The Ontological Argument and the S5 Objection

The Ontological Argument for the Triune God is the most popular video I have released so far. Generally the feedback has been positive. And even many skeptics have admitted that they cannot find a flaw, but simply choose to be unconvinced. And that is their right. You can get out of the conclusion of any argument in philosophy if you just bite the bullet and deny one of the premises. The job of the proponent of the argument is to make that intellectual price as high as possible. What do you have to give up in order to rationally deny the conclusion?

One skeptic has decided to avoid the conclusion of this argument by attacking the entire system of modal logic (called the S5 system) upon which this argument is based. This is so incredible that at first I thought it was a joke. S5 is well-accepted as a fully coherent logical system by logicians, and an accurate representation of certain kinds of modality in the philosophy journals. I mentioned to potential detractors that it is just about impossible to avoid the conclusion of this argument unless you are willing to deny the laws of logic themselves. This guy apparently took me up on the offer!

So let's go through a recap of modal logic. This is a formal logic which adds two additional truth qualifiers. Something can be true (T), something can be possibly true (◊T), and something can be necessarily true (□T). The diamond denotes possibility and the box denotes necessity. Modal logics use a semantic called "possible worlds." A possible world is a description of the way reality might be, kind of like a hypothetical situation. The actual world is a possible world where the entire description is true. In other words, it is a description of the way reality is.

There are different systems of modal logic, where the axioms are different. There are also different modal logics, where the application is different. The term possible and the term necessary have different meanings in different modal logics. So the first question is, what modal logic are we using for the Ontological Argument, and which of the available systems should we use?

Modal logics include deontic logic (logic of obligation), temporal logic (logic of time), epistemic logic (logic of knowing things) and alethic logic (logic of truth). Since the argument is an ontological or metaphysical argument, we are going to use an alethic logic, and deal with metaphysical modalities. Alethic modalities and epistemic modalities are often expressed in English using the same words. "It is possible that bigfoot exists" can mean either "Bigfoot could exist (it is metaphysically possible), whether or not bigfoot does in fact exist" (alethic), or more likely, "For all I know, bigfoot exists" (epistemic).

In alethic modal logic something "necessary" is true in all possible worlds, something "possible" is true in at least one possible world. To say a statement (we'll call this statement X) is possibly necessary (◊□X) is to say that in at least one possible world it is true that X is true in all possible worlds. This is the same as saying that X is true in all possible worlds. And this means that saying possibly necessarily X is equivalent to saying necessarily X (◊□X → □X). This means that when we pick a system of modal logic, it must account for this. It must be able to derive this formula in order to accurately represent metaphysical possibility and necessity.

To think of it another way: when we are dealing with metaphysical possibility and necessity, we are dealing with ultimate (absolute) possibility and necessity. By ultimate, I mean possibilty and necessity that does not vary over different possible worlds, for there has to be at least one truth that holds for absolutely all possible worlds without exception (if it didn't, then the statement "there is no truth that holds for all possible worlds" would hold for all possible worlds). So we know that there is at least one of these truths that has this ultimate necessity. Hence, the proposition "necessarily T" (□T) and the proposition "necessarily necessarily T" (□□T)have to mean the same thing (□T is equivalent to □□T). If it didn't, then the term "necessarily" isn't talking about ultimate necessity.

Same goes for possibility. If we say that something is metaphysically possible, we are talking about ultimate possibility, and therefore "possibly possibly T" is equivalent to "possibly T" (◊◊T is equivalent to ◊T). Otherwise, we are not talking about ultimate possibility. And not only is this true, but is must be necessarily true in order for us to be talking about ultimate possibility. If "possibly T" did not entail "necessarily possibly T" in all worlds (◊T → □◊T), then we would not be talking about ultimate possibility. In short, only systems of modal logic that have what is called a universal accessibility relation can represent ultimate metaphysical truths.

Another way of saying it is that in order for our system to reflect metaphysical truth, it must define necessity and possibility in the following way:
□P holds if and only if ∀w(P is true at w)
(Necessarily P holds if and only if for every world, P is true in that world)
◊P holds if and only if ∃w(P is true at w)
(Possibly P holds if and only if there is a world where P is true in that world)

S5 is the only system that can do this. Now S5 is not appropriate for all modal logics. It would not represent temporal modal logic, for example. But regarding the ultimate possibility and ultimate necessity needed to address metaphysical statements, only S5 will do. It is the only system that can address ultimate modality.

It states that under S5, you can stack modal operators (diamonds and boxes) as much as you want and they collapse into the last operator.
◊◊□◊□◊□◊□◊◊□□□□□□◊P → ◊P
But it is essential that this does happen if we are really talking about the fundamental laws of metaphysics.

To quote Alexander Pruss:
Broad logical possibility cannot have been different, since it is a matter of what propositions follow from what propositions, and what follows from what could not have been different. . .
. . .the collection C0 of all the fundamental laws of metaphysics could not have been different from what it is – that is central to its being the collection of the fundamental laws of metaphysics – and the ‘could not’ here surely is metaphysical. Moreover, what C0 allows cannot have been different. If it were different, that would presumably be because a collection of laws might permit different things in different circumstances. Suppose that C is some collection of fundamental laws that permits different things in different circumstances. But then there would need to be further metaphysical laws as to what the laws in C collectively permit under what circumstances, and barring a vicious regress of more and more basic laws, there would have to be fundamental laws specifying what laws in C permit. And these laws couldn’t be in C, since then the laws in C would not permit different things in different circumstances. Therefore, if C permits different things in different circumstances, then C does not contain all the fundamental laws, in the way that C0 does. Thus, what C0 permits could not be different, and hence modality could not have been different, and this is what S5 says.
-Actuality, Possibility, and Worlds: (pp. 16-17)

There has to be at least one fundamental truth that does not vary over possible worlds. Because if there are none of these truths, then that truth is the fundamental truth that does not vary over possible worlds.

This backwards E (∃) represents the existential quantifier. Whatever comes after this symbol is said to exist. When the modal operator is placed before the quantifier, we are talking about existence de dicto. So with the diamond before the quantifier (◊∃xAx), we can say it is possible that an alien exists. When the modal operator is placed after the quantifier, we are talking about existence de re. An example would be "A possible alien exists." (∃(x)◊Ax) De dicto modality states that the proposition is necessarily true or possibly true. De re modality states that a certain state of affiars possibly holds or necesarily holds. De re is about the thing itself.

Quine objected to de re modality, because he believed it that it could not be expressed without resulting in absurdities. Quine believed in what's called the substitutivity of identity, meaning that given a statement of identity (x = y), the two terms (x and y) are interchangeable. Susan Haack cites his objection in her book: The Philosophy of Logics. Quine objects to de re modality with the following argument which I will call Quine's Argument:

1. Necessarily, 50 is greater than 7
2. 50 = the number of states in the US
3. Necessarily, the number of states in the US is greater than 7

Quine states that 1 and 2 are true, while 3 is false. Yet if 50 is equal to the number of states, then we should be able to substitute one term for the other with no change in the resulting statement's truth value.

Quine says that the only way to get these equal terms to substitute for one another is to accept a premise which I will call Quine's Solution.

Quine's Solution: necessarily, terms are identical if and only if all conditions specifying them are equivalent:
(∀y (Fy ≡ y = x) & ∀y (Gy ≡ y = x)) → □∀y (Fy ≡ Gy)

I would like to point out here that there are no independent reasons for accepting Quine's Soltuion as a valid principle. It is a formula contrived specifically to address Quine's Argument. It would be an ad hoc solution even if it worked without any negative repercussions. The problem is, even if we were to accept Quine's solution, it would lead to modal collapse.

Let p be any true sentence, and let F be ' p & y = x' and G be 'y = x'; then from Quine's Solution it follows that:
□∀y (p & y = x ≡ y = x), whence, in particular
□(p & x = x ≡ x = x) and so p → □p and all truths become necessary truths!

When people talk about de re modality leading to modal collapse, this is what they are talking about. The collapse only occurs if we accept Quine's Solution. So obviously we can't accept Quine's Solution!

The problem with Quine's Argument, as Saul Kripke pointed out, is that there is a difference between rigid and non-rigid designators. A rigid designator picks out the same thing in all worlds. "The square root of 16" picks out the number 4 in all possible worlds. "The number of states in the US" picks out different values in different possible worlds (if it is indeed a proposition about states, and not about a specific number; more about this later). When we are doing modal logic, two equal terms can only be substituted for one another only if either one is a rigid designator for the other, or if they are both rigid designators for a third term. Once we recognize the distinction, the problem disappears.

This is why Genoveva Marti concludes:
Quine's argument does not succeed in proving that modal distinctions collapse. A crucial step in Quine's reasoning relies on the assumption that whenever two expressions determine the same object, those two expressions should be intersubstitutable [exchange one for the other] salva veritate [with no change in truth value] even when they occur under the scope of a modal operator.
-Quine and Modal Distinctions, p. 289

*Also note that Plantinga in The Nature of Necessity writes that Quine's case is based on a flawed view of identity.
Quine believed that: "Two terms are the same (eadem) if one can be substituted for the other without altering the truth of any statement (salva veritate)."
Plantinga noted that principle does not hold for such excellent examples of language such as English. Instead, identity should be defined as: For any property P and any objects x and y, if x is identical with y, then x has P if and only if y has P.
x=y → ∀P(Px ↔ Py)

Once we correct Quine's definition of identity, Quine's entire objection crumbles.

What's even worse for Quine is that even under his own criteria, his argument commits a fallacy of ambiguity (technically, a fallacy of amphiboly). It is like the joke: "I once met a man with a wooden leg named Smith and asked him 'what is the name of the other leg?'"

It appears the statement "50 = the number of states" is a statement about a number while the statement "the number of states in the US is greater than 7" is about states. If we interpret the third statement so that it is about numbers, then the third statement is equivalent to "Necessarily, the number that numbers the states (as things in fact stand) is greater than 7." And this is a true statement, as it is a statement about numbers. If we instead leave the third statement as it is and reinterpret the second statement so that "50 = the number of states" is about states, then the second statement is equivalent to "however many states in the US there happen to be, that's what 50 is" and this is clearly false. Either way, Quine's objection fails.

Furthermore, Quine's most significant blunder in his argument is that he tries to draw metaphysical conclusions out of an argument that only addresses semantics. This is one reason that Quine's argument against de re modality has been almost universally rejected in the philosophy journals, even by philosophers who themselves reject de re modality.

For further reading on this subject, Alvin Plantinga also shot down the Quine's objection in chapters 2 and 3 of The Nature of Necessity.

For a very thorough and technical critique of Quine's argument, please see Genoveva Marti's article.

So there is no problem with de re modality. It is fully consistent with the S5 system, and does not lead to any sort of modal collapse. Objections surrounding it have been answered half a century ago, which is why Richard Swinburne could say confidently:

If a state of affairs is necessary, then the proposition which states that the state of affairs holds will be necessary; and conversely, if a proposition is necessary, then the state of affairs which it states to be the case is necessary. Necessity de re entails necessity de dicto, and conversely.
-Richard Swinburne, The Coherence of Theism, p. 235

It's sort of a backhanded compliment, that someone would go to such extraordinary lengths to avoid believing in the Triune God. It's not every day that someone decides to jettison an entire system of logic just to avoid the conclusion of one argument. I wonder what the New Atheists would say if we resorted to such extreme measures to avoid a conclusion of one of their arguments. I am sure it would not be nice.


Bonus Section
A simpler argument for divine necessity is this. The term being means: something that exists. If abstract objects like numbers and universals really do exist, then these beings have to exist in every possible world. If this is the case, then God, defined by the philosopher as the metaphysical ultimate or the ultimate being or the greatest (metaphysically) possible being would have to exist in every possible world. After all, it would be really strange to think that some beings exist in every possible world but the ultimate being does not.

This also answers the objection that the Ontological Argument mixes de dicto and de re modality. It does not. God is defined as the metaphysical ultimate. If you could coherently describe something that is greater than God then that something would be God. If a being had omnipotence, but only existed in some possible worlds, then we could describe a greater being: one that has omnipotence in all possible world, which entails that such a being exists in all possible worlds. Only a being that exists necessarily can have maximal greatness.

Notice the definition of God here. We are using the philosopher's definition. I do not mean a tribal deity that I am specifying. I do not even mean Yahweh, the deity of Israel. People often get this confused when discussing the moral argument. They ask, Why would the existence of a deity make the difference as to whether objective moral values exist or not? Zeus was not morally perfect. This is, of course, an equivocation fallacy. I am not using the word God as a proper name, but as a term meaning the metaphysical ultimate.

The inquirer may then ask: why believe that Yahweh is this God? And I would answer that given God's moral perfection, he would want to intervene in human history and reaveal himself to someone. Of all the deities in the ancient world, only Yahweh's description is compatible with him being the metaphysical ultimate. Hence, Yahweh is the only candidate for such at title.