Modal Logic and Ontological Arguments (Column 580)

With God’s help

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A few months ago Ariel posted on an American analytic Christian website, bringing a formulation of the well‑known philosophers Alvin Plantinga’s ontological argument (see e.g. here and here; I mentioned it briefly in Column 561 and a bit more in Column 301). This is a good opportunity to get to know modal logic, and that’s what we’ll do. We’ll start with a quick primer in logic (this will require some acquaintance with the formalism) and then move on to ontological arguments in general; in particular, we’ll return to Plantinga’s argument and examine how people attack it.

Modal Logic: Basic Notation and Relations

Modal logic deals with the notions of possibility and necessity. The starting point is that there’s a difference between saying that a proposition P is (now) true and saying that it is necessarily true. For example, the statement that the sun is currently shining is true, but not necessary; the state of affairs could have been otherwise. A truth of that sort is called a contingent truth. By contrast, the disjunction “either the sun is shining or it isn’t” is a necessary truth; there cannot be a different situation in which that does not hold. Likewise, the statement that massive bodies are attracted to the earth (gravity) is, for us, always true, but it could have been otherwise in some other world (loosely speaking). There’s also a difference between a proposition that is impossible (e.g., if it contains a logical contradiction) and one that is merely false in fact. In between, we have what is possible.

Two basic modal operators are customary:

(1) — “necessarily P”;    — “possibly P”.

These operators are interconnected. For example, to say that P is not necessary is the same as saying it is possibly not (equivalently: “it’s not the case that necessarily P” = “possibly not‑P”):

(2) and .

Another useful connection (in the strong S5 system that’s often assumed in these discussions) is:

(3)    (“If P is possible, then it is necessarily possible”).

Note also the distinction between “it is necessary that (if P then Q)” and “if P then (necessarily Q)”. These are not the same claim:

(4) .

This very distinction underlies Judith Ronen’s proposed solution to the problem of foreknowledge and free choice (see Column 301) and my critique there.

Possible‑Worlds Semantics

A common interpretation of the modal operators is in terms of possible worlds. We assume there are countless conceivable “worlds,” each with a (perhaps) different reality. Any scenario we can coherently imagine is, in principle, a possible world. Of those worlds, one is actual (ours), but the others are worlds that could have been actual. Within this framework, to say that a proposition P is necessary means: P is true in all possible worlds. To say that it is impossible means: it is true in none of them. And to say it is possible means: there is at least one possible world in which it holds (one can also talk about degrees of plausibility by counting in how many worlds it holds, though we won’t need that here).

This semantics allows us to translate reasoning about necessity and possibility into ordinary reasoning about truth and falsity across worlds. Many of the useful logical connections we saw above follow immediately from this picture.

Quantifiers and the Analogy (and Its Limits)

At first glance the relation between modal operators and the quantifiers of predicate logic is very close. In predicate logic we talk about a subject x having property P: written as P(x). The quantifiers are: “for all x” (∀x) and “there exists an x” (∃x). There are well‑known relations between them (Boethius’s “square of opposition,” echoed by Maimonides). For example, the negation of “for all x, P(x)” is “there exists an x such that not‑P(x),” and the negation of “there exists an x such that P(x)” is “for all x, not‑P(x).” These mirror exactly the relations (2) above between ◊ and □.

This is unsurprising under possible‑worlds semantics: saying that a proposition is necessary is like saying it is true for all worlds; saying that it is possible is like saying it is true in some world. Yet the analogy is not perfect. There is a classic debate (already in Aristotle, with challenge from George Boole) about whether an existential claim follows from a universal one. For instance, “all aliens have wings” does not entail “there are (winged) aliens” if, in fact, there are no aliens. Boole spoke of vacuous truths of the form “if there were aliens, they would have wings.” In possible‑worlds semantics, by contrast, quantification is over the space of worlds (including merely imaginary ones), so the vacuum issue doesn’t arise in the same way; the quantifiers range over the class of worlds, not over objects that may or may not exist in ours. The upshot: the similarity is instructive but incomplete; one cannot naively translate modal logic into plain predicate logic merely by replacing □ with ∀ and ◊ with ∃.

Ontological Arguments: A General Glance

Kant coined the label “ontological argument” for proofs that derive a claim about what exists from mere definitions and conceptual analysis, without empirical premises. Anselm proposed such an argument for the existence of God (I discuss him extensively in my first notebook and in my book The First Meditation), and Descartes offered an ontological argument for our own existence (see Column 363). It is widely held—largely following Kant—that such arguments cannot succeed: definitions are arbitrary and, by themselves, cannot yield facts about the world. To evaluate an ontological argument, you must either locate a hidden premise or show an invalid step. I tend to the camp that says there are no successful ontological arguments: in every case, either a premise has been smuggled in, or the argument is invalid. Sometimes it takes work to expose this.

Plantinga’s Modal Ontological Argument

(See the sources linked above and in my Q&A: The Modal Ontological Proof.) Plantinga presents a proof of God’s existence via modal logic, as an upgrade to Anselm’s. Roughly:

1. Definition: God is the perfect being whose existence is necessary.
2. Premise: It is possible that God exists (i.e., the divine concept is coherent; not self‑contradictory).
3. From (2): There is a possible world in which God exists.
4. From (1): If God exists in any world, He exists in all worlds (that’s what it means to exist necessarily). Hence, He exists in our world.
5. Conclusion: God exists—and, indeed, His existence is necessary.

The definition in (1) is a definition; as long as it harbors no contradiction there is no bar to defining a concept that way (definitions alone aren’t claims about the world). The interesting step is (2). Atheists typically don’t claim that the very concept of God is contradictory; the argument thus seems to corner them: either embrace contradiction, or accept the conclusion. That would be a significant philosophical achievement for Plantinga (and shifts, to an extent, the burden of proof). Yet, in my view, this argument fails—not because it smuggles a premise but because it is invalid in an important way. Let’s see how.

Counterexample as Methodological Clue

You can produce parallel “proofs” for all sorts of things: a necessary fairy with wings, a perfect island, or even a necessarily salty sugar. For example: perhaps there is a world containing necessarily sour sugar; but if in that world sourness is essential to sugar, then in every world there must be sour sugar. QED. Familiar attacks on Anselm (like Gaunilo’s “perfect island”) go in this spirit. Such counterexamples do not, by themselves, refute an argument; they warn that something is amiss. To truly refute an ontological argument one must locate the precise flaw in the reasoning.

The Real Flaw: Two Senses of “Necessity”

Logic—modal logic included—talks about the truth status of propositions, not about facts themselves. Plantinga’s argument, at best, shows that the proposition “God exists” holds necessarily (i.e., in every world). But when we ordinarily say that God is necessary existence or a necessary being, we mean something metaphysical: that God’s mode of being is such that He cannot fail to exist—His existence is intrinsically necessary. That is a claim about reality, not about our sentences. Mixing these is the same confusion that underlies the problem of “logical determinism”: conflating the necessity of our knowledge/statement about the future with the necessity of the future itself.

I have elsewhere distinguished between the claim “it is necessary that (P → Q)” and “P necessarily brings about Q.” The first is a logical claim about propositions across worlds; the second is a metaphysical claim about the way the world is. One can symbolize the difference by introducing a special “necessary entailment” arrow to distinguish it from ordinary entailment. The metaphysical necessity of God’s existence—if true—speaks about the world, not merely about sentences that are true in all possible worlds.

Returning to Plantinga: his definition treats God’s necessary existence as a metaphysical necessity (an intrinsic feature of the being). But the reasoning proceeds in the logical register of modal truth across worlds. The crucial move—“there is a world in which a necessarily existing being exists; therefore the being exists in all worlds”—slides from one register to the other. If we keep the semantics straight, the premise already smuggles in the conclusion: to assert “in some world there exists a being whose existence is necessary (in our metaphysical sense)” is simply to assert that there exists a being that exists in all worlds. That is question‑begging.

Put differently: the argument doesn’t prove that God exists; it proves—by reductio—that the premise “the divine concept is coherent and allows for a possibly necessary being” is not an innocuous, modest premise. It embeds the conclusion. That’s why the same pattern lets you “prove” perfect islands and salty sugar. The modal language hides the leap.

Further Reading

• Q&A: The Modal Ontological Proof
• Column 301: On Foreknowledge and Choice
• Column 561
• SEP: Ontological Arguments
• Wikipedia: Alvin Plantinga

(For additional background see also: here, here and here.)

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