Modal Logic and Ontological Arguments (Column 580)
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:
- Definition: God is the perfect being whose existence is necessary.
- Premise: It is possible that God exists (i.e., the divine concept is coherent; not self‑contradictory).
- From (2): There is a possible world in which God exists.
- 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.
- 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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This translation preserves the original structure and style and updates all direct links to mikyab.net so that they include /en immediately after .net.
Discussion
Hi
Thank you for an excellent, well-crafted column.
I think your criticism of Plantinga is mistaken, though I admit I’m not sure I understood it properly.
As a first step, I’ll formulate the gist of your argument in my own words. If you think I’ve done it justice, you can go on and address the conclusion I draw from it (namely, that Plantinga’s argument still works after all).
You claim that Plantinga conflates a logical argument with a metaphysical one: the former is strict and necessary, but at the same time empty of descriptive content (it makes no factual claims about the world); the latter describes possible facts, but its ontological status is not strict and necessary. Did I characterize your position accurately?
I would suggest that Plantinga does not “conflate” them, but דווקא distinguishes between an ontological argument and a logical one, except that he insists on a relation of dependence or entailment between the two. So he would answer you that, regarding the concept of a necessary being (God), one cannot ignore that dependence the way you are trying to do. Ignoring it misses the force of the whole move.
I’m not sure I understood you, but I don’t think that’s what I wrote. I spoke about claims, not arguments. The claim “God is a necessary being” is a metaphysical claim, and the claim “the claim ‘God exists’ is necessarily true” is a logical claim, different from the former. The metaphysical claim should not be applied to possible worlds. Modal logic deals with logical claims, not metaphysical ones.
You claimed that Plantinga “conflates” the two kinds of claims. Try thinking about it differently (as I suggested): he does not conflate them, but argues that there is a dependence between the two kinds. I think the problem with your view is that you assume logic is always formal and empty, and therefore logical claims never touch content (facts). Plantinga is telling you: usually you are right, but there is one anomaly you must examine. It is the anomaly of the concept of an absolutely necessary being. There, for the logic to work, you have to assume a “meta-logical” reality.
I don’t know if I’m being clear…
No.
In any case, Plantinga is trying to prove a claim, so it is not enough for him to speculate. He is supposed to convince me.
This reminds me a bit of the logic of the Kuzari’s witness argument:
Similar to the proof here, the Kuzari also proves a claim from its conclusion by way of possibility:
In Rabbi Cherki’s wording: Where does the assumption come from that divine revelation is impossible in reality? From the fact that no nation in history claimed a collective divine revelation.
Since there is one nation that did claim such a revelation—the Jewish people—that means such a revelation is not impossible.
Since it is not impossible, and there is a nation that claimed it, it is necessarily true.
Admittedly, the witness argument goes one logical step further, and thus remains “strong.”
Not only is it not impossible, it must be true. Because a national event that founds a nation cannot be false. We have found no nation founded on a false formative basis.
One can argue about it. I only brought the principle, which seems to me similar and logically well grounded as an argument. After 3 points that prove why the thing falls within the realm of possibility, evidence is needed to show why it is necessary and proven also in actual reality.
And here that thinker you cited would have been wise to use the fact that after he showed in 3 points that the concept of God is possible in all worlds, he should also have had a fourth point showing real confirmation of why He actually exists in the world known to us as well.
I can’t discern any connection or similarity between the arguments.
Note that I am not claiming there is any circumstantial (inspirational) or causal-logical connection (that this necessarily leads to that).
Rather, I’m saying there is a similarity in the intuitive mode of thinking shared by the two arguments.
Both begin with an “attack” (in quotation marks) against the negating claim (“there cannot be revelation” / “there cannot be God”), and then move to a claim that supposedly crushes the basic assumptions behind the negating claim (or tries to expose a positive intuition for their own claim that is latent within the negating claim).
In Amsalem’s case (and I read the column; I know it is not about him but about a thinker who developed or reformulated his argument), he attacks the heretical claim that there is no God / that it is unlikely that He exists, from the direction that if He is conceivable, and everyone agrees that it is possible for Him to have existence in one of the possible worlds, then the required conclusion is that He exists.
The Kuzari (or more precisely Rabbi Cherki’s formulation of the Kuzari’s argument) attacks the philosophers for claiming there is no divine revelation because He is above the dimensions of time and human affairs—and tells them that they themselves think this only because the concept of revelation is foreign to them. But everyone would agree that if there is a nation whose tradition is based on revelation, then that revelation is real even by their own method (or ought to be real according to their claim).
The Kuzari himself did not formulate it this way. But Rabbi Cherki did. And this is one of the modern interpretations of his argument. (And if I may allow myself to speculate, I would say I am convinced that the French rabbis who translated the witness argument into modern philosophical language made use of Amsalem’s perspective.)
But as I said: this is, first, my speculation (the last point). And second, the first point identifies a similarity between the modes of reasoning of the arguments. It does not say there is any logical or necessary connection between them, or even that they belong to the same genre of proofs.
Obviously Plantinga is not merely speculating, and he thinks he has a winning argument (even if not a conclusive proof). Who said otherwise? My claim was that if one interprets his move in the way I interpreted it, it really is persuasive.
The gist of the move: the case of the concept of an absolutely necessary being is a special case in which logic is breached and leads us to the conclusion that such a being exists in a real way (and not only within logic).
By the way, it seems to me that Relatively Rational understands it that way too.
This is not Anselm’s argument but Plantinga’s. He is not attacking anything (and neither is Anselm), but trying to prove a claim. And I still do not see a similarity. Here we are dealing with a logical argument, and you have to show that it contains no flaws (I claimed that it does). The assumption that something plausible and supported by tradition should be accepted is an assumption of common sense. Something entirely different.
He thought he did, but as I showed, he does not.
Forgive me for troubling you. The subject interests me. Would it be fair to describe the gist of your position as claiming that one must not mix synthetic claims with analytic claims (in Kant’s language)? In your view, did Plantinga make that mix-up?
I don’t think so. In my view he conflated two meanings of “necessity”: physical and logical. This can be mapped onto analytic (logical) versus synthetic (physical), but that is not the focal point of the dispute. I think he simply did not notice that there are two different meanings here. I assume that if he had seen this distinction, he too would have agreed with it.
I assumed you would agree that there is a common denominator between your description and Kant’s. Except that I think the Kantian distinction between analytic and synthetic judgments precedes your distinction (between the two kinds of necessity). Your distinction seems to me secondary and derived from the first distinction. Therefore I think the criticism of the Kantian move also undermines your position, and does so precisely at the root level (which he recognized and you did not). So my move will be to show that the assumption Kant makes in the background of the distinction between analytic and synthetic judgments is a mistaken assumption, and from this it follows that his whole move is mistaken (and consequently also drags down your move, which relies on it).
So what is the implicit Kantian assumption that I am attacking? What is, in his view, the focal point of the discussion behind the distinction between analytic and synthetic? In my opinion, it is the assumption that the content or meanings of sentences/claims cannot come from intuition. Kant denies the existence of such a capacity. Therefore, in his view, a “synthetic” claim of a metaphysical character such as “God exists” expresses a breach of the bounds of meaning and should be judged empty of content, what he calls “theoretical” content (a logical claim in your language). Therefore Kant would also agree with you that Plantinga is too quick to confer necessity on his claims unjustifiably, since analytic claims really are necessary.
To refute Kant (and you), all one has to do is show that we do possess an intuitive capacity, and that Plantinga makes use of it here as well. The insight regarding an absolutely necessary being (or at least regarding its concept/image) is planted in us intuitively, and only afterward comes the inference called “the ontological argument,” which describes the relation between the concept and its real existence (the content of the concept). This description itself is not analytic or “logical,” and therefore does not purport to carry necessary validity in the way you attributed to it. Of course, it may be that Plantinga himself does claim that his move is “necessary,” but then I would argue, by the principle of charity, that he is “spoiling” things for himself. In summary, note that I am not claiming that my move proves that the ontological argument proves God’s existence. All I wanted to show is that your criticism fails because it misses the main point.
It seems to me that this is the point where we part as friends. 🙂
All good 😀
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