
Gödel's ontological proof - kdavis
http://en.wikipedia.org/wiki/Gödel%27s_ontological_proof
======
gjm11
The difference between Anselm's original ontological argument and modern ones
like Goedel's or Plantinga's is that while Anselm's is full of terrible
reasoning, Goedel's reasoning is (unsurprisingly) watertight and all the
terribleness is concentrated in the axioms.

Specifically, I think the idea of classifying all properties as "good" or "not
good" is hopeless: I do not believe there is any such classification that fits
anyone's intuitive ideas about goodness well enough that calling something
"godlike" if it has all "good" properties is credible. More specifically, I
suspect it's consistent to suppose -- hence there are no humanly-
comprehensible counterexamples -- that the _only_ notions of goodness obeying
all Goedel's axioms are ones that look like "P is good iff P(x)" for some
fixed object x. (And, like Krishnaswami, I think the system of modal logic
Goedel needs is awfully strong.)

In the spirit of Anselm's ontological argument, however, I offer the following
proof of the _nonexistence_ of God:

Consider a really bad argument for the existence of God. In fact, consider one
so bad that no worse argument can be conceived.

Obviously a bad argument for something fails to prove what it purports to
prove. But merely failing is a pretty mediocre kind of badness. The worst
possible argument, surely, has to be much worse than that: it must
_conclusively prove the opposite_ of what it's meant to prove.

Now, the worst conceivable argument for theism clearly "exists in the
understanding", as St Anselm put it. But it can't exist only there -- because
a bad argument is more damaging to the premise it's meant to support if it's
_actually made_.

Therefore, there is an argument for the existence of God which is actually a
conclusive proof of the nonexistence of God.

And, of course, any proposition that can be conclusively disproved is false;
therefore there is no God.

(This argument is in my opinion almost exactly as strong as the original
ontological argument for the existence of God. Which is to say, it's
absolutely hopeless. But I think it's fun.)

~~~
logicallee
wow! did you come up with this?

~~~
gjm11
What, the anti-ontological argument? Yes.

There's another very-slightly-similar inverted ontological argument that may
have been in the back of my head. It's due to a chap called Douglas Gasking,
and it begins with a standard ontological argument -- which, as you know,
delivers us a Maximally Excellent Being. Now, of course God's signature
accomplishment is the creation of the universe; but any creative achievement
is more impressive when it is accomplished under greater constraints (so,
e.g., while it is impressive to come up with a proof of Fermat's Last Theorem
it would be even more impressive to come up with a proof of Fermat's Last
Theorem _in the form of a perfect Shakespearean sonnet_ ). Well, what would be
the most impressive obstacle for God to have overcome when creating the world?
His own total nonexistence, obviously. So our maximally impressive being must
be a god who created the universe _without existing_ ; therefore God does not
exist.

I have to confess that I like mine better.

------
michaelfeathers
When I think of Godel, the first thing I think of (based on accounts) is
compulsion. He was really driven to explore the edges of reality using logic
as a probe.

Here's an account of his discovery of a logical flaw in the US Constitution:
[http://morgenstern.jeffreykegler.com/](http://morgenstern.jeffreykegler.com/)

~~~
SilasX
Not really an account of his discovery, just a story about how he claimed to
have found one; no one knows what flaw he thinks he found (but one can guess).

------
lisper
Godel's proof is essentially Anselm's with some mathematical window dressing
added. It is worth noting that Anselm's argument applies equally well to Satan
as to God: The most evil thing we can conceive of would be more evil if it
actually existed, therefore Satan must actually exist. If you're into this
sort of mind game, it makes an interesting exercise to translate this version
of Anselm's argument into formal modal logic.

~~~
michaelsbradley
Would it really apply to Satan? Historically, most Christian thinkers consider
him to be a creature and thus finite in every respect; likewise, evil is
generally considered by the same thinkers to be insubstantial, i.e. it is the
absence of good and not some kind of entity (or put another way, it's not a
matter–anti-matter kind of thing). So it seems evil would have a kind of local
maximum, or floor value, depending on your perspective, which would be "zero
good". At the same time we could speak of evil being compounded over time,
i.e. an accumulation of depraved acts and persons.

Of course, you may have other definitions of Satan and evil in mind, and I
don't mean to imply that those concepts are solely the province of Christian
minds.

~~~
lisper
One must not get too hung up on terminology here. The point is that the same
argument that applies to "the greatest good that can be conceived" applies
just as well to "the greatest evil that can be conceived." You can attach
whatever labels to these things you like. The point is that if a great good
becomes greater by actually existing, then so does a great evil, and so you
can apply Anselm's rhetorical trick just as well to the latter as to the
former.

~~~
michaelsbradley
But a "semantics of evil" was the point I was trying to bring to the fore. If
the maximum local evil (e.g. in a person) is "zero good", i.e. the complete
absence of good, then the greatest evil is a fixed value and doesn't seem
subject to the same line of reasoning.

~~~
lisper
Nope, doesn't work. If you admit the existence of even a single affirmative
evil act (torture, say) then one can always increase the amount of evil by
doing more of that affirmative evil act.

Mathematically it's analogous to trying to reason from the premise that zero
-- the "absence of anything" \-- is the smallest possible number. You can do
it, but it leads to incredibly messy math. As soon as you try to define
subtraction, all hell (pun intended) breaks loose. And as soon as you admit -1
into your system you can no longer have a smallest number (if you have
addition and induction).

~~~
michaelsbradley
Just a thought experiment:

What if we define The Good as unity, and evil as the ratio of The Good to
relative attainment of the The Good. We might then define compounded evil as a
ratio of The Good to the multiplication of the denominators of each ratio-evil
consider on its own.

So an act of torture might be (/ 1 ½) and three acts of torture would be (/ 1
⅛).

~~~
lisper
(/ 1 ½) == 2, and (/ 1 ⅛) == 8, so you're just mapping n onto 2^n. How is that
supposed to change anything? You still have a total order and no upper or
lower bound, which is all that matters.

~~~
michaelsbradley
The multiples of ½ were arbitrary, just an illustration re: denominators.

Also, your points above are well taken; I was simply trying to come at the
matter (and my previous points) with a bit more precision than I had
previously, but I cut my elaborations short as I needed to run out the door.

So with my idea above, you would basically have two notions for assigning
magnitude to evil. Absolute evil would be defined as the complete absence of
good in a person or other locus of circumstances and acts. So the greatest
real evil would a fixed value, "zero good", and "negative amounts of good"
would be a void construct.

The other notion of evil, "evil considered", would be a metaphor for
understanding how evil gets compounded according to our usual perception that
some lives and acts are more evil than others. It's basically a "score" which
doesn't correspond directly to any substance.

~~~
lisper
> Absolute evil would be defined as the complete absence of good in a person
> or other locus of circumstances and acts.

But evil is more than the mere absence of good. There are affirmative evil
acts (rape, murder, torture) so you can always become more evil by performing
more evil acts.

------
vilhelm_s
Neel Krishnaswami wrote a quite readable description of the proof on his blog:
[http://semantic-domain.blogspot.com/2014/06/g-ontological-
ar...](http://semantic-domain.blogspot.com/2014/06/g-ontological-
argument.html)

------
KierkegaardE
This proof is a mix between Anselm and a little bit of math.

Disclaimer; I've only read Anselm and Leibniz, and am just trying to piece
together the discrepant sources around.

Premise 1: There are many worlds.

Definition 1: "x necessarily exists if and only if every essence of x is
necessarily exemplified" (From the article)

Premise 2: Existence is good.

Definition 2: God is the being Good-er than whom cannot be conceived.

Sub-Conclusion 1: God if he exists, by definition is good, and by supposition
necessarily exists because existence is good. He necessarily exists because if
he did not, then a being good-er than him could be conceived (ie, one that
existed), and that being would then not be god.

sub-conclusion 2: Because there are many worlds, there exists one with a being
good-er than whom cannot be conceived. Therefore, this is a being whose
existence is necessary.

Consider then, that God's existence would not be necessary, if the world that
god governed was not necessary, because that world could, or could not exist,
and so God could, or could not exist, and god's existence would not be
necessarily be exemplified.

Therefore The world in which God exists must be necessary to our world, and by
consequence, God's existence must be necessary to our world.

If someone who has actually read him could give me some feedback as to whether
this is fairly close to what he means, I'd appreciate it.

~~~
s_baby
>Premise 1: There are many worlds.

The premise is called the plenitude principle. "The principle of plenitude
asserts that the universe contains all possible forms of existence."[1] You
first see it in Plato's work like in the Cave Allegory. You can make a
compelling argument that the plenitude principle was the precursor to the
ideas of a multiverse and evolution.

1\.
[http://en.wikipedia.org/wiki/Principle_of_plenitude](http://en.wikipedia.org/wiki/Principle_of_plenitude)

------
michaelsbradley
Readers might also enjoy the audio recording and transcript of a famous debate
between Bertrand Russell and Fr. Frederick Copleston:

 _The Famous 1948 BBC Radio Debate on the Existence of God_

[http://www.biblicalcatholic.com/apologetics/p20.htm](http://www.biblicalcatholic.com/apologetics/p20.htm)

~~~
jonahx
"Your general point, then, Lord Russell, is that it's illegitimate even to ask
the question of the cause of the world"

"Yes, that's my position"

"Well, if it's a question that for you has no meaning, it's of course very
difficult to discuss it, isn't it?"

Russell, almost with a sigh: "Yes, it is very difficult." Then: "What do you
say, shall we pass on to some other issue?"

"Let's."

What struck me most was the calmness and politesse they both maintained, and
the steadfast articulateness at the speed of thought. There seemed to be an
unspoken, shared assumption that you never, ever raise your voice or betray
any hint of frustration, let alone insult your opponent in public. It's hard
to imagine hearing something similar on the radio today.

I'm also curious how much of the decorum was a sign of the times, and how much
was specifically British.

~~~
philwelch
Russell and Copleston were titans of philosophy as well, and the philosophers
I've seen tend to discuss philosophy in a calm, polite, and considered
fashion.

------
chroma
Note that even if you accept this proof, it says nothing about a god that
answers prayers or otherwise intervenes in daily life. It also says nothing
about an afterlife. Unless you're a deist, ontological arguments are
practically useless. They've been used in the defense of many gods now
relegated to mythology.

~~~
gjm11
I don't know about that. If you accept an ontological argument as valid, what
it gives you is a god who's maximally perfect in every way. So then if you
want to show (e.g.) that he answers prayers, all you have to do is to argue
that answering prayers is more perfect than ignoring them.

Descartes gives a kinda-ontological argument for the existence of God, and
then he does indeed very rapidly move to deducing useful properties of God
from it -- e.g., that he wouldn't allow us to be systematically deceived about
everything. It's the basis of his whole epistemology, at least ostensibly.

Whereas, say, a first-cause argument tells you there's something that's in
some sense the cause or origin of the universe, but you've got a whole lot of
work to do to get from there to _anything_ relevant to actual religious
belief. (Though, e.g., William Lane Craig does somehow manage to keep a
straight face when transitioning at lightning speed from "something caused the
universe to begin to exist" to "and that something must be personal,
enormously powerful, etc.)

(Lest I be misunderstood, I'll add that I think both ontological and
cosmological arguments fail badly, and that neither of them either manages to
give much reason to believe in a maximally perfect being / first cause /
whatever, or to believe that such a being if it existed would have anything
much to do with the gods of traditional religions. But it's not my purpose to
litigate any of those claims here -- Goedel's argument is much more
interesting than any I'd make.)

~~~
sjwright
> Goedel's argument is much more interesting than any I'd make.

Hardly, it's just obscurantist gobbledegook dressed up as modal logic. It uses
symbols and math to make the same basic errors that all other proofs make.

~~~
sjwright
It's disappointing that I'm being voted down. The argument only proves that
perhaps some vague thing which Goedel slaps the 'god' label on _might_ exist,
or might have existed. The fancy looking methodology only serves to obscure
this obvious limitation of logical argument proofs.

~~~
chroma
On the disagreement hierarchy[1], your downvoted comment is DH3 at best. Also,
it's needlessly disparaging. I think you'd fare better if you avoided such
digs at others' opinions in the future.

Also, don't worry about downvotes. HN isn't that big, and individual comments
are quickly forgotten. Think of it as feedback, not condemnation of you
personally.

1\.
[http://www.paulgraham.com/disagree.html](http://www.paulgraham.com/disagree.html)

~~~
sjwright
I hadn't seen the disagreement hierarchy. Very cool.

I'm not that fussed about the downvote, I'm just innately curious as to
people's reasons.

------
riffraff
and an implementation (I think unfinished) in Coq
[https://github.com/FormalTheology/GoedelGod](https://github.com/FormalTheology/GoedelGod)

------
tunesmith
I only dabble in formal logic so I haven't come across this before, but it
sort of feels like this rests on a circular argument. You know, the "can God
make a rock so heavy that he can't lift it" or "does God have the power to
make himself even more Good". I wonder if it's possible to prove anything if
one of your axioms is actually a hidden circular argument, much like how one
can prove anything from setting true to false.

~~~
davidgerard
All ontological arguments for God are of the form:

1\. I carefully define God as having to exist.

2\. Therefore, God exists.

All the work goes into the first part, however many volumes of obfuscation
(hi, Dr Plantinga!) one hides it behind.

There are a number of problems with this approach.

------
sjwright
We see our world in terms of cause and effect, but these are temporal concepts
that might have no meaning outside of linear time. It may therefore be
nonsensical to ask what caused the universe.

------
elwell
We're still talking about God? There must be something to this "God" idea that
is important / common to humanity.

~~~
sjwright
The gullibility of children, usefully instilled into us by natural selection.

The fear of death, usefully instilled into us by natural selection.

Human minds have no experience of not existing, and tend to find the concept
difficult to comprehend. We desire answers, but unfortunately the human
condition is to prefer wrong answers over no answer.

~~~
amelius
Human brains also deceive us into believing we are distinct entities, whereas
in reality the universe is probably a single entity.

~~~
sjwright
Cute, and in some respects insightful, but as a contribution it's arguably an
equivocation on the definition of entity. You could similarly argue that human
brains deceive us into believing we are distinct entities, whereas in reality
we are billions of entities experiencing the emergent properties of multi-
cellular life.

------
amelius
It seems to me that he left out a couple of "if it is possible to exist"
clauses here and there.

~~~
juped
This is exactly the opposite of what is the case. Anselm implicitly assumed
that it was possible for God to exist, but Gödel proves it as a theorem
(theorem 2 on the Wikipedia page).

------
eli_gottlieb
This assumes that "goodness" is an irreducible, ontologically basic property.

