The Timpson Lyceum · History of Philosophy (2024)

Kurt Gödel’s Ontological Argument – Line By Line Explanation

Overview

This lecture is a line-by-line reconstruction of Kurt Gödel's ontological argument — a formalization, in second-order modal logic, of the classic Anselmian proof that God necessarily exists. Ontology being the study of being as such, the argument reasons from the concept of a supremely perfect being to its necessary existence; Gödel's contribution is to make every assumption an explicit axiom and every step a checkable inference, so that the conclusion follows mechanically. The system is built on a single primitive: a predicate $P$ of positive properties, meaning properties that are good, perfective, or morally/aesthetically positive rather than privative.

From five axioms and three definitions, the proof derives three theorems, culminating in $\Box\,\exists x\, G(x)$ — necessarily, a God-like being exists. Two features carry the weight. First, a God-like being is defined as one possessing every positive property, and positive properties turn out to be internally consistent (so such a being is possible). Second, the proof runs in the modal system S5, whose characteristic principle $\Diamond\Box p \to \Box p$ lets the argument convert the possibility of a necessary being into its actuality. The proof has been machine-verified as valid: granting the axioms and definitions, the conclusion genuinely follows.

Key Concepts

• Positive property predicate $P$ — A second-order predicate taking a property and returning true iff that property is positive (e.g. $P(\text{courageous})$ is true, $P(\text{cowardly})$ is false). The proof's sole primitive notion.

• Property variables $\varphi, \psi$ — Range over one-place properties objects may have; $\varphi(x)$ reads "$x$ has property $\varphi$." Object variables $x, y$ range over individuals.

• Modal operators — $\Box$ ("necessarily") and $\Diamond$ ("possibly"). Standard bridge: $\Box p \to \Diamond p$ (the necessary is possible), but not conversely.

• Axiom A1 (positivity is coherent under negation) — Exactly one of a property and its negation is positive:

  $P(\neg\varphi) \leftrightarrow \neg P(\varphi)$. So no property is neutral, and lacking a positive property is the same as having a negative one.

• Axiom A2 (positivity is closed under entailment) — A property necessarily entailed by a positive property is itself positive: if $P(\varphi)$ and $\Box\,\forall x\,[\varphi(x) \to \psi(x)]$, then $P(\psi)$. Positive properties can only entail positive properties.

• Theorem T1 (positive properties are possibly instantiated) — If $P(\varphi)$ then $\Diamond\,\exists x\,\varphi(x)$. Proved by reductio from A1 and A2 (see the worked example).

• God-like property $G$ — Definition: $G(x) \equiv \forall\varphi\,[\,P(\varphi) \to \varphi(x)\,]$. A being is God-like iff it possesses every positive property (and, via A1, thereby only positive properties).

• Axiom A3 (being God-like is positive) — $P(G)$. Together with T1 this yields Theorem T2: $\Diamond\,\exists x\, G(x)$ — possibly, a God-like being exists.

• Essence $\varphi\,\text{ess}\,x$ — A property $\varphi$ is the essence of $x$ iff $x$ has $\varphi$ and $\varphi$ necessarily entails every property of $x$: $\varphi(x) \wedge \forall\psi\,[\psi(x) \to \Box\,\forall y\,(\varphi(y) \to \psi(y))]$. An essence fixes everything about its bearer.

• Axiom A4 (positivity is necessary) — Positive properties are necessarily positive: $P(\varphi) \to \Box\,P(\varphi)$.

• Theorem T3 (God-likeness is an essence) — If $x$ is God-like, $G$ is its essence: any property of a God-like $x$ is positive (by the definition of $G$ and A1), hence — by A4 — necessarily held by every God-like being.

• Necessary existence $NE(x)$ — Definition: $NE(x) \equiv \forall\varphi\,[\,\varphi\,\text{ess}\,x \to \Box\,\exists y\,\varphi(y)\,]$. A being necessarily exists iff each of its essential properties is necessarily instantiated.

• Axiom A5 (necessary existence is positive) — $P(NE)$. The final axiom, which drives the possible God-like being to a necessarily existing one.

• S5 and the modal collapse step — In the modal system S5, $\Diamond\Box p \to \Box p$: if something is possibly necessary, it is necessary. This is the linchpin that turns T2's possibility into the concluding necessity.

Worked Example

The derivation of $\Box\,\exists x\, G(x)$ (the final theorem, T4). The scaffold is: A1–A2 $\Rightarrow$ T1; A3 + T1 $\Rightarrow$ T2; T3 from A4; then A5 + S5 close the argument. First the key lemma, then the conclusion.

Lemma — T1 (positive $\Rightarrow$ possibly instantiated). Assume $P(\varphi)$; show $\Diamond\,\exists x\,\varphi(x)$.

1. Suppose for contradiction $\neg\Diamond\,\exists x\,\varphi(x)$, i.e. $\Box\,\forall x\,\neg\varphi(x)$ — nothing possibly has $\varphi$.
2. Then $\varphi(x)$ is necessarily false, so every conditional $\varphi(x) \to \psi(x)$ holds vacuously; in particular

$\Box\,\forall x\,[\varphi(x) \to \neg\varphi(x)].$

1. Applying A2 with $\psi := \neg\varphi$ (and $P(\varphi)$) gives $P(\neg\varphi)$.
2. But A1 says $P(\neg\varphi) \leftrightarrow \neg P(\varphi)$, so $\neg P(\varphi)$ — contradicting $P(\varphi)$.
3. Hence the supposition fails: $P(\varphi) \to \Diamond\,\exists x\,\varphi(x)$. $\;\checkmark$

Main derivation — T4.

1. By A3, $P(G)$; applying T1 to $G$ gives T2:

$\Diamond\,\exists x\, G(x).$

1. Take any possible God-like $x$. By A5, $P(NE)$, and since $x$ has every positive property, $NE(x)$ holds.
2. By T3, $G$ is the essence of $x$; unfolding $NE(x)$ with $\varphi := G$ in the definition of necessary existence gives $\Box\,\exists y\, G(y)$. Thus a possible God-like being entails a necessary one:

$\exists x\, G(x) \;\rightarrow\; \Box\,\exists y\, G(y).$

1. Combining with step 6 and distributing the possibility operator yields possibly necessary existence:

$\Diamond\,\Box\,\exists x\, G(x).$

1. Invoke the S5 principle $\Diamond\Box p \to \Box p$ with $p := \exists x\, G(x)$. Intuitively: whether a necessary truth holds cannot vary from one possible world to another, so if it is even possible that $\exists x\,G(x)$ is necessary, it is necessary outright.

$\Diamond\,\Box\,\exists x\, G(x) \;\longrightarrow\; \Box\,\exists x\, G(x).$

1. Therefore a God-like being necessarily exists:

$\Box\,\exists x\, G(x). \qquad \blacksquare$

The proof is formally valid — the axioms, definitions, and S5 do entail the conclusion — so any dispute must target the axioms (especially the coherence of "positive properties" and A5), not the logic.

Summary

• Gödel's argument formalizes the ontological proof in second-order modal logic (S5), built on one primitive: the predicate $P$ for positive properties.
• A1–A2 govern positivity (exactly one of $\varphi, \neg\varphi$ is positive; positivity is closed under entailment) and together yield T1: every positive property is possibly instantiated.
• A being is God-like ($G$) iff it has every positive property; since $P(G)$ (A3), T2 gives that a God-like being is possible.
• With A4 (positivity is necessary) and the essence and necessary-existence definitions, T3 makes $G$ the essence of any God-like being, and A5 ($P(NE)$) forces a possible God-like being to be a necessary one.
• The S5 step $\Diamond\Box p \to \Box p$ converts possible necessity into necessity, delivering $\Box\,\exists x\, G(x)$; the derivation is machine-verified valid, so critique falls on the axioms rather than the inference.