipulate that for those properties there is no confusion, and they are perfect. You are totally allowed to say “hmm, seems suspicious,” but by the definition of these properties, there’s nothing contradictory about asserting that these properties are perfect.
But for a property like “is omniscient” or whatever, we could be more unsure. Since maybe “being omniscient” implies you also “know what it’s like to kick a baby.” And so it implies that you have a property that is not perfect, for example.
The suggestive names error.
Mistaking the structure of one’s language for the structure of the world.
Indeed! I do discuss this in the conclusion, fwiw.
Axioms 4 and 6 look like type errors to me. Could you please explain how they are not?
Not sure how they are? It’s been formalized in lean (which checks types).
EDIT: whoops, Isabelle/HOL, not Lean
P is a second-order predicate; it applies to predicates. The english word “perfect” applies to things, and it’s a little weirder to apply it to qualities, as least if you think of \phi and \psi as being things like “Omnibenevolent” or “is omnibenevolent.” If you think of \phi and \psi as being “Omnibenevolence,” it makes more sense—where we type-distinguish between “qualities as things” and “things per se.” It’s still weird not to be able to apply P to things-per-se. We want to be able to say “P(fido)” = “fido is perfect”, but that’s not allowed. We can say “P(is_good_dog)” = “being a good dog is perfect”.