Modal Logic and Ontological Arguments (Column 580)

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With God’s help

Modal Logic and Ontological Arguments

Several months ago Ariel posted a question to the site, in which he presented a formulation by the well-known American Christian analytic philosopher Alvin Plantinga,^[1] of the ontological argument (see, for example, here and here). This formulation is based on modal logic, and this is an opportunity to become acquainted with it (see also column 301, and somewhat in column 561, and in greater detail in the third book of the Talmudic Logic series). After a short lesson in logic, in which we will become acquainted with it (this will require a bit of formalism), we will return to his argument. Through it we will discuss ontological arguments in general, and in particular learn how they are examined and attacked.

Modal Logic: Notation and Basic Relations

Modal logic deals with the concepts of possibility and necessity. The point of departure is that there is a difference between saying that a proposition P is true and saying that it is necessarily true. For example, the proposition that the sun is currently shining is true, but not necessary, since things could have been otherwise. Such a truth is called a contingent truth (incidental, not necessary; or dependent on something, as opposed to a truth that depends on nothing). By contrast, the proposition that either the sun is shining or it is not is necessary. There can be no different state of affairs in which this does not hold. The proposition that there is a force of gravity in the world that pulls objects toward the earth is also contingent. In our world this is always so (apparently independently of anything), but there could have been a world in which the law was different. Similarly, there is a difference between saying that proposition P is false and saying that it is impossible (for example, if it contains a logical contradiction: the sun is both shining and not shining at the same time). Between saying that P is necessary and saying that P is impossible, there is also the claim that P is possible.

In modal logic several modal operators are defined, and in particular two basic ones: necessary and possible. Their customary notation is:

□P – it is necessary that P ; ◊P – it is possible that P.

If you think about it, you will see that there are relations among these operators. Thus, when I say that P is not necessary (= it is not true that P is necessary), this means that not-P is possible. In logical notation (the sign ~ denotes negation): ~□P = ◊~P, and ~◊P = □~P. Note that saying that P is necessarily not true (□~P) is not equivalent to saying that there is no necessity that it is true, or that it is not ‘necessarily true’ (~□P).

Likewise, there is a difference between the statement ‘it is necessary that P follows from Q’ (□(Q→P)) and the statement ‘if Q is true, then P is necessarily true’ (Q→□P). This distinction underlies the solution Judith Ronen proposes to the problem of foreknowledge and free choice (see column 301, and my criticism there of her solution).

By the way, the claim that P follows necessarily from Q is ostensibly a third kind of claim, and its formalization would have to replace the implication arrow with necessary implication. One might symbolize it as follows: P<==Q. What is necessary here is the implication, not P and not Q. The meaning of such an implication (if it has any distinctive meaning at all, since in a simple sense it is like the first statement in the distinction above) is a complex philosophical matter, and I will need it later.

Similarly, when I say that P is not possible, this means that it is necessary that it is not true: ~◊P = □~P.^[2]

Another relation between these operators is the following implication: if P is necessary, then it is certainly possible. This is denoted as follows: □P → ◊P.

As an exercise, try to think about the meaning of the claim that P is not necessarily untrue, or that not-P is possible. This too can be expressed in terms of either of these two operators.

Possible Worlds: A Modal Interpretation

The accepted interpretation of these operators is given in terms of possible worlds. The framework of the discussion is the assumption that there are countless imaginary worlds, each with a different reality. Every possibility that arises in our imagination is a world unto itself, and every such world is of course possible in principle. Our world is only one of them that happened to be actualized, but any of the others could also have been actualized.

In this terminology, to say that proposition P is necessary means that it is true in every possible world. To say that it is not possible means that it is true in none of the possible worlds (there is no possible world in which it is true). To say that it is possible means that there is at least one world in which it is true (one can also speak of the probability of a proposition in terms of the number of possible worlds in which it is true). Notice that this interpretation allows us to get rid of the operators ‘necessary’ and ‘possible,’ which it is not always clear how best to handle. Instead, we can formulate all our claims in terms of simple propositions for which there are only truth and falsity (as in classical logic), without necessity and possibility, but across different worlds. The multiplicity of worlds is the price we pay for translating modal concepts into a logic of truth and falsity.

One of the important uses of this interpretation is to derive and understand all the logical relations we saw above. To understand them, we had to think about the meanings of necessity and possibility and the relation between them, but in the possible-worlds interpretation these claims basically become ordinary logical claims (examined in terms of truth or falsity), and the relations among them are relations of inclusion or identity (they can be represented by Venn diagrams over worlds). For example, the last implication above (that the necessity of proposition P entails its possibility) now becomes completely trivial: if P is true in every possible world, then certainly there is at least one world in which it is true. This is an ordinary deduction from the general to the particular (from the truth of a general proposition to the truth of a particular case included within it: as in, if all human beings are mortal, then Socrates is mortal). So too, the relation between the claim ‘it is not necessary that P holds’ and the claim ‘it is possible that P is not true’ emerges quite simply. The utility becomes even greater with more complex claims and relations. In the third book of the Talmudic Logic series, we use this to understand the relations among the various kinds of commandments in Jewish law (positive commandment, prohibition, a prohibition inferred from a positive commandment, a prohibition reparable by a positive commandment, existential and obligatory positive commandments, a prohibition that involves no action, and so on).

Before I move on to Plantinga’s argument, one surprising remark. But before that, I need to present quantifiers in predicate logic.

Quantifiers in Predicate Logic

At first glance, the relation between the modal operators is very similar to the relation between quantifiers in predicate logic. In predicate logic, one deals with a proposition that assigns a property P to some subject (= individual; not necessarily a subject in the grammatical sense) x, and denotes the sentence as follows: Px. That is, the subject x has the property P. For example, Moses (=x) is kind-hearted (=P). In addition, two quantifiers are defined there:

∀x – for every x ; ∃x – there exists an x

We can now write the sentence ‘for every x, property P holds’ as follows: (∀x)(Px). In the example above: ‘all individuals are kind-hearted.’ The sentence (∃x)(Px) means ‘there exists an x that has property P,’ and in the example above: ‘there exists a kind-hearted individual.’

There are relations between these two quantifiers (this is essentially Boethius’ square of opposition, which also appears in Maimonides’ Words of Logic), and they look very similar—indeed identical—to the relations between the modal operators. For example, to say that it is not true that property P holds for every x is like saying that there exists an x for which this property does not hold (there is an individual who is not kind-hearted): ~(∀x)(Px) = (∃x)(~Px).

Similarly, the negation of ‘there exists a kind-hearted individual’ is ‘there is no kind-hearted individual’ (or: ‘all individuals are not kind-hearted’): ~(∃x)(Px) = (∀x)(~Px). This is exactly identical to the relations between the modal operators.

This similarity should not surprise us, because according to the possible-worlds interpretation, to say that some proposition is necessary is like saying that it is true in every possible world. Here, then, is the connection between a modal operator (necessity) and a quantifier (universal) in logic. To say that some proposition is not necessary is like saying that there exists a world in which it is not true. Here is another connection between a modal operator (possibility) and a quantifier (existential). But I will now show you that this similarity is not complete.

A Note on the Relation Between Quantifiers and Logical Operators

There is a philosophical dispute among logicians about the relation between a universal statement and an existential statement. It is customary to think that if I say that every X is Y, then it necessarily follows that there exists an X that is Y. Aristotle indeed held this view. But the logician George Boole disagreed, arguing that this need not be correct. Consider, for example, the proposition ‘All aliens have wings.’ Can one infer from this that ‘There exists a winged alien’? In fact, there are no aliens at all. And yet, George Boole claims, one can still say that the universal proposition is true (if there are aliens, then they necessarily have wings; an alien without wings is impossible), but this does not mean that aliens exist, and certainly not that winged aliens exist. The universal proposition is hypothetical, contrary to fact (if there were aliens, they would have wings), and it can be ‘vacuously true’ (this is the mathematical term for this kind of hypothetical truth).

At first glance, we would expect that according to George Boole, a similar relation would hold between modal operators as well; that is, if a proposition is necessary, this would not necessarily mean that it is possible. According to this, the implication I defined above—□P → ◊P—would apparently not be correct on his view. But if you think again, you will see that this is not so, and in the context of modal logic George Boole would probably accept Aristotle’s claim. This implication is correct according to everyone.

To see this, think within the possible-worlds interpretation. To say that every alien has wings means that in every possible world in which there are aliens (and there are imaginary worlds of that kind, even if not our world), they have wings. It therefore follows clearly that there are possible (imaginary) worlds in which there are winged aliens. Here the implication is valid, and there is no room for Boole’s claim about vacuous truth. There is no option in which aliens do not exist in all possible worlds, for by virtue of their being possible there are (imaginary) worlds in which they exist. In modal logic the quantification is over all possibilities, including imaginary ones, and therefore there is no room there to speak about objects that do not exist (like aliens in our world). This means that, contrary to the initial intuition suggested by the similarity to quantifiers, modal logic cannot be translated into ordinary predicate logic (using a universal quantifier to describe necessity and an existential quantifier to describe possibility).

We have learned that the similarity in the relations between modal operators and quantifiers is not complete. The lesson is that it is important to pay attention and be careful about the conclusions we draw from external similarity. Logical notation can confuse us, but it can also help us in this.

We now arrive at Plantinga.

A General Look at Ontological Arguments

‘Ontological argument’ is a term coined by Kant to describe an argument that assumes no premises. It is an argument based on definitions and conceptual analysis, whose conclusion is a claim about the world. Anselm proposed an ontological argument for the existence of God (I discussed it at length in the first booklet and the first conversation of my book The First Existent). Descartes proposed an ontological argument for our own existence (see column 363).

Mainly בעקבות Kant, it is customary to think that ontological arguments are impossible, since it cannot be that from definitions alone one can derive any claim about the world. Definitions are arbitrary, and therefore they cannot say anything about the world. But that is only a declaration. When one examines an ontological argument, it is not enough merely to declare that it is impossible; one must put one’s finger on the point at which it fails.

Since I too belong to the camp of those who think ontological arguments are impossible, I have already developed a toolbox that helps prove this. As a rule, it always turns out either that the argument in question is invalid, or that it does contain some assumption without our noticing it. In every ontological argument I have examined, I found that it either contains some assumption or is invalid, although sometimes nontrivial analysis is required to expose this. Therefore, there really are no ontological arguments.

The interesting question is whether the problem lies in the validity of the argument or in the existence of a hidden premise. This matters because if the problem is validity, then the argument is worthless. But if the problem is that it contains a premise, that in itself does not disqualify the argument. Every argument has premises. In that case, we must form a view regarding the premise that underlies it. If we accept it, then we must also accept the conclusion, and that is the argument’s value. It may not be ontological, but it is certainly useful and instructive. If the premise is false, then the argument is worthless—but not because it is ontological; rather because it is based on a mistaken premise (in our view).

The Modal Ontological Argument

Alvin Plantinga offers an argument that proves the existence of God through modal logic (see the sources linked above). This is an ontological argument, and he presents it as an upgrade of Anselm’s argument. The argument goes roughly as follows:

1. Definition: God is the perfect and necessary being.
2. Premise: It is possible that God exists.
3. Conclusion: There is a possible world in which God exists (from the definition of modal logic under the possible-worlds interpretation).
4. Conclusion: God exists in every possible world, and in particular in our world (from his definition as necessary it follows that he exists in every possible world).
5. Additional conclusion: His existence is necessary (because whatever exists in every possible world is necessary).

Line 1 is a definition. Definitions are arbitrary, and so long as they contain no contradiction there is nothing to prevent us from defining any concept we wish. In line 2 there is a premise, but notice that this is not a premise about the world; it is about the concept of God (that it is free of contradiction, and therefore possible). Moreover, this even seems like a reasonable assumption, and not merely a possible one. It is not plausible that the atheist who denies God’s existence must also assume that his existence is self-contradictory. In the terms of the previous column, this is an unnecessary ‘widening of the front’ from his point of view (this is in fact exactly what Ariel argued, rightly, in his second message in the original thread). This is very similar to the situation in Anselm’s original formulation, where he too assumed that the concept of God is free of contradiction, and yet it still falls under the heading of an ontological argument.

If the atheist claims that the concept of God contains an internal contradiction, he is making a very strong claim. Most atheists do not claim this, but only the contingent claim that God does not exist. If this argument were to succeed in backing atheists into a corner such that all of them had to claim that the concept of God is contradictory, that itself would be a significant philosophical achievement for Plantinga. For if that is indeed their claim, then it would seem that the burden of proof is on them and not on the believer (as people are accustomed to say: whoever claims that something exists is the one who must prove it). If you, the atheist, claim that a concept that appears reasonable on its face is actually contradictory, and you say this without pointing to the contradiction itself, then the burden of proof is on you. Such a claim is an artificial and forced ad hoc move, and your position is weaker. After all, someone who is losing a debate can always deny a very reasonable premise that underlay the victorious side’s arguments, and in this way ostensibly escape an embarrassing defeat. That is not serious.

So for the moment, it does indeed seem that we have here an ontological argument (without premises about the world). If I wish to maintain my view that there is no argument that is both ontological and valid, I must show that this argument is flawed—that is, invalid. If I manage to show this, it will mean that I have found in this argument a flaw of the second kind; we saw that such a flaw shows the argument to be worthless as well. Can this be shown with respect to Plantinga’s argument? I think very much so.

A Methodological Insight: The Meaning of a Counterexample

I began my discussion with Ariel by giving a counterexample. In this way one can prove the existence of anything whatsoever, such as the necessary winged fairy, or the perfect island. My words were:

By the same token, you can prove the existence of ‘necessarily salty sugar.’ There may be a world in which there is sugar that is necessarily salty. From this it follows that in all worlds there exists salty sugar (whose saltiness is not necessary in every such world). QED.

I will mention that with respect to Anselm’s formulation as well, the first attack by the monk Gaunilo was exactly of this sort. He wanted to prove the existence of ‘the existing island’ or ‘the perfect island.’ But as I explain in the book and the booklet, counter-demonstrations by themselves are not a sufficient argument. So long as we have not pointed to the flaw in the argument itself, this is merely a reductio ad absurdum, but the conclusion would then be that the perfect island and the salty sugar, etc., have indeed also been proved to exist. To attack an ontological argument, one must point directly to the flaw in the argument. A counterexample is at most a hint that there must be a flaw (otherwise one could prove in this way masses of implausible things). But by itself it is not a refuting argument.

Similarly, I explained that Kant’s declaration attacking Anselm by claiming that he proved the existence of God (a factual claim about the world) on the basis of definitions alone is not, in itself, an attack. True, that is exactly what he did. So long as you have not pointed out where the flaw lies in his argument, you cannot reject an argument merely because its conclusion seems implausible to you. In an ordinary argument, you can say that if the conclusion is implausible, that means that probably one of the premises is problematic. But in an ontological argument, which is a premise-free argument, there is no such option. Therefore, there you must either show that it is invalid or bite the bullet and accept its conclusion.

The Counterargument

In what I wrote there, I went on to point to the place where, in my opinion, the flaw lies in the argument itself. The three formulations I proposed complement one another. In formulation 1, I point to a confusion between two meanings of ‘necessary’ that exists in Plantinga’s formulation. Modal logic, like logic in general, deals with the truth of propositions and not with facts. Plantinga’s argument proves that the proposition ‘God exists’ is necessarily true, but that is not the same as saying that his existence is necessary. This is a subtle distinction, so I will elaborate a bit.

Usually, when we speak of God as the necessity of existence, we do not mean that we necessarily know of his existence. Our knowledge depends on us and on our capacities, but is not connected to him. God’s necessary existence is a claim in metaphysics, which says that he has a different kind of existence. I exist in an incidental way, not a necessary one (contingently, dependent on circumstances), whereas he exists necessarily. Metaphysics compels his existence, and it is impossible that he not exist. There is something within him that compels his existence (which is why this is sometimes described by the unfortunate expression ‘the cause of himself’; see column 435 and my article here). This is metaphysical necessity, not logical necessity. The claim here is a claim about the world, not a claim about the truth of propositions.

By contrast, the logical determination according to which the proposition ‘God exists’ is necessarily true is a claim in logic and not in metaphysics. It is a statement about the truth of a proposition and not about some fact. One may say that it concerns us and our knowledge, and not the world as such.^[3] The same problem exists in the argument of ‘logical determinism,’ which also confuses the necessity of my knowledge about the future (a logical necessity that deals with propositions) with the necessity of the future’s actual realization (a metaphysical necessity that deals with facts). See columns 301 and 459.

This is probably the meaning of the philosophical distinction I made above. There I distinguished between the claim □(Q→P), whose meaning is that in every possible world it is necessary that Q implies P—that is, that the sentence ‘Q→P’ is true in every possible world—and the claim that P follows necessarily from Q, whose formalization replaces the implication arrow with necessary implication: P<==Q. What is necessary here is the implication or derivation of the fact P from the fact Q. One may say that necessary implication expresses intrinsic-ontic-metaphysical necessity, that is, a claim about the world, whereas ordinary implication expresses the derivation of the truth of one proposition from another (the necessity operator merely indicates that this is true in every possible world). The first claim (the logical-modal one) speaks of logical necessity regarding the truth of the proposition ‘God exists,’ whereas the metaphysical claim speaks of the fact that his existence belongs to the type of necessary existence and not ordinary existence (here too people use the term ‘contingent,’ but in a metaphysical rather than a logical sense).

Let us now return to Plantinga’s argument. Notice that it mixes these two meanings. His definition of God as a necessary being concerns God himself and the necessity of his existence; that is, it is an assumption on the metaphysical plane. But modal logic speaks about the necessity of the truth of a proposition—that is, that it is necessary that the proposition ‘God exists’ is true. This is logical necessity, not metaphysical necessity. That kind of necessity can be translated, according to the interpretation we saw, into the claim that the proposition is true in every possible world. But metaphysical necessity cannot be translated in that way, because that necessity is a property of our world and not necessarily of every imaginary world we may conceive. In the same way, one can say that the proposition that bodies with mass are attracted to the earth is necessary in the metaphysical sense (in our world, by its nature, there is no escaping it), but there is no necessity that it be true in every imaginary world one can think of.

In my opinion, Plantinga’s argument mixes these two meanings of necessity, and therefore tries to ride two horses at once. He assumes that in one particular possible world there exists an X necessarily. But the necessity here is metaphysical (otherwise it would exist not only in that world but in every possible world), and then suddenly he jumps from this to the conclusion that because of its necessity it exists in every possible world. Once you are speaking within the modal interpretation, you can no longer speak about existence by necessity in the customary sense (the intrinsic-metaphysical sense).

If we translate everything into the modal interpretation, what you are really saying is that in one possible world there exists something that exists in every possible world. Then you have simply assumed directly that it exists in all worlds, and in effect you have assumed the conclusion. Where is the argument here? Now you can understand why the argument is invalid, or alternatively why the assumption that the concept contains no contradiction is an unreasonable assumption that can prove anything you want. This explains the difficulty that makes it possible to prove everything that appeared in the counterexamples.

In another formulation, one can say this as follows. This argument does not prove that God exists; rather, by reductio it proves that assumption 1—that his concept contains no contradiction—is not a correct assumption. The argument shows that this assumption leads to an absurd result, and that itself is a reductio proof that this assumption is not correct. This is so even though on its face it did not seem to be so (as I noted at the beginning of the analysis).

In yet another formulation, this can be put as follows (formulation 3 in my reply to Ariel). This argument is really saying that there necessarily exists a possible world in which a necessary being exists. But you inserted necessity here, so it is no wonder that you obtain necessity. When you speak of something possible, you cannot say that there necessarily exists a possible world in which it exists, even though apparently this is precisely the modal interpretation of possibility. Again, in my opinion, the reason is that this argument uses the term ‘necessity’ in two different senses. And of course, if it is not necessary that such a possible world exists, the proof collapses on its own.

1.

Footnotes

1. I wanted to give a link here to the Hebrew __Wikipedia__, but to my surprise there is no entry on him there. This seems to indicate some sort of bias in that encyclopedia.
2. These operators behave very similarly to quantifiers in predicate logic (and to the relations between them and their negations). Below I will comment on the difference between them.
3. Perhaps one can derive the logical conclusion (that the proposition ‘God exists’ is necessarily true) from the metaphysical claim (that ‘God is a being whose existence is necessary’), but that is a rather involved philosophical discussion, and this is not the place for it.