The Ontological Argument

The ontological argument is the most audacious and most disputed of all the arguments for God: a purported a priori proof that God exists from the bare concept of God, with no appeal to any observed feature of the world. If it works, the existence of God is — like a theorem of logic — knowable by thought alone, and God's non-existence is not merely false but impossible. Its fascination is that it seems to conjure existence out of a definition; its notoriety is that almost everyone suspects this cannot be legitimate, yet locating the precise flaw has occupied logicians from Gaunilo to Gödel.

This page gives Anselm's original two-stage argument, Descartes' and Leibniz's versions, the two classic objections (Gaunilo's parody and Kant's "existence is not a predicate"), and the modern modal reformulations of Plantinga and Gödel, where the action now is — and where the debate connects to modal logic and the metaphysics of possibility.

Anselm's argument

Anselm of Canterbury (Proslogion, 1078) defines God as

$$\text{God}{=}_{\text{df}}\text{that than which nothing greater can be conceived}(\text{aliquid quo nihil maius cogitari possit}).$$

Note the definition is available even to the "fool" who denies God — to deny God, one must understand the concept, so God at least exists in the understanding (in intellectu). The argument then runs:

1. God (the greatest conceivable being) exists at least in the understanding — even the fool grants this.
2. It is greater to exist in reality (in re) as well as merely in the understanding than to exist in the understanding alone.
3. Suppose God existed only in the understanding.
4. Then we could conceive of a greater being — one just like God but also existing in reality.
5. But then we could conceive something greater than that than which nothing greater can be conceived — a contradiction.
6. Therefore God exists in reality as well as in the understanding. ∎

The structure is a reductio: the assumption that the greatest conceivable being lacks real existence generates a being greater than the greatest conceivable being, which is incoherent.

Proslogion III adds a second, stronger argument often regarded as the more interesting: it is greater to exist necessarily (so as to be unable not to exist) than contingently. So the greatest conceivable being must exist necessarily — its non-existence is not merely false but inconceivable. This modal Anselm is the ancestor of the twentieth-century versions below.

Gaunilo's reply: the perfect island

Anselm's contemporary, the monk Gaunilo, objected with a parody ("On Behalf of the Fool"). Run the same reasoning on the concept of the most perfect island: an island than which no greater can be conceived. It is greater to exist in reality than in the understanding alone; therefore the perfect island must exist. Since this conclusion is absurd, the form of reasoning must be invalid.

Anselm's reply — and the standard defence — is that the parody disanalogy: an island has no intrinsic maximum (there is no greatest possible number of palm trees; any island can be bettered), so "greatest conceivable island" is an incoherent notion that never gets the argument started. God, by contrast, is defined by attributes — power, knowledge, goodness — that plausibly do have intrinsic maxima. Whether this reply succeeds depends on whether the divine perfections really are maximisable and compossible — exactly the worry pursued in the coherence of theism.

Descartes and Leibniz

Descartes (Fifth Meditation) gives a cleaner version that makes the argument's logic explicit. God is a supremely perfect being; existence is a perfection; so existence belongs to the essence of God as surely as having three angles summing to two right angles belongs to the essence of a triangle. To conceive God lacking existence is as contradictory as conceiving a triangle whose angles fail to sum to $180°$ — the predicate is contained in the subject.

Leibniz noticed a gap: Descartes' argument shows only that if God's existence is possible, then God exists — it presupposes that the concept of a supremely perfect being is coherent. Leibniz tried to supply the missing premise by arguing that the perfections, being "simple, positive, and absolute" qualities, cannot conflict, so a being possessing all of them is possible. This possibility premise is precisely the load-bearing assumption that the modern modal versions isolate — and that critics attack.

Kant's objection: existence is not a predicate

The most influential objection, due to Kant (Critique of Pure Reason), targets the shared assumption of every version — that existence is a property or "perfection" that a thing can possess or lack, and so can be packed into a concept. Kant denies it:

"Being is obviously not a real predicate; that is, it is not a concept of something which could be added to the concept of a thing."

To say "$x$ exists" is not to describe $x$ — not to add a feature to its concept — but to posit that the concept is instantiated. Kant's example: a hundred real thalers contain no more coins-as-described than a hundred possible thalers; existence adds nothing to the concept, only to my finances. Hence the predicate "exists" cannot be a great-making property whose presence in the concept of God guarantees an instance. In modern terms (Frege, Russell): existence is a second-order property — a property of concepts/propositional functions, expressed by the quantifier $\exists$ — not a first-order property of objects. "God exists" says the concept God is instantiated; you cannot make a concept self-instantiating by definition.

This is widely regarded as fatal to the classical (Anselmian/Cartesian) argument. The modal versions are, in part, attempts to evade it — by trading on necessary existence (or modal status), which is more plausibly a genuine property of things, rather than on bare existence.

The modal ontological argument (Plantinga)

Alvin Plantinga (The Nature of Necessity, 1974) recasts Anselm's Proslogion III using possible-worlds modal logic. Define:

• Maximal excellence: a being is maximally excellent in a world $W$ iff it is omnipotent, omniscient, and morally perfect in $W$.
• Maximal greatness: a being is maximally great iff it has maximal excellence in every possible world.

The argument is then strikingly short:

1. It is possible that there exists a being with maximal greatness. (possibility premise)
2. So there is a possible world $W^{\ast }$ in which a maximally great being exists.
3. A maximally great being is, by definition, maximally excellent in every world — so it is omnipotent, omniscient, and morally perfect in every world, including the actual one.
4. Therefore a maximally great being exists in the actual world. ∎

Formally, with $◊$ for possibility, $\square$ for necessity, and $G$ for "a maximally great being exists":

$$◊\square G⊢\square G\text{(hence }G\text{)},$$

which is valid in the modal system S5, whose characteristic axiom $◊\square p\to \square p$ ("whatever is possibly necessary is necessary") does exactly the work: if a necessary being exists in any world, it exists in every world, because necessary existence does not vary across worlds. The argument is valid; everything rides on premise 1.

The parity / symmetry objection

Plantinga himself concedes the argument does not compel assent, because the possibility premise can be mirrored. Consider "no-maximal-greatness": it is possible that there is no maximally great being. But a maximally great being exists necessarily or not at all — so if its existence is possible, it is actual, and if its non-existence is possible, it is impossible. Thus:

$$◊\square G⊢\square G\text{but equally}◊\neg \square G(\text{i.e. }◊\square \neg G\text{ blocked, and }◊\neg G)⊢\neg \square G(\text{hence }\neg G).$$

Both possibility premises are prima facie plausible, exactly one can be true, and the argument gives no independent reason to prefer the theist's. So Plantinga's verdict: the modal argument establishes not that theism is true but that belief in God is rational — it shows the existence of God follows from a premise (the bare possibility of a maximally great being) that a reasonable person may accept. Critics regard this as an admission that the argument cannot do the one thing an argument is for: move someone who does not already grant the conclusion.

Gödel's ontological proof

Kurt Gödel left a formalised version (circulated 1970, published posthumously) that supplies the Leibnizian possibility premise with an axiomatic theory of positive properties in third-order modal logic. Sketch:

• A property may be positive ($P$); the positive properties form an ultrafilter-like, logically coherent collection (axioms: positivity is closed under entailment, and for any property exactly one of it and its negation is positive).
• God-like ($G$): $x$ is God-like iff $x$ has every positive property — $G(x)\leftrightarrow \forall \varphi (P(\varphi )\to \varphi (x))$.
• Being God-like is itself positive; a positive property is possibly instantiated (from the coherence of the positive properties); God-likeness is an essence of any God-like being; and necessary existence is positive.
• The theorems follow: possibly God exists, hence (the essence machinery plus an S5 step) necessarily God exists: $◊\exists xG(x)⊢\square \exists xG(x)$.

Gödel's proof is rigorous and has been machine-verified (Benzmüller and Paleo, 2013, who checked the derivation with automated theorem provers). But the same work exposed its Achilles' heel: modal collapse. The axioms entail that every truth is necessary ($p\to \square p$) — collapsing the distinction between the contingent and the necessary, and with it any room for free or contingent divine action. Repairs (Anderson, Hájek, Fitting) weaken the axioms to block the collapse, at the cost of complicating the once-elegant system. As with Plantinga, the formal validity is not in doubt; the contested premise is, once again, the claim that God-likeness (or maximal greatness) is possible.

Where the ontological argument sits

The ontological argument is the purest test of whether existence can be reached by reason alone. Its history is a slow migration of the disputed premise: Anselm and Descartes were stopped by Kant's charge that existence is not a predicate; the modal turn (Plantinga, Gödel) evades Kant by trading on necessary existence and the resources of S5 modal logic, producing arguments that are demonstrably valid — but at the price of relocating the entire weight onto a single possibility premise ($◊\square G$) that is no more obvious than the conclusion and can be met by a symmetric atheist premise. The honest verdict is that the argument, in all its forms, is conditionally sound — if a maximally great being is possible, it is actual — but cannot independently establish the antecedent, and so functions less as a proof than as a demonstration that theism, if coherent, is necessarily true. That makes the coherence of the divine attributes — whether the concept of God is so much as possible — the real battleground, and it leaves the a posteriori cosmological and teleological arguments to carry the empirical case.