I think the modal ontological argument is a much stronger target of discussion.

1. If God exists then it is necessary that God exists
2. It is possible that God exists
3. Therefore, it is possible that it is necessary that God exists
4. If it is possible that it is necessary that God exists then it is necessary that God exists
5. Therefore, it is necessary that God exists

In formal logic:

$\exists xGx\to\Box\exists xGx\\\Diamond\exists xGx\\\therefore\Diamond\Box\exists xGx\\\Diamond\Box\exists xGx\to\Box\exists xGx\\\therefore\Box\exists xGx$

How does 2 differ from 3? — Michael

3 is not an axiom, just a definitional fact. 2. isn't necessary, I just left it there because you put it there. See:

If some X is TTWNGCBC, then X necessarily exists
God is an X.
Therefore, God (necessarily) exists. — Hallucinogen

1. If there exists something which is TTWNGCBC then this thing necessarily exists
2. If there exists something which is TTWNGCBC then this thing is God
3. There exists something which is TTWNGCBC
4. Therefore, God (necessarily) exists.

But 3) is an empirical claim that needs to be shown. It's not something that's true a priori. — Michael

Oh right. But I was assuming that by "If there exists something which is TTWNGCBC", you meant the same thing as "If some X is TTWNGCBC," in the arguments you gave when you were previously attacking it. I wouldn't try to defend this argument beginning with "If there exists something which is TTWNGCBC", because of the flaw you've pointed out, so I would have to insist on going back to the way you originally simplified it, to "If some X is TTWNGCBC (...)"

Showing that "there exists something which is TTWNGCBC" is the intent of the argument, so I suppose you would have to express it in a way that does not require the satisfaction of an empirical claim, as I have done above.

But I was assuming that by "If there exists something which is TTWNGCBC", you meant the same thing as "If some X is TTWNGCBC," in the arguments you gave when you were previously attacking it. — Hallucinogen

Yes, they mean the same thing.

3 is not an axiom, just a definitional fact. 2. isn't necessary, I just left it there because you put it there. — Hallucinogen

Then the argument is invalid. To make your argument more precise:

1. If there exists something which is TTWNGCBC then this thing necessarily exists
2. God is defined as TTWNGCBC
3. Therefore, God (necessarily) exists

It's still missing the premise that asserts that there exists something which is TTWNGCBC, which as you say is the very intent of the argument.

To make this clearer by analogy:

1. If there exists something which is the greatest conceivable vampire then this thing necessarily exists
2. Dracula is defined as the greatest conceivable vampire
3. Therefore, Dracula exists

The conclusion doesn't follow. I'd need as a premise that the greatest conceivable vampire exists.

Yep. A shame it's invalid.

It's still missing the premise that asserts that there exists something which is TTWNGCBC — Michael

It doesn't need one; the argument is still valid.

1. If there exists something which is TTWNGCBC then this thing necessarily exists
2. God is defined as TTWNGCBC — Michael

Line 1 connects anything that fits the definition of TTWNGCBC (since you're allowing it to mean the same thing as "if some X is...") with it necessarily existing. So once a concept is identified as fitting that definition, it is shown to in fact exist.

Your objection seems to me to undermine the very capacity of logical and mathematical generalizations to prove anything about the world. All I need is for God to fit the definition of something which is a valid generalization of the logical and mathematical relationships between things that I already know to exist.

1. If there exists something which is the greatest conceivable vampire then this thing necessarily exists
2. Dracula is defined as the greatest conceivable vampire
3. Therefore, Dracula exists

The conclusion doesn't follow. I'd need as a premise that the greatest conceivable vampire exists. — Michael

It's a false analogy. Vampires aren't non-contingent entities.

It's a false analogy. Vampires aren't non-contingent entities. — Hallucinogen

The greatest conceivable vampire is.

A vampire that exists is greater than a vampire that doesn’t exist.

A vampire that necessarily exists is greater than a vampire that non-necessarily exists.

Therefore, the greatest conceivable vampire is one that necessarily exists.

Replacing the word “vampire” with “intelligence” or “entity” or “thing” doesn’t change the logic.

If when you said "the greatest possible vampire" you ultimately didn't mean something uniquely vampiric, but just meant a non-contingent entity upon which everything else is contingent and we're calling it "Dracula", then your analogy is just the same argument and it is valid.

I don’t know how accurate that website is at parsing modal logics, but my understanding is that modal ontological arguments are commonly accepted to be valid, and so opponents must challenge the premise(s).

https://plato.stanford.edu/entries/ontological-arguments/#PlaOntArg

The “victorious” modal ontological argument of Plantinga 1974 goes roughly as follows: Say that an entity possesses “maximal excellence” if and only if it is omnipotent, omniscient, and morally perfect. Say, further, that an entity possesses “maximal greatness” if and only if it possesses maximal excellence in every possible world—that is, if and only if it is necessarily existent and necessarily maximally excellent. Then consider the following argument:

There is a possible world in which there is an entity which possesses maximal greatness.
(Hence) There is an entity which possesses maximal greatness.

Under suitable assumptions about the nature of accessibility relations between possible worlds, this argument is valid: from it is possible that it is necessary that p, one can infer that it is necessary that p. Setting aside the possibility that one might challenge this widely accepted modal principle, it seems that opponents of the argument are bound to challenge the acceptability of the premise.

I don’t know how accurate that website is at parsing modal logics — Michael

The counter model looks right.

There might be something in the " suitable assumptions about the nature of accessibility relations between possible worlds", butI don't see it. It's invalid in all six, according to the tree proof generator.

So I'm left to supposing that it's down to how we pass

• There is a possible world in which there is an entity which possesses maximal greatness.
• (Hence) There is an entity which possesses maximal greatness.

◇∃xGx→∃xGx looks invalid. there can be a world in which ◇∃xGx is true, and yet ∃xGx false - It is possible to have green cows, but there are no green cows.

◇∃xGx→∃xGx looks invalid. — Banno

Given the definition of “maximal greatness” as being necessarily “maximally excellent”, the argument is ◇□ ∃xGx→ □∃xGx.

It is possible that something is necessarily maximally excellent, therefore it necessary that something is maximally excellent.

The counter model looks right. — Banno

I think it’s interpreting this as an inference rather than as a premise:

∃xGx → □∃xGx

◇□ ∃xGx→ □∃xGx. — Michael

But that's just an instance of ◇p→p, which is pretty clearly invalid.

∃xGx → □∃xGx — Michael

p→□p. Invalid.

But that's just an instance of ◇p→p, which is pretty clearly invalid. — Banno

◇□p → □p is valid.

p→□p. Invalid. — Banno

This is given as a premise, not an inference. It’s either true or false. If I gave this argument, would you reply by saying that 1) is invalid?

1. p→q
2. p
3. q

I call bullshit on this badly constructed argument (Cosmological, Kalam, Contingent), it should go more like this.

Firstly it over complicates things, if cat, fish, dog is impossible (non-existent) then by fact of existence cat, fish, dog exist then cat, fish, dog exist.

Where does contingency come into it ?

A simpler reformulation of the previous.

Either it is not possible that God exists or it is necessary that God exists
It is possible that God exists
Therefore, it is necessary that God exists

¬◇∃xGx ⊻ □∃xGx
◇∃xGx
□∃xGx

equivocation of necessity to possibility.

It’s like saying it’s possible that my next coin flip will be tail. So if I do flip it it will be tails. (By necessity)

That works.

https://www.umsu.de/trees/#((~3~9~7xGx~2~8~7xGx)~1~9~7xGx)~5~8~7xGx

The question then is whether or not there is a satisfactory definition of God of which both ¬◇∃xGx ⊻ □∃xGx and ◇∃xGx are true.

The second premise is true if the definition doesn't contain a contradiction.

The first premise appears to be an application of the Buridan formula, ∃x□Dx → □∃xDx, where Gx is defined as □Dx, and Dx is defined as something like "x is the demiurge".

The full argument then is:

1. ∃x□Dx → □∃xDx
2. ◇∃x□Dx
3. ∴ ◇□∃xDx
4. ∴ □∃xDx

What's interesting is that according to that website 1-3 is valid, 3-4 is valid, but 1-4 is invalid. That strikes me as a contradiction.

The countermodel is:

Worlds: { w0 }
Individuals: { 0 }
@: w0
D: { }

The second premise is true if the definition doesn't contain a contradiction, which I think is an easy condition to satisfy. — Michael

If the definition is "a something a greater than which cannot be conceived", I'm not convinced. There's the obvious comparison of "A number a larger than which cannot be conceived" - the idea is not coherent.

If the definition is "a something a greater than which cannot be conceived", I'm not convinced. There's the obvious comparison of "A number a larger than which cannot be conceived" - the idea is not coherent. — Banno

For the sake of argument I'm using the more simplistic definition "the demiurge of all possible worlds".

Hmm. The we are now a long way from Canterbury.

The we are now a long way from Canterbury. — Banno

Yes, I've already argued with others that Anselm's argument is invalid. I'm now trying to find the strongest kind of ontological argument. It's a more worthy topic of discussion.

Yep.

But "the demiurge of all possible worlds" might need some work...

Let Gx mean "x is God" and Fx mean "x created the world" (or anything else).

∃xGx ↔ ∃x□Fx
◇∃x□Fx
∴ ∃xGx

Proof

But given that ◇∃x□Fx ↔ ∃x□Fx is valid, ◇∃x□Fx begs the question.

Also as a counterargument:

∃xGx ↔ ∃x□Fx
◇¬∃xFx
∴ ¬∃xGx

Proof

But given that ◇¬∃xFx → ¬∃x□Fx is valid, ◇¬∃xFx also begs the question.

So at least with respect to the modal ontological argument there is no reason to believe either that God exists or that God doesn’t.

I'll use an argument formed from Anselm's text to show the argument isn't sound. It's taken from the English text of the Proslogion chapter 2. I think this is a better version of the argument, but it still doesn't work.

Premise (1) The fool hears and understands "a being than which none greater can be thought."

Premise (2) If the fool hears and understands, then it exists in his understanding.

Intermediate Conclusion: (3) The being than which none greater can be thought exists in the fool's understanding. Follows from (1) and (2) Modus Ponens

Premise (4) The being than which none greater can be thought exists in the understanding alone. (This would be the atheist's position according to Anselm.)

Premise (5) A being of which none greater can be thought can exist in reality.

Premise (6) A being that exists in reality is greater than a being existing in the understanding alone.

Intermediate Conclusion: (7) The being of which none greater can be thought existing in reality would be greater than the same being existing in the understanding alone. Follows from (5) and (6)

Final Conclusion: (8) A being than which none greater can be thought existing in reality would be greater than the same being existing in the mind/thought. Follows from (4) and (7)]

This presumably shows that the atheist position is contradictory, and therefore false. Why? Because the atheist's position is that a being than which none greater can be thought exists in the understanding alone. However, this cannot be so, because one could add a further attribute that would make it greater, viz., existing in reality. Thus, it can't exist in the understanding alone. It would be similar to saying that 10 is the greatest number. However, someone replies, no, I can add 1 to the number 10 and get 11, so 11 is the greatest number.

Since this argument is deductive it is valid. However, it must also be sound, i.e., the premises must be true, to be a good argument. The first premise with a problem is (2). How can a being exist in the understanding? Only a concept can exist in the understanding. We have the concept unicorn, but that doesn't mean unicorns are running around in my mind. Moreover, if you change the premise to only the concept existing in the understanding, then the argument is no longer valid. Why? Because if you add to the concept existing in reality you would still just have a concept existing in reality, not the being itself.

Finally, many find the argument dubious for other reasons, viz., trying to prove the existence of something from the concept alone, which others have pointed out in this thread, is very problematic to say the least.

As a plain language person I can't get past the word "greater". To my way of thinking, "greatness" implies some mechanism to measure some observable or measurable property. But even this definition falls apart outside strictly defined parameters. Who is the greatest artist? Who is the greatest athlete of all time? Who (or what) is the greatest (fill in the blank)? Any criteria you choose to measure "greatness" in these examples is arbitrary.

Perhaps a naive question here, but does the word "greater" have some special meaning/usage in a philosophical discussion apart from the plain language meaning/usage?

Why? Because if you add to the concept existing in reality you would still just have a concept existing in reality, not the being itself.

Finally, many find the argument dubious for other reasons, viz., trying to prove the existence of something from the concept alone, which others have pointed out in this thread, is very problematic to say the least. — Sam26

I believe Anselm is trying to distinguish between two different ideas, "understanding that something exists in reality" and "experiencing that something exists in reality." As he says in Chapter 2 "And so. O lord, since thou givest understanding to faith, give me to understand-as far as thou knows it to be good for me-that thou exist, as we believe, and that thou art what we believe thee to be."

Unlike the painter and painting example, where producing a painting is the reason he understands the painting exists in reality, it is the idea of "a being than which none greater can be thought" and its deductive implication that Anselm understands such a concept of a being "exists in reality." So, when you say, "....you would still just have a concept exist in reality, not the being itself.", what is this idea trying to express? That the deductive argument should produce some experience of "the being of God"? Demonstrate some experience we had corresponds to this idea of "a being than which none greater can be thought."? It is a deductive argument, it is about ideas. Geometric proofs are about ideas, which does not mean it will have any successful application in reality.

Whether the argument is sound, how can we fairly access this? How does one evaluate the "truth" of "God is a being than which no greater can be thought", and "Existing in reality" is greater than "existing in understanding" in order to determine soundness. What should we do, take a poll on how many people agree with these premises?

Maybe what this argument ultimately demonstrates is the vacuousness of using general concepts and deductive reasoning whether one thinks something exist or not.

That experience is the final arbiter.

Goats eat everything; therefore there is something that eats everything. therefore It is possible that something eats everything.

So you have a proof of the Great Goat:

Either it is not possible that something eats everything or it is necessary that something eats everything.
It is possible that something eats everything.
Therefore it is necessary that something eats everything.

And this we all call the Great Goat.

A few more small steps and we have that everything is a goat.

Goats eat everything; therefore there is something that eats everything. therefore It is possible that something eats everything.

So you have a proof of the Great Goat:

Either it is not possible that something eats everything or it is necessary that something eats everything.
It is possible that something eats everything.
Therefore it is necessary that something eats everything. — Banno

Well, it's not a proof, but it is a valid argument.

I've never liked this one. Possibly necessary => necessary doesn't seem, to me, to be an adequate model of what a Godlike "necessary being" would look like.

Roughly, possibility means "exists in one (connected) possible world", and necessary means "exists in all (connected) possible worlds". It isn't exactly an account of what it would mean for a God to be a necessary existent in a world where they exist.

If you read premise one as "If god exists (in a world) then god exists in all worlds connected to that world", that's quite different from "If god exists (in a world), then their existence in that world is an essential property of it". If we take the sense of necessity of God's existence as "one whose essence includes existence", that sense of essentiality does not resemble necessity as a quantifier over possible worlds. Why? Essentiality concerns one entity in the world it's in - as a property of that entity. Necessity concerns one entity's behaviour in all worlds.

In other news, it doesn't tell you if the necessary existent is a god. Just that if a predicate behaves like premise 1, then it exists in all possible worlds. Could be the goat.