Gödel's ontological proof

Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel.

Can someone please explain what are the symbols in the proof and elaborate about its flow:

$$\begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \to P(\psi) \\ \text{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi) \\ \text{Th. 1.} & P(\varphi) \to \Diamond \; \exists x[\varphi(x)] \\ \text{Df. 1.} & G(x) \iff \forall \varphi [P(\varphi) \to \varphi(x)] \\ \text{Ax. 3.} & P(G) \\ \text{Th. 2.} & \Diamond \; \exists x \; G(x) \\ \text{Df. 2.} & \varphi \text{ ess } x \iff \varphi(x) \wedge \forall \psi \left\{\psi(x) \to \Box \; \forall y[\varphi(y) \to \psi(y)]\right\} \\ \text{Ax. 4.} & P(\varphi) \to \Box \; P(\varphi) \\ \text{Th. 3.} & G(x) \to G \text{ ess } x \\ \text{Df. 3.} & E(x) \iff \forall \varphi[\varphi \text{ ess } x \to \Box \; \exists y \; \varphi(y)] \\ \text{Ax. 5.} & P(E) \\ \text{Th. 4.} & \Box \; \exists x \; G(x) \end{array}$$

Does it prove both existence and uniqueness ?

Edit: these are modal logic symbols.

• It would be useful if you point out where you found this proof and where the image is taken from. – Asaf Karagila Dec 1 '12 at 11:45
• Have you read the Wikipedia article and the related articles? Can you point at a particular point that was not clear to you? – Asaf Karagila Dec 1 '12 at 11:52
• @Amr: Where it is accompanied by a plain English paraphrase and full explanation. – Brian M. Scott Dec 1 '12 at 11:53
• @Asaf: The OP will get it automatically, and I directed it to Amr as well because it is an extension of his comment. – Brian M. Scott Dec 1 '12 at 11:59
• hahaha, it's funny, because 0x90 is assembly code for NOOP, and Asaf is trying to tell people to direct comments to the OP, but 0x90 means NOOP. Sorry I had my fun [= – eazar001 Jan 12 '14 at 5:16

The modal operator $\square$ refers to necessity; its dual, $\lozenge$, refers to possibility. (A sentence is necessarily true iff it isn't possible for it to be false, and vice versa.) $P(\varphi)$ means that $\varphi$ is a positive (in the sense of "good") property; I'll just transcribe it as "$\varphi$ is good". I'll write out the argument colloquially, with the loss of precision that implies. In particular, the words "possible" and "necessary" are vague, and you need to understand modal logic somewhat to follow their precise usage in this argument.

• Axiom $1$: If $\varphi$ is good, and $\varphi$ forces $\psi$ (that is, it's necessarily true that anything with property $\varphi$ has property $\psi$), then $\psi$ is also good.
• Axiom $2$: For every property $\varphi$, exactly one of $\varphi$ and $\neg\varphi$ is good. (If $\neg\varphi$ is good, we may as well say that $\varphi$ is bad.)
• Theorem $1$ (Good Things Happen): If $\varphi$ is good, then it's possible that something exists with property $\varphi$.

Proof of Theorem $1$: Suppose $\varphi$ were good, but necessarily nothing had property $\varphi$. Then property $\varphi$ would, vacuously, force every other property; in particular $\varphi$ would force $\neg\varphi$. By Axiom $1$, this would mean that $\neg\varphi$ was also good; but this would then contradict Axiom $2$.

• Definition $1$: We call a thing godlike when it has every good property.
• Axiom $3$: Being godlike is good.
• Theorem $2$ (No Atheism): It's possible that something godlike exists.

Proof of Theorem $2$: This follows directly from Theorem $1$ applied to Axiom $3$.

• Definition $2$: We call property $\varphi$ the essence of a thing $x$ when (1) $x$ has property $\varphi$, and (2) property $\varphi$ forces every property of $x$.
• Axiom $4$: If $\varphi$ is good, then $\varphi$ is necessarily good.
• Theorem $3$ (God Has No Hair): If a thing is godlike, then being godlike is its essence.

Proof of Theorem $3$: First note that if $x$ is godlike, it has all good properties (by definition) and no bad properties (by Axiom $2$). So any property that a godlike thing has is good, and is therefore necessarily good (by Axiom $4$), and is therefore necessarily possessed by anything godlike.

• Definition $3$: We call a thing indispensable when something with its essence (if it has an essence) must exist.
• Axiom $5$: Being indispensable is good.
• Theorem $4$ (Yes, Virginia): Something godlike necessarily exists.

Proof of Theorem $4$: If something is godlike, it has every good property by definition. In particular, it's indispensable, since that's a good property (by Axiom $5$); so by definition something with its essence, which is just "being godlike" (by Theorem $3$), must exist. In other words, if something godlike exists, then it's necessary for something godlike to exist. But by Theorem $2$, it's possible that something godlike exists; so it's possible that it's necessary for something godlike to exist; and so it is, in fact, necessary for something godlike to exist. QED.

Convinced?

• Well, if $x$ is a godlike thing, then "is the same as $x$" is a property that $x$ has. So it must be forced by $x$'s essence, which is the property of being godlike. So the property "is godlike" forces the property "is the same as $x$". We conclude that there can't be any godlike thing distinct from $x$, i.e., uniqueness also holds. – mjqxxxx Oct 26 '13 at 22:55
• How are the axioms motivated? None of them really persuades me. – Hagen von Eitzen Aug 20 '14 at 17:15
• @HagenvonEitzen Most of these axioms are things which you could find in a text by a moral realist. Especially a Kantian. They are not too far fetched in the tradition, but of course, you can doubt them. I think the most interesting thing is that it probably tells us a little bit about Godel's own metaethical stance, but that's a question for a different board. – Alfred Yerger Feb 18 '15 at 4:57
• "No Atheism" -- Not at all. Atheism is consistent with the possible existence of a flapdoodlish entity that has all flapdoodle properties ... where theists misleadingly equate "flapdoodlish" with "godlike" and "flapdoodle" with "good", even though none of the connotations of "godlike" and "good" are required by the proof. Further, even strong atheism is, barring modal collapse, compatible with the possible existence of God (as normally defined). – Jim Balter May 2 '17 at 4:40
• @PredragStojadinović Most of the circularity comes from taking words laden with connotation, like "good", and "godlike", and equating them to purely formal symbols that don't imply those connotations. It's just another bit of intellectually dishonest theist sophism. – Jim Balter May 2 '17 at 4:46

The box is a modal operator (it is necessarily true that ...), with the diamond its dual (it is possibly true that ...). '$P(\varphi)$' holds when the property expressed by $\varphi$ is 'positive' (maybe better, is a perfection). Other novelties are defined. $G(x)$ says $x$ has all perfections (so is God). $\varphi$ ess $x$ says the property $\varphi$ is the essence of $x$. $E$ is the property of necessary existence (existence in virtue of your essence). [Don't blame me, I'm just reporting ....]

You'll find out just a little more about Gödel's very strange apparent aberration here: http://plato.stanford.edu/entries/ontological-arguments/#GodOntArg

Robert Adams introduction to Gödel's original note in Kurt Gödel: Collected Works. Vol III, Unpublished Essays and Lectures is well worth reading.

Petr Hajek has an amazingly patient exploration of the current state of play in investigations of Gödel's Ontological Proof and its variants in Matthias Baaz et al. (eds) Kurt Gödel and the Foundations of Mathematics (CUP 2011). But I can't say that this changed my impression that this is little more than a curious side note in logic.

• Indeed your impression is right. A proof that some entity exists that possesses the entirety of a certain set of properties says nothing about "God" unless one cheats and arbtrarily assigns the word "good", with all its connotations, to those properties. Contrary to the BS below, this is not part of "a 900 year tradition of logical discourse", it's part of a tradition of theistic sophistry. – Jim Balter May 2 '17 at 4:53
• @JimBalter “A proof that some entity exists that possesses the entirety of a certain set of properties says nothing about "God"” Umm, what? If those are properties that define a God then of course they prove its existence. – user76284 Jul 13 at 20:10
• ^ reading comprehension problem. – Jim Balter Jul 14 at 1:49

I think dismissing an argument that stands foursquare in a 900 year tradition of logical discourse as a 'strange apparent aberration' is a little questionable. Anyway, it's worth noting that the argument's overly strong axioms and definitions lead to modal collapse. Modifications have been suggested that appear to solve this issue. For example see http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0CC0QFjAB&url=http%3A%2F%2Fappearedtoblogly.files.wordpress.com%2F2011%2F05%2Fanderson-anthony-c-22some-emendations-of-gc3b6dels-ontological-proof22.pdf&ei=cNP0U-TnKafH7AbkwoEQ&usg=AFQjCNGBuNm20ZkEB12IkNgklPHZABw58A&sig2=MJyHYb7IbeDV9QmOLb8U0A&bvm=bv.73231344,d.ZGU.

Like any proof, Gödel's Ontological Proof depends on acceptance of the axioms, and I would suggest the only argument that can be made for them is one of 'reasonableness'. Recently, work has been published looking at the application of automated theorem provers to the ontological proof. Notwithstanding the modal collapse issue of Gödel's original, it seems to have stood up quite well as an exercise in higher order modal logic. See http://page.mi.fu-berlin.de/cbenzmueller/papers/C40.pdf

• That variation is really nice. Not only does it have less extraneous consequences, but the axioms are more intuitive. – PyRulez Sep 19 '16 at 13:33

I think the most important matter in understanding Gödel's proof is the interpretation of $P$. In this respect, I would like to mention that the popular interpretation of $P(φ)$ as "$φ$ is positive" is not entirely accurate, as a result of axiom $1$. Taking both axioms $1$ and $2$ into account, it is best interpreted as "$φ$ is a property that an all-positive entity has". The negation of $P(φ)$ does not mean "$φ$ is negative" and neither does it mean "$φ$ is a property that an all-negative entity has". It means either "$φ$ is not a property that an all-positive entity has", or "$φ$ is a property that an all-positive entity does not have". (These are equivalent, which means the proposed interpretation conforms to axiom $2$. There cannot be more than one way to not have a property, as opposed to the possibilities neutral and negative for the "positive"-interpretation.)

Consider the property $ψ_1(x)$ to mean "$x$ is extreme in politeness", and consider that this means "either $x$ is very polite or $x$ is very impolite". This property is rather neutral than positive, yet, it is a property that an all-positive entity has. Indeed, $\neg ψ_1(x)$ would imply that $x$ is not very polite (apart from the fact that $x$ is not very impolite either). Note that $ψ_1$ is a property that also an all-negative entity has. We could call $ψ_1$ a two-edged property.

The given interpretation is indeed inspired by axiom $1$. Consider the property $φ_1(x)$ to mean "$x$ is very polite" and consider that $P(φ_1)$ holds. This $φ_1$ forces $ψ_1$ (it is necessarily true that anything with property $φ_1$ has property $ψ_1$). Now it follows from axiom $1$ that $P(ψ_1)$ holds, even though, as argued above, $ψ_1$ is not more positive than it is negative. Also an all-negative entity has it. A possible solution is to interprete $P(φ)$ as: "$φ$ is a property that is held by an all-positive entity". (I let it to the creativity of the reader to find an interpretation that conforms to axiom $1$, but not to axiom $2$.)

$G(x)$ means "$x$ is an all-positive entity", or "$x$ is God-like". Gödel proves that such an entity necessarily exists. The difference in interpretation of $P$ is of importance in analyzing whether the converse of the proof-scheme also proves that an all-negative entity necessarily exists. We could interprete $P(φ)$ to mean "$φ$ is a property that an all-negative entity has" and $G(x)$ to mean "$x$ is Devil-like". The truth of axiom $5$ ($P(E)$ holds) can especially depend on a correct interpretation of $P$. If $P(φ)$ meant just "positive" (conversely "negative"), then existence, according to axiom $2$, could not be a property that can be held by both a God-like and a Devil-like entity. Considering that $P(φ)$ means "$φ$ is a property that an all-positive (conversely all-negative) entity has", it is possible that existence is a two-edged property: both God-like and Devil-like entities can have it. Since a Devil-like entity that exists is more negative than a Devil-like entity that fails to exist, it appears that the converse of axiom $5$ also holds: existence is a property held by an all-negative entity. Therefore, according to Gödel's axioms, it is also necessary that a Devil-like entity exists.