Modal Logic and the Barcan Formula (Column 634)

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With God’s help: Modal Logic and the Barcan Formula

Some time ago I was asked a question about modal necessity and the Barcan formula. The problem is already mentioned briefly in the Wikipedia entry on the Barcan formula, and it is described in greater detail in this video. I did not answer there because I did not have time to get into the argument itself. I have now listened to the video critically, and here is my response.

Introduction: the advantages and disadvantages of formalization. I have more than once pointed out the double-edged character of mathematical and logical formalization: on the one hand, it enables us to sharpen definitions and arguments and to attain maximal clarity and certainty; at the same time, there is a danger that we will be taken captive by the formalization and mistakenly think that the conclusion is certain and necessary. I discussed this as well in several columns dealing with probabilistic and statistical fallacies, where there can be errors in calculation or errors in interpreting the results. In all of these cases, both in logic and in statistics, the mistake can arise because even if the argument is logically valid (or a mathematical proof), its premises require examination, and/or its meaning (its semantics) is not precise enough (semantics and meaning are not part of logical syntax itself). The application of a formal argument to a philosophical or other problem always involves additional assumptions (see also columns 50 and others). In a responsum here you can find an example of a mistake in understanding the meaning of logic and mathematics themselves.

As I wrote to the questioner in my reply, several columns have already been written here on the site about fallacies and analyses of modal-logic arguments (see columns 160 and others). You can see there that modal logic is a field that is even more prone to error, although it is not always clear where the problem lies. Sometimes the problem is in the meaning of possible worlds, which is the accepted semantics for modal logic, and sometimes the problem lies in hidden assumptions at the basis of the formalization itself. Equipped with this suspicious point of view, we can approach the problem raised by the questioner regarding the Barcan formula. But first, a reminder—with some necessary additions and emphases—about modal logic and the semantics of possible worlds.

Modal logic: a reminder. The background to modal logic and the semantics of multiple worlds was given in detail in column 580, so here I will only briefly repeat the main points. In traditional logic we speak about the truth or falsity of a proposition P. But truth can be contingent or necessary. For example, the proposition “it is now light outside” is true, but not necessary. It could also have been dark. By contrast, the proposition “either it is light outside or it is dark outside” is necessarily true. It could not have been false. So beyond the truth or falsity of a given proposition—which is what traditional logic deals with—it can also be necessary, possible, or impossible. That is what modal logic deals with. If we take a proposition P, we denote the two basic modal operators relating to it as follows: □P

necessary that—

possible that—

If you think about it, you will see that there are relations between these operators when one uses the negation operator. For example, when I say that something is not necessary (= it is not true that it is necessary), this means that it is possible for it not to be true (that is a proposition whose truth is contingent). By contrast, the statement that something is necessarily false means that it is impossible. In logical symbolism, we express these relations as follows:

~□P = ◊~P

Notice that saying that it is necessarily false (□~P) is not equivalent to saying that there is no necessity that it is true, or that it is not necessarily true (~□P).

It is not necessary that the proposition P

be true. This means that it is true but could also have been false. But to say that it is necessarily false means that it cannot be true. This is an illustration of the precision that modal formalization makes possible. The semantics of possible worlds gives modal properties meanings in terms of all the possible worlds one can imagine. Suppose we have the set of all possible worlds that can be conceived. There is a world in which there is no law of gravitation, a world in which there is such a law but with a different constant, a world with negative mass, a world without

electromagnetism, a world with people who have wings, a world in which I do not exist, and so on. Every state of affairs that is logically possible (even if it is not physically possible, that is, even if it contradicts the laws of nature in our world) is one of these worlds. There are of course infinitely many such worlds, each differing from the others in at least something. You will not find there a world in which a proposition and its negation are both true simultaneously. That is not a world one can conceive; indeed it is impossible (its description is empty). Therefore there is no possible world in which that occurs. By contrast, in every such world the proposition “P or not-P” will be true,

since it is necessarily true (there can be no world in which it is false). In terms of the set of possible worlds, the properties above can be given the following interpretations: a necessary proposition is one that is true in every possible world.

A possible proposition is one for which there are possible worlds in which it is true.

An impossible proposition is one for which there is no possible world in which it holds. Notice that this semantics transfers us from the language of necessity and possibility, with which we do not really know how to work, to the language of truth and falsehood, which traditional logic does know how to handle. For logicians this is a major advantage, because logic knows how to deal well with the truth values of propositions—true or false—and through them we can now also deal with their modal properties (necessity and possibility). The price we pay for this convenience is that truth and falsehood must be checked across all possible worlds and not only in ours. This in fact requires the use of quantifiers (“there exists,” “for all”), which are part of predicate calculus. These are the two basic quantifiers:

for every x

there exists an x

Not by accident, similar relations hold between these two quantifiers and the modal operators we saw above. When I say that it is not true that something holds for every x, this means that there is an x

(at least one) for which it does not hold. You can now see that the statement “it is necessary” translates into the statement “true in every possible world,” or: for every possible world, P holds. If we formalize this, we get: □P = ∀w P(w) (where w

is some possible world, and P(w)

means: the proposition P holds in world w.)

Notice that we translated a proposition that contains a modal operator (on the left-hand side of the identity) into a proposition containing only propositions and quantifiers. From here you can easily see the modal relations I described above between necessity and possibility. For example, the proposition that something

is necessary means that it is true in all possible worlds. Therefore the proposition that it

is not necessary means that there is a possible world

(at least one) in which it does not hold. This is one of the relations described above.

Different meanings of necessity. Above I already pointed to the difference between saying that a proposition P is true and saying that it is true necessarily. The proposition

“is true” is a statement about our world, but there may certainly be worlds or realities in which it would not be true. By contrast, P

“is true necessarily” means that it is true in every world that can be conceived. Now note the difference between the proposition “necessarily, if P”

“then Q,” and the proposition “if P then necessarily Q.” This sounds very similar to us, but these are two different propositions. In the first proposition, what is necessary is the implication of Q from P, but not P

and Q in themselves. By contrast, in the second proposition, if P

is true (not necessarily), then Q

is necessarily true. Notice: it is not the implication that is necessary. The implication is contingent; the consequent is what is necessarily true. In the first proposition, there can be no possible world in which P holds and Q does

not. In the second proposition, if P

holds in some world, then Q holds in all worlds (but there is no claim here about P

in other worlds). To see the difference more sharply, suppose we know that P holds only in our world. According to the first proposition, this means only that

Q also necessarily holds in our world, but its holding there is not itself necessary (and therefore it may or may not hold in some of the others). By contrast, according to the second proposition, Q

holds in all worlds. In column 301

I presented Judith Ronen’s argument concerning knowledge and free choice. Her claim is that even if one assumes that God necessarily knows what I will do, this is equivalent to a proposition of the first kind: necessarily, if God knows that I will do A

then I will do A (the necessity is attached to the implication), but this is not equivalent to the claim that if He knows, I will necessarily do it. Therefore God’s knowledge does not imply determinism (that doing A

is necessary). This illustrates the distinction I described above. As for the matter itself, I explained there why in my view this distinction does not really solve the difficulty, and this is not the place for that.

To sharpen further the issue of necessary existence, I will point to another distinction. Many identify the claim that God exists throughout the entire time axis with the claim that His existence is necessary. That identification is mistaken, at least in part. If His existence is necessary, then He will exist at every point in time; but the fact that He exists at every point in time does not mean that His existence is necessary. It may be that He exists contingently at every point in time. Admittedly, with regard to existence in every possible world this distinction is harder to maintain. The statement that His existence is necessary seems identical to the statement that He exists in every possible world (or in every world that can be conceived). Here there cannot be contingent existence in all those worlds, because if it is possible that He not exist, then by definition there will be at least one of those worlds in which God does not exist. In the first dialogue of the book The First Existent, I analyzed Anselm’s ontological argument for the existence of God. At the end of that dialogue I explained the meaning of chapter 4 of the Proslogion, where Anselm proves the necessary existence of God. At first glance this seems superfluous, since in the earlier chapters he had already proved His existence by a logical-conceptual proof (what Kant calls an “ontological argument”), and therefore seemingly we already know necessarily that He exists. What, then, is the added value of the last chapter if we already know from the previous chapters that God is a necessary being? I explained there that there is a difference between knowing necessarily that He exists (= epistemic necessity) and the claim that His existence is necessary (= ontic necessity). The statement “I necessarily know that God exists” is not identical to the statement “I know that God necessarily exists.” Hence even if Anselm reaches the conclusion in chapters 2-3 that the proposition “God exists” is necessarily true (because I have a logical proof for it), that is an epistemic conclusion. But that does not mean that His existence is necessary in the ontic sense, and that is what Anselm does in chapter 4. This can be formulated as follows: when I say that I necessarily know (or know with certainty) that something exists, that is a statement about me and about the certainty I have in that knowledge. When I say that the thing necessarily exists, that is a statement about it and its existence, irrespective of me and of my knowledge of it. Even if I did not know this, it would still necessarily exist. Facts about the world are not dependent on our knowledge of them. One might perhaps say that on the ontic plane it is not quite right to speak of necessity (of the proposition as such), but rather of certainty (of the person about the truth of the proposition). Let us take another example. The cogito argument proves necessarily that I exist (see column 363 on this). Does that mean that my existence is necessary? Consider whether it seems plausible to you that each of us is an ontically necessary entity like God. After all, I exist only in this world and not in others, and yet the cogito leads me to the logical (and therefore necessary) conclusion that in this world I exist. Therefore it is perhaps better to say that the cogito creates in me certainty that I exist, but it certainly does not mean that my existence is necessary in the ontic sense.

We can now examine which of the two kinds of necessity is described by the possible-worlds semantics. At first glance, it is epistemic necessity (that is, certainty), since every world I can conceive will satisfy what is certain in my eyes. By contrast, ontic necessity might perhaps be a function of the world we are discussing. It may be that in a certain world it is necessary that P hold, but that does not mean that it holds in all possible worlds. In other words, one can give a different semantics of possible worlds that would fit ontic necessity. P will be true in every world that contains the components from my world that make P necessary. The quantification here is over a subset of all worlds. For example, if I claim that God exists necessarily in a complex world like ours, then this necessity will be expressed only in sufficiently complex worlds. In a simple world perhaps God would not exist, or at least His existence there would not be necessary. In another, broader formulation, one can speak of all worlds in two different ways: if I speak of something that exists in every world I can imagine—this represents epistemic necessity; and if I speak of something that exists in every possible world—this represents ontic necessity. In column 580 I discussed the difference between these two kinds of necessity, and showed how Plantinga’s modal-ontological argument fails because it does not distinguish between them. Among other things, I showed there that he speaks of necessity in a single world and mixes it up with necessity in the sense of existence in all possible worlds. But the first necessity is ontic and the second is epistemic. The first means that the proposition is necessarily true in our world without reference to what happens in the other worlds, and the second means that the proposition is true in every possible world. That concludes the introduction to modal logic and its meanings. We now turn to the Barcan formula.

The Barcan Formula

Not all elements in modal logic are unequivocal. Some of them can be defined in different ways. This matters because certain definitions assume certain assumptions, and therefore different semantics may underlie a modal formalization. One example is the Barcan formula, which proposes the following relations: ∀x □P(x) → □∀x P(x). In our words: if for every x it is necessary that P holds, then it is necessary that for every x, P holds. Notice that x here is some object; that is, the quantification (the use of quantifiers) is over objects and not over worlds (as above). In light of the relations we saw above, we can see that the meaning of this relation in terms of possibility and existence is: ◊ ∃x P(x) → ∃x ◊P(x). In our words: if it is possible that there exists an x

for which P holds

then there exists an x for which it is possible that P holds. These two are formulas logically equivalent to one another. In the same Wikipedia entry you can find the converse Barcan formulas, which are the two converse implications (obtained by swapping the antecedent and the consequent in the formula). If one adopts those as well (this requires an additional assumption), then in the formulas above one can replace implication with equivalence (an equivalence is a two-way implication. To say that a proposition P

is equivalent to a proposition Q means that P implies Q and Q

implies P.)

It is important to understand that the Barcan formula is an assumption, not a result of modal logic. First, because you cannot prove it on the basis of the relations above, nor on the basis of the possible-worlds semantics. It follows that this is an independent assumption, one that may be accepted or rejected in our modal logic. Second, even at the conceptual level it does not necessarily follow from the meanings of the operators and quantifiers we have encountered. I will try to illustrate this through a certain model of the formula. Think of x as ranging over the class of all entities with mass. And suppose the property P is occupying space. In that case, the left-hand side of the formula says that for every massive object we choose, it is necessary that it occupy space; that is, in every possible world it occupies space. What will happen in a world in which there are no massive objects at all? No problem. What will happen in a world in which there are massive objects but no space? It is not clear. Can there be massive objects there? (After all, they necessarily have to occupy space.) Alternatively, perhaps they occupy space only if there is such a thing there. But in a world without space it is not true that they need to occupy space. Let us take a more specific example. In our world there are stones. Therefore, one of the x’s I can choose is some stone. What is the meaning of the claim that this stone necessarily occupies space? There may be other worlds in which that stone does not exist, and perhaps there are no stones there at all. Again, one can argue that we must examine this only in worlds in which there are stones, or in which there is this stone. But it seems that this necessity cannot be represented by the possible-worlds semantics unless we restrict it by additional conditions. Similarly, one can argue that in our world the principle of causality necessarily holds. Does that mean that the principle of causality is true in every possible world? Not really. It may be that the existence of the principle of causality depends on certain characteristics unique to our world, and in worlds where they do not exist the principle of causality will not be true (it is true in every world in which those characteristics exist, not in every possible world).

We now turn to the right-hand side of the formula. It speaks about the totality of possible worlds and claims that every stone we choose in every such world occupies space. The choice of the stone is made in each and every world, whereas on the left-hand side the choice is made only in our world (where, as noted, there are stones), and the statement speaks about the totality of worlds. One can propose an interpretation of the difference between the two sides and say that the left-hand side of the formula speaks about ontic necessity (which exists in our world, and not necessarily in every other possible world), and the claim is that every stone we choose necessarily occupies space. As noted, this is ontic necessity, and therefore it does not refer to other worlds. By contrast, the right-hand side of the formula speaks about epistemic necessity, and therefore it speaks about all worlds. The claim is that in every possible world a stone occupies space. Whether or not this is the correct interpretation, you can already see that this identity is far from being obviously and always true. These problems and the like show that the Barcan formula is not a simple result of modal logic

1 In my debate with Aviv Franco I pointed out that according to Hume the principle of causality is not a result of observation and is therefore true a priori. Notice that in light of what we have seen here, this does not necessarily mean that it is true in every world that can be conceived. One can of course argue that it is reasonable to think it also holds in other worlds, but it is important to understand that this is not necessary

If one wishes to adopt it, especially according to the possible-worlds semantics, one must add in the background some restrictions or assume additional philosophical assumptions. We can now move to the problem that exists in the Barcan formulas if one takes the possible-worlds semantics into account, but we already understand that the formula assumes additional assumptions, and if a problem arises then they apparently do not hold (and therefore the formula is not correct).

The problem: preliminary definitions. To describe the problem, I will follow this video. It presents two philosophical approaches side by side:

(Necessitism

This is an approach that says that every object that exists here exists necessarily. It might have had other properties, but it must exist. Of course, one may already ask here what would define it as the same object if its properties were different. Suppose it is a coin: in another world it must exist, but it could be a tree. The only thing one can say about this object in every world is that it is an object that can be a coin (or a tree).

(Contingentism

This is the opposite approach, which claims that not everything is necessary; that is, there is at least one object whose existence is contingent. He then explains that every object in every world is either concrete (actualized) or possible. For example, the coin in my world that would be a tree in another world: in my world it is a concrete coin, but it is also a possible tree. He also treats being identical with itself as a property that it always instantiates concretely, which in my opinion is incorrect (because identity and existence are not properties, but statements that refer to the thing as such. See below).

He now explains that necessitism, which at first glance seems puzzling and contrary to intuition, is not really so. His claim is that even necessitists agree that the coin could have failed to be a coin—which is what our intuition says—but they add that if it had not been a coin it would have been something else, a possible coin. About that we have no intuition, and so that can be accepted. I did not understand this distinction at all. If it had been something else, then it would not have been it. This is just wordplay. My intuition says that it could simply not have existed, and therefore the claim that it would have existed in another form is also counterintuitive. The two claims are of course related, but they are not identical. As he himself explains, necessitism assumes the existence of infinitely many objects, for every object that can exist does in fact exist, only perhaps in another form. Even about that I am not sure, because my coin is also a possible tree and a possible star, and that is the same object itself. There may be one object in the world that can be infinitely many different objects in infinitely many different worlds. Admittedly, if there is a world in which infinitely many objects can exist, then infinitely many objects must exist in my world. But up to this point this is conceptual hairsplitting that is not very convincing, together with strange theses, even if they are possible. But now he proceeds to prove necessitism from the Barcan formula. If he succeeds in this, then we have a real problem: a very implausible ontological thesis can be proven by means of modal logic

The problem: the proof from the formula. If we look at the second version of the formula above, we see that it says that if an object satisfying the property P is possible, then there exists an object for which it is possible to have the property P. We now understand that this moves us from the possibility of a certain object to the existence of some object (not necessarily the same object). The contingentist will reject this formula, because it says that every object that is possible actually exists. But the contingentist argues that there are objects that are possible yet do not actually exist. The formula shows us that the number of possible objects and the number of existing ones is identical. I do not agree with this claim. As I noted above, two different possible objects can correspond to one existing object. For example, on the left-hand side let us suppose that an object satisfying the property P is possible, and from this it follows that in fact there exists an object that can have P. Now take another formula that speaks about the property Q, and the possibility of an object satisfying Q implies that there exists an object that can satisfy Q. But I see no reason why this should not be the same object. It can satisfy both P and Q. This is of course true also for infinitely many such possible entities,

2 The source of his argument is this article by Williamson.

all of which are realized in the very same object. Therefore it does not follow from this formula that the number of possible objects is identical to the number of existing objects. There is another mistake in his formalization. He is essentially taking the property P to be “an object that does not actually exist.” He now argues that the contingentist assumes ◊ ∃x P(x), that is, that there is an object such that it is possible that it does not actually exist. But according to the Barcan formula we get ∃x ◊P(x), that is, there exists an x such that it is possible that it does not actually exist. But if it does not actually exist, in what sense does it “exist”?! This is plainly an empty semantic trick. You assign to the property P the meaning “does not actually exist” and arrive at a contradiction. That only means that this substitution is problematic. And indeed, a closer look immediately reveals the problem: P is not a property. As I noted above, existence or nonexistence is not a property but a statement about the thing itself (about the object), not about its characteristics. I explained this in detail in the discussion of the ontological proof in the first dialogue of my book The First Existent (and briefly in a responsum here). Kant already pointed out that if this were a property, then one could prove the existence of a perfect being (God) from its definition, and that is not plausible (in his view, impossible).

A general look. At the end of the lecture he moves on to point out another problematic issue. It seems that formal logical manipulations succeed in proving a metaphysical claim. That itself is problematic, since logic is not supposed to contain content. It is a relation between propositions that assumes nothing about the propositions themselves. He argues that from here we see the opposite: logic can teach us about the world, and the distinction between logic and metaphysics does not exist. But this is of course mistaken, because we have seen that he is indeed making assumptions, even if without being aware of them. One can say that this is in effect an ontological proof, and the problem it presents is very similar to the general problem in ontological arguments, in Kant’s formulation: one cannot prove synthetic claims about the world on the basis of logical manipulations alone (without assumptions). And if he has succeeded in doing so here, then he has simply smuggled hidden assumptions into the field without noticing. Here he falls into the same error into which Anselm fell, and into which many others fell when they tried to present ontological arguments (such as Descartes in the cogito and Anselm in his ontological argument). But as I explained in my aforementioned book, Kant’s critique begs the question. He assumes that it is impossible to prove propositions about the world (synthetic propositions) on the basis of logic and definitions alone. That is the emptiness of the analytic. But so long as you have not pointed out the flaw in the argument, this is an empty declaration. On the face of it, every such argument proves that this assumption itself is false, for it does succeed in proving a synthetic claim on the basis of definitions and logic alone. Only if we identify the fallacy in the argument or expose the hidden assumptions it makes can we return and conclude that indeed there is no way to prove synthetic claims on the basis of definitions and logic alone. That is what I did in the first dialogue with regard to Anselm’s argument, and in column 363 with regard to the cogito argument. Here I have done so with regard to Williamson’s argument. I found in his argument at least three flaws: 1. His argument assumes the Barcan formula, and as I showed, it itself is not pure logic; it tacitly assumes metaphysical assumptions. 2. Beyond that, he conflates ontic and epistemic necessity. 3. Finally, he repeatedly assumes that the existence or nonexistence of an object are its properties, and they are not.