Gödel's ontological proof

Question

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.

Answer 1

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?

Answer 2

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.

Answer 3

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

Answer 4

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.