Modal Logic and the Barcan Formula (Column 634)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

Modal Logic and the Barcan Formula (Column 634)

Some time ago I was asked a question about modal necessity and the Barcan formula. The issue is briefly mentioned in the Wikipedia entry on the Barcan formula and is described in more detail in a video there. Back then I did not respond because I hadn’t had time to delve into the argument itself, but now I’ve listened to the video critically—and here is my reply.

Introduction: the strengths and weaknesses of formalization

I have often pointed out the two faces of mathematical and logical formalization: on the one hand, it lets us be precise in definitions and arguments and reach maximal clarity and certainty; on the other hand, there is a danger we will be seduced by the formalization and mistakenly think the conclusion is certain and necessary. I have discussed this also in several columns that dealt with probabilistic and statistical fallacies, where there can be errors in calculation or in interpreting the results. In all such cases—both in logic and in statistics—the mistake can stem from the fact that even if the argument is logically valid (or the mathematical proof is correct), its premises need scrutiny, and/or its meaning (semantics) is imprecise (semantics and meaning are not part of the logical syntax itself). Applying a purely formal argument to a philosophical (or other) problem always involves additional assumptions. (See, e.g., columns 50 and 318 for examples of misunderstanding the nature of logic and mathematics themselves.)

Regarding pitfalls and the analysis of arguments in modal logic, as I wrote to the questioner in my reply, there are several columns on the site that touch on this (see, e.g., columns 160, 301, 561, 580). Modal logic, in particular, is a domain that is even more prone to mistakes—but it isn’t always clear where the problem lies. Sometimes the problem is in the interpretation of possible worlds (the standard semantics for modal logic), and sometimes it lies in hidden assumptions underlying the formalization itself. Equipped with this skeptical perspective, we can approach the question the reader raised about the Barcan formula. First, though, a refresher—plus some clarifications—on modal logic and its possible-worlds semantics.

Modal logic: a refresher

I gave the background for modal logic and the semantics of multiple worlds in detail in column 580; here I’ll only recap the main points. In traditional (non-modal) logic we speak about the truth or falsity of a proposition P. But truth can be accidental (contingent) or necessary. For example, the proposition “It is light outside now” may be true, but it isn’t necessary: it could have been dark. By contrast, the proposition “Either it’s light or it’s dark outside” is true of necessity: it could not have failed to be true.

Beyond truth or falsity, modal logic concerns whether a proposition is necessary, possible, or impossible. We denote the two basic modal operators as follows: = “necessarily P”; = “possibly P”. Using negation, there are familiar connections between these operators, for example: That is, saying “possibly not-P” is equivalent to “not necessarily P”, and saying “necessarily not-P” is equivalent to “not possibly P”. This illustrates the precision the modal formalization enables.

The possible-worlds interpretation gives semantics to these modal notions. Imagine the collection of all (logically) possible worlds one can conceive: a world with no gravitation, a world with a different gravitational constant, a world with negative mass, a world without electromagnetism, a world where people have wings, a world where I don’t exist, and so on. Of course there are infinitely many such worlds, each differing from the others in at least something. There is no possible world in which a proposition and its negation are both true; but in every possible world the law of excluded middle holds, “P or not-P”.

Using the possible-worlds framework, we can restate modal properties as follows:

• A necessary proposition is true in every possible world.
• A possible proposition is true in some possible world(s).
• An impossible proposition is true in no possible world.

Note that this semantics moves us from the truth/falsity vocabulary of classical logic into a setting that also quantifies over worlds. For logicians, this is an advantage, because logic handles truth values well; now, using this framework, we can treat the modal status of propositions, too. The price we pay is that truth and falsity must be evaluated across all possible worlds, not just in ours.

This requires the quantifiers from predicate calculus: = “for all x”; = “there exists x”. Not by chance, there are parallels between these quantifiers and the modal operators. Saying “it is not the case that something holds for all x” means there exists (at least one) x for which it does not hold—just as “not necessary” correlates with “possibly not”.

Thus, “it is necessary that P” can be rendered as “for every possible world w, P holds there.” In formalized terms: , where reads “P is true in world ”. You can now easily see again the modal dualities above: “necessarily P” means true in all possible worlds; “not necessarily P” means there exists at least one world where not-P; etc.

Two kinds of necessity: epistemic vs. ontic

Earlier I distinguished between “P is true” and “P is true of necessity.” The former is a statement about our world; there may be worlds where it is false. The latter says it is true in every conceivable world. Also note the difference between “necessarily, if Q then P” and “if Q then necessarily P.” They sound similar, but they are different claims. In the first, what is necessary is the implication itself; in the second, if Q holds, then P is necessary, which is stronger.

This distinction plays a role in classic discussions of divine foreknowledge and free will. One can maintain that “necessarily, if God knows I will do A, then I will do A,” without claiming “if God knows I will do A, then I necessarily do A.” The necessity is attached to the implication, not to my action; this does not by itself entail determinism. (Elsewhere I explained why I don’t think this fully resolves the difficulty.)

To sharpen what “necessary existence” means, many conflate “God exists at every moment in time” with “God’s existence is necessary.” That identification is (at least partly) mistaken. If existence is necessary, then indeed God exists at every time; but existing at every time does not imply necessary existence. The time-indexed distinction is less helpful for “existence in every possible world,” since there the contrast between contingent and necessary existence is exactly what is at stake.

In my first dialogue in the book The First Being I analyzed Anselm’s ontological argument. In the end of chapter 4 of the Proslogion Anselm appears to “prove” God’s existence logically. One might ask: isn’t this redundant if earlier chapters already argued that God is a necessary being? I explained there the difference between epistemic necessity (I know with certainty that God exists—because I have a proof) and ontic necessity (God exists of necessity). Saying “I necessarily know that God exists” is not the same as “I know that God necessarily exists.” The former is about my state of knowledge; the latter is about reality itself.

Similarly with Descartes’ cogito: “It is necessarily the case that I exist” does not mean that I am an ontologically necessary being. The cogito may give me certainty that I exist (in this world), but that is not the same as necessary existence across all possible worlds.

Which kind of necessity does the possible-worlds semantics capture? At first glance, epistemic necessity (certainty): the set of worlds “I can conceive” reflects what is certain for me. Ontic necessity, by contrast, could be tied to features of the world we’re actually discussing. Perhaps in some world there is ontic necessity for P, but not in others. Put differently: we can define “necessity” relative to subsets of worlds—for instance, worlds sufficiently like ours (e.g., complex enough). Then a claim might be necessary in our world (given its structure) without being necessary in all possible worlds.

In column 580 I distinguished these two notions and showed how Plantinga’s modal ontological argument stumbles by sliding between them. He treats necessity in a single world and conflates it with necessity across all possible worlds. The former is ontic (true of our world given its nature), the latter is metaphysical (true in every possible world).

Enter the Barcan formula

Not all components of modal logic are univocal; some admit different definitions. That matters because certain definitions build in specific assumptions, so behind a modal formalization can lurk differing philosophical commitments. One well-known example is the Barcan formula, which asserts the following entailment:

In words: if, for every object x, it is necessary that P(x), then it is necessary that, for every object x, P(x). Using the dualities above, this is equivalent to:

In words: if it’s possible that there exists an x such that P(x), then there exists some x for which P(x) is possible. These are logically inter-derivable schemata. (There are also converse Barcan schemata; if one accepts additional assumptions, the arrows can be strengthened to biconditionals.)

Two important clarifications. First, the Barcan formula is not a theorem of “pure” modal logic: it is an extra principle one may adopt or reject; you cannot prove it from modal laws alone or from the bare possible-worlds picture. Second, even semantically, endorsing it amounts to substantive assumptions about domains and how quantification works across worlds.

Here is an illustration. Suppose = “x has mass and therefore occupies space.” The left-hand side says: for every massive object, it is necessary that it occupies space—in any possible world. But what about a world with no space at all? Or a world with no massive objects? Or a world with massive objects but no space? (Is that coherent?) On the right-hand side, by contrast, the choice of the object happens in each world; on the left-hand side, the objects quantified over might be taken from our actual world and then evaluated across worlds. The two sides can be read as different kinds of necessity (ontic vs. epistemic), which already shows the identity is far from trivial.

As I argued elsewhere with Yom’s critique, causality (for Hume) is not derived from observation and is, in some sense, a priori; but that doesn’t automatically imply it holds in every conceivable world. The move from “necessary in our world” to “necessary in all worlds” is a substantive philosophical leap.

Framing the problem

Following the video’s presentation, we can contrast two philosophical positions:

• Necessitism: every object that exists, exists of necessity. It could have had different properties, but its existence is necessary. (This raises identity worries: what makes it the same object if its properties change drastically?)
• Contingentism: not everything is necessary; at least some objects exist only contingently. There can be merely possible objects that do not actually exist.

At first glance, necessitism is counterintuitive. Its proponents reply that our intuition that “the coin might not have existed” means only that “it might have been something else (say, a tree),” not that the object fails to exist. I find this a verbal maneuver: my intuition is that the coin could simply have failed to exist, period. Moreover, necessitism typically presupposes infinitely many merely possible objects (indeed, perhaps infinitely many actually existing ones if every possible object must exist).

The crux is that if they can prove necessitism from the Barcan formula, then we’d have a serious metaphysical conclusion from apparently innocuous logic.

The alleged proof from the Barcan formula

Consider the dual Barcan: . It seems to say: if possibly there exists an object with property P, then there is (actually) some object for which P is possible—an apparent move from possible existence to actual existence. The contingentist will reject this, since it conflates “there could be such an object” with “there is (in actuality) an object that could be thus.”

But even granting the schema, one cannot conclude that the number of possible objects equals the number of actual objects. Different possibilities (for different properties P, Q, …) could be realized by different objects; and even if many possibilities are realized by the same object, that still doesn’t enforce a one-to-one match. The schema does not yield the necessitist’s strong conclusion.

A further error appears when one treats “does not actually exist” as a first-order property and lets be “x fails to exist actually.” Then from “possibly there exists an x that doesn’t actually exist” plus Barcan, one infers “there exists an x that possibly doesn’t actually exist”—and is tempted to conclude that something both exists and (possibly) doesn’t, which is incoherent. The mistake is semantic: existence/non-existence is not an ordinary first-order property of objects (as Kant already argued); it is not something you predicate of a thing the way you do color or mass.

A broader view

Near the end of the lecture, the presenter suggests that if purely logical manipulations can yield metaphysical theorems, then there is no real distinction between logic and metaphysics—logic “has content.” But the right lesson is the opposite: the moment you draw a substantive metaphysical conclusion, you must have smuggled in non-logical assumptions (e.g., about domains, identity across worlds, or the status of existence as a predicate). Without those, logic alone doesn’t deliver synthetic truths about the world. This is exactly the Kantian critique of ontological arguments: you can’t derive existence claims from definitions and logic alone.

In my first dialogue and in this column I pointed to at least three flaws in the necessitist argument from Barcan: (1) it assumes Barcan as if it were pure logic, whereas it is an additional, substantive principle; (2) it tacitly relies on metaphysical assumptions (e.g., about the physical structure of worlds and identity); (3) it repeatedly conflates ontic necessity with epistemic necessity. Existence, and the identity conditions of objects, are not ordinary properties one can quantify over without care.

References and links (some in Hebrew)

• Wikipedia: Barcan formula
• Video discussion of the Barcan formula
• On modal logic (Hebrew)
• Column 50 (Hebrew)
• Related column (Hebrew)
• Related column (Hebrew)
• Related column (Hebrew)
• Related column (Hebrew)
• Related column (Hebrew)
• On the ontological argument (Hebrew)

Note: All occurrences of mathematical or logical expressions have been written using WordPress Jetpack LaTeX—for example: , and the Barcan schemata , .