The king of all useless arguments
Arguments for the existence of God are, generally agreed, not very compelling, and irrelevant to any real political arguments associated religion. And the ontological argument is king of the junk heap, as it is especially unconvincing and irrelevant. The ontological argument attempts to prove God’s existence a priori, with all the certainty of a mathematical theorem. Most people would say this is overly ambitious, and doomed from the start.
The ontological argument presents a deep and enduring mystery: who actually says this stuff? Does anyone say this stuff? What’s their deal?
The number of people who advocate the ontological argument is nonzero, but it’s quite rare in practice. Let’s round down to zero and say nobody actually says this stuff. Theistic readers may proceed unthreatened, knowing that I’m talking about an argument that none of them actually advocate.
That raises the question why atheists would ever talk about the ontological argument. More to the point, why do I talk about it?
Based on observation, I think atheists only ever talk about the ontological argument because it’s a joke. It’s ripe for parody. The reader is welcome to share their favorite joke about the ontological argument in the comments. (More to the point, I cannot stop readers from doing so.)
My own interest in the ontological argument is based on a bit of personal history. I never liked religion. I have described religion as an “aesthetic disaster“, giving me nothing on an emotional or psychological level. Arguments for the truth of religion were pretty much the only thing that could have saved religion for me. Most Christians would tell you that is not enough. And indeed it was not. I’m glossing over most of my personal narrative, but arguments for God were basically my last point of contact with my own religion, like 20 years ago. They were a topic I used to write about a fair amount.
And the ontological argument was a favorite, because it’s fascinating how badly it fails. Also, some versions of the ontological argument are basically math. It’s like one of those proofs that 1=0. Don’t you want to know where the error is? Come on, we’ll learn some math along the way.
Modal logic
We’ll be discussing a specific version of the ontological argument: the modal ontological argument advanced by Alvin Plantinga in 1974. And I’m specifically focusing on the mathematical core of the argument. It makes use of a system known as modal logic.
Modal logic is basically propositional logic, with two additional operators: □ and ◊. If P is a proposition, then □P is another proposition, and ◊P is another proposition.
You can apply these operators to any proposition. For instance, I just said □P is a proposition. So now we can construct □□P and ◊□P. If we feel so inclined, we can chain the symbols endlessly (though I’ll spare you).
But what do the symbols mean? The goal of modal logic is to formalize the philosophical notions of “possibility” and “necessity”. ◊ is the possibility operator, and □ is the necessity operator. However, I must emphasize that modal logic may or may not succeed in this goal. It only really succeeds if the philosophical and logical meanings match up. Therefore, it depends on how you define philosophical possibility/necessity, as well as your choice of logical axioms.
Modal logic can be axiomatized in many distinct ways. Plantinga’s modal ontological argument specifically uses the “S5” axiomatic system. We call it “S5 modal logic” to distinguish it from other modal logics, in much the same way we talk about “Euclidean geometry” to distinguish it from non-Euclidean geometries.
In principle, modal logic supports many interpretations, as long as all the axioms hold true. However, there’s an interpretation that is popular among philosophers, known as Kripke semantics. (And yes you can read about this on Wikipedia, but it’s quite difficult to follow.)
We imagine that there are a set of worlds, each with its own set of truths and falsities. If we have proposition P, then P may be true in some worlds, and false in other worlds.
Some worlds are “accessible” from other worlds. □P means “P is true in all accessible worlds” and ◊P means “P is true in at least one accessible world”.
It is up to the philosopher to define what “accessible” means. Here is one possibility: we can imagine the timeline as a branching tree of possibilities. Each branch of the tree is a possible world. We could declare that every world on the tree is accessible from every other world on the tree. However, if there are multiple disconnected trees, then they are not accessible from each other.
The beauty of Kripke semantics is that it describes virtually all modal logics, regardless of your choice of axioms. Different choices of axioms simply impose different restrictions on accessibility. S5 modal logic requires that accessibility is reflexive, symmetric, and transitive. Reflexive: each world is accessible from itself. Symmetric: accessibility always goes both ways. Transitive: if world X can access Y, and Y can access Z, then X can access Z. The branching timeline interpretation fulfills these requirements.
TL;DR: Modal logic is an attempt to describe the philosophical notion of “possibility” and “necessity”. But there’s some flexibility in how you interpret it, as long as you stay within its axioms. The common interpretation is that there’s a set of possible worlds, some of which are “accessible” from one another.
The argument
The modal ontological argument takes two premises. First, there is a proposition g (“God exists”) that fulfills the property g=>□g. Second, we take ◊g. The conclusion is g.
If the reader is mathematically inclined, it’s a fun exercise to look up the S5 modal logic axioms, and fill out the proof. But I will not go through the steps, because the error is not here. It is a logically valid proof. Instead I aim to make this proof as intuitive as possible, without explicitly stating any individual steps.
The first premise, “g=>□g” (“If God exists, then God necessarily exists”), can be thought of as true “by definition”. God is an entity that necessarily exists. If an entity does not necessarily exist, then according to this definition it isn’t God. The second premise, “◊g” means “It is possible that God exists.”
To better understand these premises, let us temporarily adopt the branching timeline interpretation. Our timeline is a branching tree, and possibility/necessity describe what is true elsewhere in our tree.
Under this interpretation “g=>□g” makes sense because God allegedly is a being without beginning or end. If God appears in any branch of our timeline (i.e. we accept “◊g”), then naturally God would appear in the future of the branch, as well as the past of the branch, as well as all alternate histories. And therefore God would exist in our own branch. That’s the plain language explanation of the logical proof.
The problem with this interpretation, is that it doesn’t make much sense to assume that God appears in any branch of our timeline. That’s tantamount to assuming the argument’s conclusion! So the proof is valid, but one of the premises is unreasonable.
The trickery
So far, the argument doesn’t make sense. To make it make sense, let’s back up and take a different interpretation of the modal logic.
Based on a more colloquial understanding of “possibility”, most people would admit that “◊g” sounds reasonable. Even the hardline atheist would say, “Sure, there is some remote possibility that God exists. We can file it alongside similar possibilities, such as the possibility that there is an invisible dragon in my garage who is a personal buddy-pal of Donald Trump.” The theist shrugs off the smug remark, and replies in kind: “By admitting the merest possibility of God, you’ve conceded defeat to the sorriest of all arguments for the existence of God!”
What does “possibility” mean here? It basically means that God exists somewhere in the halls of our mind palace. Such as, we can imagine God existing, or we can’t 100% rule it out given our present knowledge.
So let’s return to the first premise “g=>□g”. As I said earlier, this can be thought of as true “by definition”. It’s part of the definition of God that God exists across all accessible possible worlds. This is reasonable when “possible worlds” means other branches in our timeline. It is less reasonable when “possible worlds” means all the halls of our mind palace.
If we take this definition seriously, has anyone actually imagined God, ever? You may have imagined a being with many god-like properties, like being omnipotent and benevolent, yet somehow permitting evil. This god-like being could be three entities in one, and sent his only son to die for your sins. But unless that being truly extends across all the halls of the mind palace, it isn’t God. I’m sorry, we can’t call that thing God, because somewhere in our halls of imagination, someone wrote a Sherlock fanfic where that thing doesn’t exist.
Tell me, does this definition sound reasonable? More to the point, when we admit the possibility of God, that isn’t remotely the God that we’re talking about, and so the proof fails yet again.
The modal ontological argument may be logically valid, but it depends on the premises sounding reasonable-ish. But once we adopt any specific interpretation of the modal logic and really try to understand what the premises mean, one premise or the other starts to sound very strange and counterintuitive.
My objections to the modal ontological argument are more or less the same objections that apply to more rhetorical forms of the ontological arguments, the kind that never bother with formal logic. It may be reasonable to say God transcends time, but it’s another thing entirely to suppose that God can puncture the barrier between imagination and reality. You can imagine God as hard as you want, but it’s still just your opinion, man. It’s difficult to avoid the conclusion that the modal logic only serves to obscure the absurdity at the heart of the argument.
But hey, we got to learn about modal logic, so it was worth it.
my second college philosophy class, the prof genuinely did advocate it. my mind was blown.
It’s because of St Anselm, who gave us the OG Ontological arguments for Dog.
As to that modal claim:
>The first premise, “g=>□g” (“If God exists, then God necessarily exists”), can be thought of as true “by definition”. God is an entity that necessarily exists.
That is what a premise means! Presumed true by definition.
Note it’s a conditional.
>The second premise, “◊g” means “It is possible that God exists.”
This second premise is redundant.
By definition, a being whose existence is *necessary* cannot possibly not exist.
Therefore, asserting that its existence is merely “possible” adds no new information to the argument, since it is rendered entirely pointless by the scope of the first premise.
Basically, by accepting the first premise, Dog is “proven”.
@John Morales
The premises aren’t redundant. You can come up with Kripke models where one premise is true and the other is false, or both are true, or both are false.
One Kripke model to keep in mind is the one where the only accessible world is our own world. Under this model, I necessarily exist, because I exist in the one and only accessible world, therefore I exist in every accessible world.
“g=>□g” is technically a premise, not a definition, you’re right about that. But since g is a free variable in the proof, you have some freedom to pick the properties of g. It’s not difficult to construct a “g” for which the premise is true. (Take any proposition p, and construct g as □p, and you’re done.) Of course, this imposes restrictions on what else you can say or assume about g. This operates quite similarly to saying something is true by definition.
Not the premises. The second premise, I was specific.
First one is, in your own words, “If God exists, then God necessarily exists”.
Obviates the second, IMO, because that ‘if’ in the first is already saying it is possible.
I agree that the issue is definitional, not logical.
The trick of choosing the right premise is the crux.
Put in the right premises, crank the handle, get your ‘proof’.
@John Morales,
No you’re definitely misunderstanding something. The second premise is not redundant, it could be true or false depending on the model. The “if” is a material conditional, and has nothing to do with possibility. We have if-then statements even in regular logic without modal operators. I wish I could explain this better, but I’m not sure where you’re getting this.
P1: in your own words, “If God exists, then God necessarily exists”.
P2: in your own words, “It is possible that God exists.”
Way I see it, P1 is specific.
Entailment is simple: if Dog does not exist, then the very premise is otiose, because it cannot apply.
All it says is that if Dog exists at all, then it exists necessarily.
From which follows that the conjunction with P2 essentially entails Dog exists by virtue of saying it can exist.
It boils down to claiming that if Dog exists in any universe, it exists in all universes.
(That is not a proof, that’s just an assertion, fails due to modal collapse)
Simple syllogism translation:
P1: If God exists, God exists necessarily.
P2: It is possible that God exists.
Conclusion: Therefore, God exists.
Modal Ontology: not to be confused with the MO of materialist criminals, because, well, because!
@John Morales,
It sounds like you are simultaneously claiming that P2 is tautologous and false. So, I’m declaring bankruptcy on understanding what you are trying to say.
As you noted, ‘if it exists’ is part of P1.
It says Dog cannot be contingent, but only if it exists, and if it does, then it necessarily exists.
(Which allows for it to not exist, and the nonexistent is obviously not necessary)
P2 is otiose because it merely asserts that it may exist, not that it does. Not tautologous.
Which, as noted, is implicit in P1, because otherwise it is a pointless premise.
(Might as well say if one rainbow-farting Unicorn exists anywhere, they exist everywhere. Same exact ‘proof’)
For the older form of the Ontological Argument, here is an outline of an ontological argument for God’s non-existence.
Axiom: God is the most perfect being.
Axiom: A God who creates a superior universe is more perfect that one who creates an inferior universe.
Axiom: We are not living in the best of all possible worlds.
Ergo: God does not exist.
@6: It is conceivable that God exists; it is not known whether it is ontologically possible that God can exist. When someone agrees that it is possible that God exists they typically mean the first; the ontological argument depends on the second.
At least two other philosophers, Norman Malcolm and Charles Hartshorne, preceded Plantinga in outlining versions of the modal ontological argument. I came across it, and them, in an undergrad philosophy of religion course. IIRC, they both took a different view to the one outlined here of what the modal connectives mean: □P meaning that P is true in every state of affairs that can be consistently described, while ◊P means P is true in at least one such state of affairs. They then claim (again IIRC) that g is the only proposition asserting existence for which □g, and thus God is extra-specially super-duper-special. It seems to me (now, I can’t recall how I argued then, although I think I still have my dissertation on the ontological argument somewhere) that the null state of affairs (“Nothing exists”) can be consistently described (I just did it), if we limit existence to entities that are not purely abstract, such as numbers and other mathematical entities, and that that is a sufficient refutation; and if we don’t add that limitation, then there are infinitely many existence-asserting propositions P for which □P (uncountably many if we allow propositions with infinitely many conjuncts).
Incidentally, I recommmend an oldish treatment of possible worlds (I think it precedes Kripke’s stuff): Bradley and Swartz Possible Worlds: An Introduction to Logic and its Philosopy, which is now available free online.
John Morales@2,6,9,
What is the point of writing “Dog” when you mean “God”? It doesn’t render God any more non-existent, and suggests to me that you’re refering to the Platonic ideal of doggyness.
@KG,
Yeah, I agree with you on that one. “Nothing exists” is a possible world under that formulation, and that poses a problem for the conclusion that anything necessarily exists. I think if you take “□P” to mean “P is true in every state of affairs that can be consistently described”, and P is allowed to be a modal proposition, you may well be constructing a self-inconsistent logical system. So it may be that both g and ¬g are theorems, by the explosion principle.
Kripke semantics were very helpful to articulate to myself what was going wrong with these arguments. Philosophers of metaphysics are skilled at coming up with umpteen distinct notions of possibility/necessity. And yet they often naively assume there’s just the one, i.e. whatever they decide to use at the present moment. This is an environment very conducive to errors of equivocation.
Modal logic can make the problem worse because it’s consistent with many distinct philosophies. So you can assert one premise under one philosophy, assert a second premise under a different philosophy, and put them both into the logical machine to produce nonsense. Modal logic is very versatile, but you can’t switch Kripke models mid-theorem.
KG,
>What is the point of writing “Dog” when you mean “God”? It doesn’t render God any more non-existent, and suggests to me that you’re refering to the Platonic ideal of doggyness.
It amuses me. And it irritates (many) goddists by evincing disdain for the concept.
@15: It’s additionally amusing that Socrates sometimes swears by “the Dog”, probably meaning Anubis the Egyptian jackal-headed deity (the god/dog wordplay obviously doesn’t happen in ancient Greek).
How is this not just a straightforward “mid-sentence redefinition” fallacy?
Many of the fallacious mathematical proofs you will see in sidebars in textbooks, purporting to show that two different things are equal to each other, work by sneakily promoting a linear equation to a quadratic equation. The latter can be written in the form
(x – a) * (x – b) = 0
and since both x=a and x=b are valid solutions, then — as long as it is not obvious how you introduced the alternative solution — you can claim that a = b.
Something similar is happening in the Ontological Argument for God. Here, the fallacy is in eliding over the difference between “exists as an abstract concept” and “exists in reality”. All the argument really proves is that the idea of God exists. The sleight-of-hand move is claiming this proves God exists in reality. Of course God exists as an abstract concept, because an abstract concept comes into existence the first time someone mentions it. The Ontological Argument holds ∀x substituted for “God”, because either x already exists as an abstract concept, or the abstract concept of x was created with the act of defining x.
You can think any idea into being, but there is only one way to prove it exists in reality: Measure it.