From Anselm to Plantinga: A Modus Tollens Reformulation of the Modal Ontological Argument
Few arguments in the philosophy of religion have generated as much fascination, criticism, and reformulation as the ontological argument for the existence of God. Unlike cosmological and teleological arguments, the ontological argument does not begin with an observation about the physical universe. Its starting point is instead a concept of God and an analysis of what follows from that concept. For this reason, ontological arguments have traditionally been understood as a priori arguments: arguments that seek to establish God's existence through reason rather than empirical observation.
The argument has taken substantially different forms over the centuries. Anselm of Canterbury formulated the earliest and most famous version in the eleventh century. Later philosophers, including René Descartes, Gottfried Wilhelm Leibniz, Charles Hartshorne, Norman Malcolm, and, most famously in the contemporary period, Alvin Plantinga, developed increasingly sophisticated versions of the argument. Plantinga's contribution was to recast the argument in the language of modern modal logic and possible-world semantics.
The purpose of this article is not merely to rehearse Plantinga's argument. It is to examine a closely related formulation in which the modal structure is made explicit through modus tollens. The central idea is simple:
If God does not exist, then it is impossible for God to exist
It is not impossible for God to exist.
Therefore, God exists.
This formulation does not attempt to bypass the controversial premise that God's existence is metaphysically possible. Rather, it makes that premise and its logical consequences unusually transparent. The decisive question becomes whether the concept of a necessarily existing God is coherent, or whether some genuine contradiction can be identified in it.
Anselm and the Beginning of the Ontological Argument
Anselm of Canterbury (c. 1033–1109) presented what became the paradigmatic ontological argument in his Proslogion, written in the late eleventh century. His famous characterization of God is id quo maius cogitari nequit: "that than which nothing greater can be conceived."
Anselm's argument proceeds by considering what would follow if such a being existed only in the understanding and not in reality. If the greatest conceivable being existed merely as an idea, Anselm argues, then a greater being could be conceived, namely, one possessing the same greatness while also existing in reality. But this would contradict the definition of God as the greatest conceivable being.
The argument therefore moves from a conception of God's greatness toward God's existence.
Anselm's argument immediately provoked criticism. Gaunilo, a contemporary of Anselm, attempted to demonstrate that similar reasoning could apparently be used to establish the existence of an ideal island. Later, Thomas Aquinas rejected Anselm's reasoning, while Immanuel Kant famously objected that existence is not a predicate that can simply be added to a concept to make the thing greater.
These criticisms are important because they reveal a central difficulty in the classical ontological argument: how can a merely conceptual analysis yield a conclusion about what actually exists?
The modern modal versions of the argument attempt to answer this problem in a different way.
From Anselm to Modal Logic
The crucial development is the introduction of modal concepts, possibility and necessity.
A proposition is possibly true if it is true in at least one metaphysically possible world. A proposition is necessarily true if it is true in every metaphysically possible world. The "possible worlds" language should not be understood as referring to alternate physical universes that someone has literally discovered. It is a semantic framework for evaluating claims about what could or could not have been the case.
This distinction is essential.
Consider the proposition:
"There is a unicorn."
If unicorns are metaphysically possible, then there is at least one possible world in which unicorns exist. That does not imply that unicorns exist in the actual world. A contingent entity may exist in some possible worlds while failing to exist in others.
A necessary entity is different. If an entity is genuinely necessary, its existence is not contingent upon which possible world is actual. If such an entity exists in one possible world, it exists in every possible world, including the actual world.
This distinction provides the foundation for the modal ontological argument.
Plantinga's Modal Ontological Argument
Alvin Plantinga's formulation is one of the most influential modern versions of the ontological argument. Plantinga defines maximal excellence in terms of properties such as omnipotence, omniscience, and moral perfection. A being possesses maximal greatness when it possesses maximal excellence in every possible world. Consequently, maximal greatness includes necessary existence.
In simplified form, Plantinga's argument can be represented as follows:
It is possible that a maximally great being exists.
If a maximally great being exists in a possible world, it exists necessarily in every possible world.
Therefore, a maximally great being exists in every possible world.
Therefore, a maximally great being exists in the actual world.
Therefore, God exists.
The modal logic underlying the argument is important. The crucial transition is from:
Possibly, necessarily, God exists
to:
Necessarily, God exists.
In the modal system commonly called S5, this transition is valid. Plantinga's argument therefore does not depend upon simply asserting that whatever can be imagined must be real. It depends upon a much more specific proposition: if the existence of a necessarily existing being is genuinely possible, then that being exists necessarily.
Plantinga himself was cautious about describing this as a straightforward proof that compels belief in God. His conclusion was that, if the central modal premise is rationally acceptable, then the argument establishes at least the rational permissibility of accepting its conclusion.
That qualification identifies the true pressure point of the argument.
The controversial premise is not the conclusion. Nor is it primarily the modal inference. It is the claim that God's existence is metaphysically possible.
The Real Question: What Would Make God's Existence Impossible?
Suppose God is defined for purposes of the argument as a being whose existence is necessary if God exists.
This definition does not simply assert that God exists.
The distinction is analogous to an ordinary conceptual definition. To define a bachelor as an unmarried man is not to assert that any bachelors exist. The definition tells us what something would have to be if it were a bachelor.
Likewise, to define God as a necessarily existing being does not, by itself, establish that God exists. It establishes a modal condition:
If God exists, God cannot fail to exist.
Symbolically, let G represent "God exists."
The definition can therefore be expressed as:
G → □G
If God exists, then God exists necessarily.
The important question is whether G is possible.
If God's existence is impossible, then:
□¬G
God does not exist in any possible world.
But if God's existence is possible, then:
◇G
God exists in at least one possible world.
Given the definition of God as a necessary being, those alternatives have radically different consequences.
If God exists in even one possible world, then God exists in every possible world. If God exists in no possible world, then God's existence is impossible.
There is therefore no stable middle category in which God is a possible necessary being who nevertheless exists in some possible worlds but not others.
A necessary being, if possible, is necessarily actual.
The Modus Tollens Formulation
The argument can now be expressed in a form that makes its logical structure particularly clear.
Let:
G = God exists.
□G = God exists necessarily.
◇G = God possibly exists.
□¬G = God necessarily does not exist.
◇¬G = God possibly does not exist.
The first premise follows from the definition of God:
1. ¬G → □¬G
If God does not exist, then it is impossible for God to exist.
Why?
Because God has been defined as a being whose existence is necessary if God exists. If God fails to exist, there is no possible world in which God exists. Otherwise, God would exist in some possible world and, being necessary, would exist in every possible world, including the actual world.
The second premise is:
2. ◇G
It is not impossible that God exists.
But possibility can be expressed in terms of the negation of necessity:
◇G ≡ ¬□¬G
If God possibly exists, then it is not the case that God necessarily does not exist.
Thus premise 2 can be restated:
2. ¬□¬G
Now compare the two premises:
1. ¬G → □¬G
2. ¬□¬G
The structure is unmistakably modus tollens:
If P, then Q.
Not Q.
Therefore, not P.
Here:
If God does not exist, God necessarily does not exist.
God does not necessarily fail to exist.
Therefore, God does not fail to exist.
Thus:
∴ G
God exists.
The argument is therefore not simply a restatement of the conclusion in the definition. The definition supplies the conditional relationship between God's existence and necessary existence. The modal premise supplies the denial of God's necessary nonexistence. The conclusion follows by modus tollens.
Why the "Possible World" Objection Misses the Point
One of the most persistent misunderstandings surrounding modal ontological arguments is the claim that establishing God's existence in a possible world would establish nothing about the actual world.
That objection is correct for contingent beings but not for a necessarily existing being.
Suppose a unicorn exists in some possible world. Nothing follows about the actual world because unicorns, if they exist at all, are contingent beings. Their existence does not entail existence in every possible world.
The modal ontological argument does not reason in that manner.
Its claim is instead:
If a necessary being exists in any possible world, it exists in every possible world.
The actual world is itself one of the possible worlds under consideration. Therefore:
God exists in some possible world
→ God exists in every possible world
→ God exists in the actual world.
The argument does not infer actuality merely from possibility.
It infers actuality from possible necessary existence.
This distinction is decisive.
A possible contingent being need not be actual. A possible necessary being must be actual.
Correcting a Common Confusion About Coherence
At this point, however, a significant qualification is necessary.
It would be too strong to say simply:
"Every coherent concept exists."
That formulation confuses conceptual coherence with metaphysical possibility.
A coherent description can fail to correspond to an actual entity. The concept of a unicorn may be coherent, for example, without any unicorn actually existing. But this does not mean that unicorns are metaphysically impossible. It means only that their possible existence does not entail their actual existence.
The relevant claim concerning God is stronger:
If the concept of God is metaphysically coherent in such a way that God's existence is genuinely possible, then there is a possible world in which God exists.
That is what metaphysical possibility means in possible-world semantics.
The important question is therefore not whether the word "God" can be defined without contradiction at the level of ordinary language. The question is whether the properties constituting the proposed divine being can jointly obtain.
If they cannot, God's existence is impossible.
If they can, then God's existence is possible.
And if the being in question is defined as necessarily existing, possible existence entails necessary existence.
Thus the debate is concentrated in a precise location: is the concept of a necessarily existing God coherent, or can an actual contradiction be demonstrated?
The Burden of an Alleged Incoherence
This gives the argument a somewhat different character from the way it is sometimes presented.
The defender of the argument does not need to demonstrate God's existence independently before asserting the possibility premise. Nor does the defender need to produce an empirical observation of God existing in a possible world. Possible worlds are not empirical locations.
Instead, the defender maintains that God's existence is metaphysically possible unless there is sufficient reason to conclude otherwise.
An opponent can therefore challenge the argument in at least two fundamentally different ways.
First, the opponent may challenge the modal framework itself. Perhaps the relevant system of modal logic does not license the transition from possible necessary existence to necessary existence.
Second, and more directly, the opponent may challenge the possibility premise:
◇G
That challenge requires more than merely saying, "We do not know whether God exists."
Ignorance about God's existence does not establish God's impossibility.
The relevant claim would have to be that:
□¬G
God cannot exist.
And to establish that, one would need to identify some contradiction in the concept of the proposed God.
This is why objections such as the problem of evil require careful analysis. The mere existence of evil does not formally demonstrate that an omnipotent, omniscient, and perfectly good being is impossible. To defeat the modal premise, the objection would need to establish an incompatibility between the relevant divine attributes and the existence of any possible reality.
Likewise, divine hiddenness may constitute an evidential objection to the existence of God, but it does not automatically demonstrate the metaphysical impossibility of God.
The distinction between evidential disconfirmation and modal impossibility is therefore critical.
The Asymmetry Between Possibility and Impossibility
The argument becomes particularly interesting when possibility and impossibility are considered together.
For a contingent entity, it exists in some possible worlds, but not others (possibly exists).
For a being whose defining characteristic is necessary existence, however, the only options are:
Either God is impossible, or God is necessary.
If God's existence is possible, then God exists necessarily.
If God does not exist, it is impossible for God to exist.
This can be represented as:
◇G → □G
and
¬◇G → □¬G
The question, therefore, is not simply whether God happens to exist in the actual world. It is whether God's existence is metaphysically possible.
Once necessary existence is built into the concept, the modal question determines the actual question.
That is the distinctive force of the modal ontological argument.
Plantinga's familiar presentation emphasizes the positive route:
It is possible that a necessarily existing God exists.
Therefore, God exists necessarily.
Therefore, God exists.
The reformulation presented here emphasizes the same modal relationship from the opposite direction:
If God does not exist, then God's existence is impossible.
But God's existence is possible.
Therefore, God does not fail to exist.
Therefore, God exists.
The difference is not merely rhetorical.
The second formulation makes explicit that the argument can be understood as a reductio of God's nonexistence.
Assume:
¬G
Given the definition of God as necessarily existent:
¬G → □¬G
But the possibility premise states:
◇G
And:
◇G ≡ ¬□¬G
Therefore:
¬G → □¬G
¬□¬G
∴ ¬¬G
∴ G
The conclusion follows by modus tollens.
This formulation also exposes what an objection must accomplish. It is not enough to demonstrate that God's existence has not been empirically established. The argument does not claim that empirical evidence is what establishes the modal premise.
The relevant challenge is stronger:
Show that God's existence is impossible.
If that cannot be done, and if metaphysical possibility is rationally warranted, then the modal conclusion follows.
What the Argument Does and Does Not Establish
The argument should nevertheless be stated with precision.
The conclusion is not:
"We can imagine God."
Nor is it:
"God has not been proven impossible."
Nor is it:
"We lack evidence against God."
Its conclusion depends upon the stronger premise:
God's existence is metaphysically possible.
If that premise is true, and if God is properly understood as necessarily existent, the conclusion follows within the relevant modal framework.
The argument therefore has the familiar structure of a conditional proof:
If God is possible, God exists.
This does not make the argument trivial. It identifies the precise philosophical dispute.
The central question becomes:
Is there any incoherence in the concept of a necessarily existing God?
If there is, the argument fails at its possibility premise.
If there is not, then denying God's existence generates a modal contradiction: God's nonexistence would entail that God's existence is impossible, while the possibility premise entails that God's existence is not impossible.
The Significance of the Reformulation
The historical significance of the ontological argument lies partly in its remarkable ability to survive successive transformations in philosophical vocabulary.
Anselm framed his argument in terms of greatness and conception. Later philosophers recast the argument through perfection, necessity, and existence. Contemporary modal versions replace much of the original conceptual machinery with the formal language of possible worlds.
Plantinga's version represents one of the most sophisticated forms of this development. It avoids some of the traditional difficulties associated with treating existence as an ordinary property and instead concentrates upon the modal status of a maximally great being. The Stanford Encyclopedia of Philosophy identifies Plantinga's argument as a paradigmatic modern modal ontological argument and notes that its validity depends upon the modal framework employed, while the principal controversy concerns the acceptability of its central possibility premise.
The modus tollens reformulation makes that structure even more explicit.
It begins not with the assertion that God exists, but with a conditional contained in the very definition under examination:
If God does not exist, then God cannot possibly exist.
It then introduces the modal premise:
God can possibly exist.
The two claims cannot simultaneously be true if God is necessarily existent.
Consequently, God's nonexistence is untenable under the stated premises.
Conclusion
The ontological argument has traveled a remarkable philosophical distance from Anselm's eleventh-century meditation on "that than which nothing greater can be conceived" to Plantinga's formal treatment of maximal greatness and possible worlds. Its essential question, however, remains remarkably stable: what follows from the concept of a being whose existence cannot fail?
The modern modal formulation gives that question precise logical form.
If God is a necessarily existing being, then God's existence cannot be contingent. God cannot exist in some possible worlds while failing to exist in others. Either God's existence is impossible, or God exists necessarily.
The crucial premise is therefore the possibility of God:
◇G
If this premise is true, then God's necessary existence follows. And because the actual world is one of the possible worlds, necessary existence entails actual existence.
The modus tollens formulation approaches the same conclusion from the denial:
If God does not exist, then God necessarily does not exist.
But if God's existence is possible:
God does not necessarily fail to exist.
Therefore:
God does not fail to exist.
And therefore:
God exists.
The force of the argument ultimately rests upon a question that is both narrower and more demanding than the broad question "Can we prove that God exists?" The relevant question is whether the concept of a necessarily existing God is genuinely incoherent.
If an actual contradiction can be demonstrated within the concept, the argument's possibility premise collapses.
If no such contradiction can be demonstrated, and metaphysical possibility is rationally warranted, then the modal structure of necessary existence does the remainder of the work.
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