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.

The Ontological Argument for the Triune God

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The Ontological Argument is a purely a priori logical argument that operates like a mathematical proof. It derives the existence of a perfect being from the laws of modal logic themselves. It's a technical argument for the existence and properties of God. That's why this is my favorite argument for the existence of God, because it goes beyond fallible evidence and when understood correctly, is virtually impossible to argue against.

In the field of philosophical theology, there are two competing definitions of God. In creator theology, God is defined as "the creator of all that is not God." In perfect being theology, God is defined as "A Maximally Great Being" which means that God possesses all great-making, properties such as love, knowledge, and power, and possesses each in a maximal way. He also possesses no flaws, such as immorality. He would be all-powerful, all-loving, and all-knowing. I intend to show you three things about this Maximally Great Being:

This video will present and defend three points:
1. A Maximally Great Being Exists (hence, perfect being theology is correct)
2. Only one Maximally Great Being can possibly exist (hence monotheism is true)
3. A Maximally Great Being must be multipersonal (the being must exist in two or more persons)

Let's start defining things before we get into the argument. First up: possible world. A possible world is basically a hypothetical situation. Calling it a possible world is not to say that such a world actually exists. It's just a description of what reality might be; a way for philosophers to see what ideas are coherent and what ideas are not. If something is possible, then we say that it exists in some possible world.

In metaphysics, the term "possible" has a different meaning than in epistemology. In epistemology, you could look at a difficult math problem and say "it's possible that there is a solution for it" which is like saying "for all I know, there is a solution to this problem." In metaphysics, something is possible if it is logically coherent. For example, my wearing a red shirt is possible, but the existence of square circles is not metaphysically possible.

If something is possible, it exists in a possible world. Rolling a six on a die is one example. If something is necessary, it exists in all possible worlds. The laws of mathematics are examples of this. If something is contingent, it exists in some possible worlds but not in others. If something is impossible, then it exists in no possible worlds. A square circle would be an example of this.

Next up entailment. Entail means "to imply necessarily." For example, shape entails size. If something has a shape, it necessarily has a size. After that: Great-making properties. A great-making property is a property that it is better to have than to lack. A lesser-making property is a property that it is better to lack than to have. A neutral property is one that is neither. Maximal greatness is the state of having all great-making propertries, and to their maximal extent. It also means having no lesser-making properties. One of these properties is necessity. Nothing that exists contingently can be maximally great. This is not to say that something has to exist to be maximally great. Instead, if something maximally great exists, then its existence is either necessary or impossible.

Now one Immanuel Kant's objections to an obsolete version of this argument is that existence is not a property. This is true. And this argument does not assume that it is. It is immune to the classic criticisms of Anselm's ontological argument.

Necessity is a property. Existence is not. Necessity does not entail existence. Necessity just means that if something exists, it exists in all possible worlds. Numbers have this property.

Now on to the Ontological Argument itself. This argument has five premises, which if true, entail the conclusion.

Premise 1: It is possible that a Maximally Great Being (MGB) exists
Premise 2: If it is possible that a MGB exists, then a MGB exists in some possible world
Premise 3: If a MGB exists in some possible world, then a MGB exists in all possible worlds
Premise 4: If a MGB exists in all possible worlds, then a MGB exists in the actual world
Premise 5: If a MGB exists in the actual world, then a MGB exists
Conclusion: A MGB exists.

Premises 2 through 5 are pretty uncontroversial in academia. They are just restatements of the laws of modal logic, accepted by theist and atheist philosophers alike. The controversy is in Premise 1. Is it possible (metaphysically) that a Maximally Great Being exists. I think that on the face of it, we have good reason to believe that it is true. As long as the concept of a Maximally Great Being is coherent, and does not violate any known necessary truths, then Premise 1 is true. A school of philosophy called "natural atheology" has attempted to build arguments against the coherence of a Maximally Great Being, but this project has been all but abandoned. Richard Swinburne explains why in his book: The Coherence of Theism.

But I think we can advance an argument in favor of Premise 1, that will make it airtight. It was developed by Robert Maydole and is called the Modal Perfection Argument. For the sake of this argument, we can go with the following definitions:

Great-making property: A property that an omnipotent, omniscient, omnibenevolent, and metaphysically necessary being must have.
Lesser-making property: A property that an omnipotent, omniscient, omnibenevolent, and metaphysically necessary being cannot have.

Here is how it works:

1. If a property is a great-making property, its negation is a lesser-making property
2. Great-making properties do not entail lesser-making properties
3. Maximal Greatness is a great-making property
Hence, Maximal Greatness cannot entail its negation of non-Maximal Greatness.

It is perfectly within the laws of modal logic for a property (which is not a perfection) to entail its negation. If a property is an incoherent (or impossible) property (square-circleness is an example, so we'll call this S), then it's necessary that everything has the negation of S (we'll call this ~S) as a property. That's what it means for a property to be impossible. But if everything has ~S, that means that every property entails ~S. If it didn't, then something could have some other property and not have ~S, which means it would have S, which means S would be a possible property. But if every property entails ~S, then S also entails ~S. This is consistent with the Principle of Explosion, which states that if you assert a contradiction, you can logically infer anything from it.

Hence, if it is impossible that a Maximally Great Being exists, then necessarily, nothing has Maximal Greatness. Hence, all things have non-Maximal Greatness. Hence, all properties entail non-Maximal Greatness, including Maximal Greatness. But we just established that Maximal Greatness cannot entail non-Maximal Greatness.

For the minority of philosophers who deny the principle of explosion, here is another way to view the argument. If Maximal Greatness is an impossible property, then all properties entail non-Maximal Greatness. Hence, no property can be a great-making property. But this is absurd. It would mean that some properties such as goodness, intelligence, and wisdom are not better to have than to lack. So again, there is no way that Maximal Greatness is an impossible property.

But that would mean that its is possible that a Maximally Great Being exists. And therefore such a being exists in some possible world, and therefore in every world, and therefore in this world. And since some of these properties such as omnipotence, omniscience, and moral perfection entail personality, then it follows that this Maximally Great Being must be personal. But that's the definition of a personal God. Hence, perfect being theology is correct.

Let's go through a few of the common objections. The first is Dawkins' own delusional objection. In The God Delusion, Richard Dawkins attempts to use this argument to disprove the existence of God. He says that if God is the greatest possible being, then wouldn't it be greater for God to not exist and still bring the universe into being? But far from undermining the Ontological Argument, it shows its coherence.

To answer Dawkins: Greater? Maybe. Possible? No. For in what possible world does a non-existent being exist?

What about a Maximally Great Pizza or a Maximally Great Bird? The problem with positing anything physical, is that physical things are dependent on space for their existence. And we know from modern cosmology that space once existed as a singularity, and no bird or pizza in such a state would even be a bird or a pizza. So the idea of a necessarily existing physical thing is undercut.

But how do we know that only one Maximally Great Being exists? Some of the properties possessed by a Maximally Great Being cannot exist in more than one being in any possible world. Omnipotence is one example. If multiple beings are omnipotent, then a logical contradiction follows if their wills come into conflict. If one omnipotent being chooses to bring about a state of affairs where a green elephant exists, then such a state of affairs will be actualized. But if another omnipotent being in the same world wants to bring about a state of affairs where a green elephant does not exist, then that state of affairs will be actualized. So in this world, a green elephant would both exist and not exist, but no possible world can contain such a contradictory state of affairs. So no possible world can have multiple omnipotent beings. Hence, there can only be one God.

Alexander Pruss presents the same idea worded differently: "Omnipotence requires perfect freedom and an e fficacious will. But there cannot be two beings with perfect freedom and an e fficacious will. For if they are perfectly free, they will be able to will incompatible propositions to be true, and then one of their wills shall have to fail to be e fficacious. (This argument assumes that we are individuating beings in such a way that distinct beings with will have their own will. If God is a Trinity, the persons of the Trinity do not have distinct wills, and hence will not count as distinct beings in our sense.)"

Robert Maydole's solution is to add the premise:
4. Being Supreme is a great-making property
Since being supreme entails that all other beings are inferior, only one MGB can exist! Again, there can only be one God.

Finally, what about a unitarian or uni-personal God? I'm afraid such a being, too would not be Maximally Great. For at least one of the properties of such a being, love, is problematic. In order to be necessarily loving, God must be loving in all possible worlds. Since all that is not God is subject to the creative and destructive power of the omnipotent one, God can be the only necessarily existing being. This means that there are possible worlds where God alone exists. But is God loving in those worlds? It would make no sense to say that God is loving in a state of affairs where only he exists. Does he love himself? Self-love doesn't sound like love. Love has to be expressed between one person and another in order to be true love. But that means that God must exist in at least two persons, if not more. Hence, unitarianism is false.

I want to remind all my viewers that both of these arguments are logically valid deductive arguments. You cannot avoid the conclusion of this argument without denying at least one of the premises. And in order to justify that denial, you have to show that the negation is more plausible than the premise itself. But I think to any fair-minded individual, all premises presented are pretty obviously true. And with this sequence, we have a logical proof of the existence of God. The Triune God of Scripture lives, and atheism is toast.

Part 2 (The S5 Objection and de re modality) is here!
Part 3: The Greatness Objection