Proof from Logical Necessity, or the Ontological Proof (3)

Kant-er Arguments

Obviously, I’m not the only one with objections. Immanuel Kant, most notably, spent eleven pages in “Critique of Pure Reason” poking holes in the Ontological proof. He uses much more robust reasoning than I do, so if nothing I’ve said so far has convinced you, I’d recommend you give his arguments a go.[28] I’ll attempt to summarize the entire thing here.

Kant’s critique comes in four separate parts. First off, he points out that “God is something greater than we can think of” has a hidden assumption: god exists. If god did not exist, then we can say anything about it without contradicting ourselves. “Unicorns are made of gold” is just as truthful as “Unicorns are not made of gold,” but only “Coelacanths[29] are not made of Gold” is true. Since we can say anything we want about non-existent beings, we can prove anything we want about them too.

Second, the ontological proof is supposed to be a proof of god’s existence, yet as noted above Anselm had to assume god existed to write his proof. This is allowed in proofs, if you use a  technique known as “reductio ad absurdum;” you assume something, derive a contradiction from that assumption, and are forced to conclude that assumption is false. Of course, any proof that applies “proof from contradiction” to the assumption “god exists” would have to conclude god does not exist, which seems counter-productive in this case.

Third, we don’t toss around “being” and “exist” lightly. We know that coelacanths exist because we’ve found fossils of them, photographed a few of them swimming about, held them in our hands, and even tasted them. [30] We don’t say they exist because they are “beings,” or have a property called “existence.” Anselm calls god a “being” and says he “exists,” but offers no evidence beyond his proof to back that up. God hasn’t earned either of those labels, yet most Ontological proofs assume he has.

Fourth, we can describe what a unicorn is in physical terms, and set up various tests and experiments to try and catch one. Since we could pin them down as “beings” in the same way as we’ve done to the coelacanth, we can debate their existence in a meaningful way even if no-one’s actually seen a unicorn in the wild. The rational god of Avicenna will never jump into a fishing net or be lured out by hay. Not only do we lack any tangible proof of its existence, we could never find any. God will never be a “being,” no matter how badly an Ontological proof wants him to be.

All of the “distilled” proofs provided by the Encyclopaedia of Philosophy trip up one at least one of Kant’s counter-arguments, and all of them trip up on the last two.

Those last two, in fact, apply to all variations of the Ontological proof. You cannot show something exists in the real world without referring to the real world in some way. Remember my kitten example from the introduction? Until I began defining the physical characteristics of a kitten, you had no way to prove its existence and no reason to take the idea seriously. Conceptual ideas can only be defined within a logical system, and only when that system relies on assumptions that are a close match to the laws of reality do those ideas happen to coincide with reality. For instance, zero-order logic is not permitted to use the concept of sets, and those seem to be an essential abstraction for understanding the real world.

As a direct example, take the third “distilled”[31] proof from the Encyclopaedia of Philosophy, which claims any god is not contingent. While this dodges most of the objections outlined above, it treats existence as if it was an arbitrary label and not something justified via tangible evidence. Since existence is contingent, and this proof says a god has the property of existence, god must be contingent after all.

The Ontological proof tries to use concepts and logic alone to prove the existence of something physical and tangible. It’s impossible, plain and simple.

Gödel’s Proof, and the Problem of Infinity

Simplicity is rare in Ontological proofs, though.

As I mentioned in the introduction, long and complicated chains of reasoning seem more impressive than short, simple ones. In reality, long proofs are more likely to suffer from small errors in logic, and less open to cleaning them out.

A perfect example of this is Gödel’s Ontological proof. To start at the beginning, his insistence on positive properties is suspicious. We tend to make negative properties the verbal negation of positive ones because we prefer to think about the positive, not because one method is inherently better; compare “non-corrupt” and “corrupt“ to “just” and “non-just.” If we apply Gödel’s argument to “negative, morally aesthetic properties” instead, we can prove a god’s existence via the same line of reasoning, since if none of the positive properties conflict then neither can the negative ones, but we’re forced to conclude this god is “perfectly corrupt,” “all-weak,” “merciless,” and an “absolutely amoral” deity.[32] The restriction on “positive” properties is in place to ensure Gödel proves the existence of a god he wants to exist, not because it’s necessary for the proof.

Speaking of which, why does Gödel go to great lengths to use the pure, rational logic to formulate his proof, yet use such a loose definition of “morally aesthetic?” That’s like trying to build a rock-solid building on a swamp. A logical proof is no stronger than its weakest part, and the definitions form the bedrock of the entire argument.

Note as well a subtle problem with Definition 1 and Assumption 3:

Definition 1: An object has the “God-like” property if, and only if, that object has every property in P.

Assumption 3: The “God-like” property is in P.

If you’ve read my take on the Cosmological proof, this should twig an alarm bell. I demonstrated that a container of things is not automatically a thing itself. If the “God-like” property is a property, then it was already in P and thus assigning an object the “God-like” property means that it must already have the “God-like” property to begin with! We could also define a “God-God” property, which requires every property in P including “God-like,” a “God-God-God” property via similar means, and so on.

Even if you object to the above lines of reasoning, Gödel’s proof has a gaping hole. The Epicurean Paradox[33] is the same size as that hole:

If God is willing to prevent evil, but is not able to, then He is not omnipotent.
If He is able, but not willing, then He is malevolent.
If He is both able and willing, then whence cometh evil?
If He is neither able nor willing, then why call Him God?

(“Dialogues Concerning Natural Religion,” by David Hume)

So is this paradox:

If God is perfectly just, no-one is punished less than they deserve.
If God is merciful, someone must be punished less than they deserve.
Therefore, God cannot be perfectly just and merciful.

Same here:

If God is omnipotent, can he perform an action that he cannot perform?

Gödel is careful to prevent simple contradictions from derailing his proof, but does nothing to keep out more complicated ones. These conflicts lead to a definition of a god that contradicts itself, rendering it nearly useless.

I say “nearly” because there is one way out; instead of combining multiple attributes into a single god, you could place a strict limit of one property per god. Most believers will reject that outright, at the time of this writing, since most believers are monotheistic. It also denies any composite property such as “good” from being a god, since that would include the contradictory properties “merciful” and “just,” among others. It also suggests that any property we could come up with has a god associated with it, including “fortitude,” “ambidexterity,” “radical-ness,” and “ability to explain mathematics without sounding condescending.” Even polytheists have their limits, and the vast majority would reject thousands of gods, let alone a potentially infinite number.

Ignoring that escape route, believers dismiss the contractions as not applying to a god because they are beyond rational thought, or as proving that the person asking the question doesn’t understand the type of infinity that the gods posses. The first reply also dismisses the Ontological proof, since it relies on the target god being rational. The second instead proves that the believer doesn’t understand what they’re asking for. Here, let me remind you of a few definitions:

Omnipotent \Om*nip"o*tent\, a. [F., fr.L. omnipotens, -entis;
     omnis all + potens powerful, potent. See {Potent}.]
     1. Able in every respect and for every work; unlimited in
        ability; all-powerful; almighty; as, the Being that can
        create worlds must be omnipotent.
     2. Having unlimited power of a particular kind; as,
        omnipotent love. --Shak.
Omniscient \Om*nis"cient\, a. [Omni- + L. sciens, -entis, p. pr.
     of scire to know: cf. F. omniscient. See {Science}.]
     Having universal knowledge; knowing all things; infinitely
     knowing or wise; as, the omniscient God. --
     {Om*nis"cient*ly}, adv.

(Webster’s Revised Unabridged Dictionary, 1913 edition)

Note that no restrictions are placed on the chosen god, in either case. If a god can do any action, then even the contradictory ones must be doable. If the god can know everything, it must know about things you’d rather keep private. If god is infinitely just, then she must punish fairly in every case, no matter how much mercy you’d like him to grant.

There’s only one escape from this quagmire: redefine the words to place a limit on your god. This is quite dishonest, because people expect the words to mean roughly what a dictionary says they do and thus will misunderstand you. It’s far better to use a different phrase to avoid confusion, though I’ll admit “effectively omnipotent” or “really, really, really super powerful” don’t have the same ring.

A limited god is still a god, mind you. The two definitions I outlined in the introduction do not make reference to infinite power, as you’ll recall. We can still bless a god with enough power to create the universe, or do any number of incredible feats. But note that all versions of Ontological make reference to infinity, either by directly describing an unlimited being, or indirectly implying an infinite number of traits that are infinitely more perfect than any other being can claim. You can’t have your infinity and eat it too.

That doesn’t stop Ontological proofs from trying. All the ones I’ve seen merely introduce more errors, above and beyond the pair related to existence.

The Proof that God Does Not Exist

I can’t leave this proof without sharing my favourite variation. Instead of spending most of a chapter developing objections, Douglas Gasking just cuts to the point by using the same reasoning to prove god doesn’t exist:

  1. The creation of everything is the most marvellous achievement imaginable.
  2. The merit of an achievement is the product of (a) its intrinsic quality, and (b) the ability of its creator.
  3. The greater the disability (or handicap) of the creator, the more impressive the achievement.
  4. The most formidable handicap for a creator would be non-existence.
  5. Therefore if we suppose that the universe is the product of an existent creator we can conceive a greater being — namely, one who created everything while not existing.
  6. Therefore, God does not exist.

The implications are pretty clear. If you can prove and disprove something using the same line of thought, there’s something wrong with your line of thought.

[28] “Critique of Pure Reason” has long since dropped out of copyright, so there are a number of translations available online. The same is true of most of the documents I’ve mentioned, so feel free to analyse them for yourself.

[29] A fish thought to have gone extinct with the dinosaurs, until one jumped into an African fishing net in 1938.

[30] From what I’ve read, they’re very fishy.

[31] As quoted from the Encyclopaedia: “These are mostly toy examples. But they serve to highlight the deficiencies which more complex examples also share.”

[32] Not even Satanists would worship this god. They value personal responsibility, knowledge, justice, and individuality.

[33] Ironically, Epicurus never came up with his paradox. A critic of his, Lactantius, incorrectly attributed it to him four centuries later. Epicurus is sometimes labelled an atheist, again thanks to Lactantius, but was more deist.