Consider this:

1. ∃xF(x) → ∃x∀y(F(y) ↔ (x = y))

If we take F(x) to mean something like "x is the only unicorn" then (1) is true.

Now consider these:

2. ◇∃x(F(x) ∧ A(x))
3. ◇∃x(F(x) ∧ ¬A(x))

If take A(x) to mean something like "x is male" then both (2) and (3) are true.

Now consider these:

4. ◇□∃x(F(x) ∧ A(x))
5. ◇□∃x(F(x) ∧ ¬A(x))

Under S5, ◇□p ⊢ □p, and so these entail:

6. □∃x(F(x) ∧ A(x))
7. □∃x(F(x) ∧ ¬A(x))

(6) and (7) cannot both be true, and so therefore (2) does not entail (4) and (3) does not entail (5):

8. ◇∃xP(x) ⊬ ◇□∃xP(x).

This is where modal ontological arguments commit a sleight of hand. To claim that it is possible that God₁ exists, where necessary existence is one of God₁'s properties, is to claim that it is possibly necessary that God₂ exists, where necessary existence is not one of God₂'s properties.

The claim that it is possibly necessary that God₂ exists isn't true a priori, and so the claim that it is possible that God₁ exists isn't true a priori. As it stands it begs the question.

Or we have to reject S5, but if we reject S5 then modal ontological arguments are invalid because “possibly necessary” wouldn’t entail “necessary”.

The true value of Gödel's work is not that it manages to reduce the belief in God to a belief in 5 complex axiomatic expressions in higher-order modal logic. The true value of Gödel's work is that it manages to prove that atheists will reject a mathematically unobjectionable proof if it proves something that they disagree with. Gödel was truly a genius.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

Just to add to Tim Wood's previous question,what is meant by the words "God" (or "God-like") and "positive"

'No sequence of words or of logical symbols, however cunningly arranged, can oblige the world to be thus and not so.'

Thus saith the unenlightened.

This is simply a sad fact of life for me, though God can famously speak, and it is so. God's words are infinitely more puissant than mine. He can speak me into existence, allegedly, but I cannot return the favour, and nor can Gödel.

God's words are infinitely more puissant than mine. He can speak me into existence, allegedly, but I cannot return the favour, and nor can Gödel. — unenlightened

:fire: :up:

Sorry for the sidetrack, this is just about Anselmian ontology:

If we suppose that existence and non-existence (the negation) can be properties of something, X, then what does it mean to say that X does not exist?
What was that X in the first place, then? :chin:
Either it's nonsense, or such a property already presupposes existence (implicitly) in some way, i.e. that X we spoke of that so happens to not exist.
As a starting point, I'm guessing that failure to differentiate imaginary/fictional and real can lead to reification; that certainly holds elsewhere.

By the way, in mathematics, a proper existential quantification form can be:
p = ∃x∈S φx
where p is the proposition, x is a (bound) variable, S a set, and φ a predicate.
Note that x is bound by S, and ∃ and φ aren't quite interchangeable.
Less confusion invited.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

And so forth. I cannot tell if the form of the argument is valid: if I convert it to truth tables, it is not. And what is meant here by "exist." — tim wood

Say that the following is provable from theory T:

xx and yy and zz --> rr

With xx, yy, zz the axioms of T.

What does that mean about rr?

In and of itself, such rr means nothing at all. It's just string manipulation.

The semantics, i.e.the truth about rr, lies elsewhere than in any of the syntactic consequences provable from T. Furthermore, it requires a specific mathematical process to unveil such semantics.

First of all, you must have some model-existence (or even soundness) theorem in T that guarantees that any provable theorem rr is indeed true in such models of T.

What is a model of T or even just a universe of T? How does it harness the truth of T?

From any (even arbitrarily) chosen metatheory, you need to construct a structure M, which is a set along with one or more operators. Every such structure M represents an alternative truth of T, i.e. a legitimate interpretation of T.

In other words, unveiling the truth cannot be done on the fly, between lunch and dinner. You also had better avoid non-mathematical methods of interpretation. They simply don't work.

It would cost an inordinate amount of work to correctly harness the truth of Godel's theorem.

This work has not yet been done at this point. The researchers have currently only spent time on investigating the consistency of his axioms and the issue of a possible modal collapse.

With this groundwork out of the way, it will still take quite a bit of time and work to develop a legitimate interpretation for Godel's theorem.

So, don't hold your breath!

I can personally certainly not do the work, because I am familiar only with PA's truth in its ZFC models. I actively avoid trying to interpret anything else, because these interpretations tend to be extremely confusing. When I accidentally get to see some advanced model theory, I run away.

This user has been deleted and all their posts removed.

Ignore the schizophrenic above. — Lionino

You do not understand enough mathematics to interpret the semantics of Godel's theorem. I have merely pointed out that you are clearly not even aware of that.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

This user has been deleted and all their posts removed.

So, the first question to consider is:

1. If something is possibly necessary, is it necessary?

Under S5 (one type of modal logic), the answer is "yes". Ontological arguments depend on this. They all reduce to the claim that because God is possibly necessary, God is necessary.

If we reject S5 then the answer is "no" and all ontological arguments fail.

But let's assume S5 and that the answer is "yes".

The next questions are:

2. Is it possible that there necessarily exists a God who is unique and performs miracles?
3. Is it possible that there necessarily exists a God who is unique and does not perform miracles?

If we accept S5 and if (2) and (3) are both true then it is both the case that there necessarily exists a God who is unique and performs miracles and that there necessarily exists a God who is unique and does not perform miracles.

This is a contradiction. Therefore, (2) and (3) cannot both be true.

Therefore, either:

4. It is not possible that there necessarily exists a God who is unique and performs miracles, or
5. It is not possible that there necessarily exists a God who is unique and does not perform miracles

Even though "God is unique and performs miracles" is not a contradiction, it might not be possibly necessary, and even though "God is unique and does not perform miracles" is not a contradiction, it might not be possibly necessary.

Therefore, one cannot claim that because some definition of God is consistent then it is possibly necessary.

Therefore, the claim that God is possibly necessary begs the question, and as such all ontological arguments fail.

Pretty clear why it wasn't published.

S5 is the logic of epidemics in which every possible world is infected by a virus whose transmission is symmetric and transitive.

As for Godel's argument, if we take the special case of his argument in which the positive properties P are taken to be the properties that are true for every possible individual, i.e by taking

$P(\psi) := \Box \forall x, \psi (x)$

and if we replace axiom A1 above with

$P(\psi) \equiv \neg N(\psi)$

where

$N(\psi) := \Diamond \exists x, \neg \psi (x)$

Then i expect that the resulting argument reduces to a trivial tautology of S5 in which all individuals are infected by the godliness virus.