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

Under S5 (one type of modal logic), the answer is "yes". — Michael

I'm very rusty in modal logic. How do you derive ('n' for necessary, 'p' for possible):

pnQ -> nQ

/

We start with:

Df. pQ <-> ~n~Q
therefore, nQ <-> ~p~Q

Ax. n(Q -> R) -> (nQ -> nR)

Ax. nQ -> Q

Ax. pQ -> npQ

And at least one easy theorem:

Th. Q -> pQ

How do you derive:

pnQ -> nQ


This is how far I get:

Suppose pnQ

Show nQ (or show ~p~Q)

Suppose ~nQ (or suppose p~Q) to derive a contradiction

?

The crank — Lionino

Hey, calling cranks 'the crank' is my schtick. Please don't steal my act!

◊~p → □◊~p (5 axiom)
◊~p → ~◊~◊~p (Definition of □)
~~◊~◊~p → ~◊~p (Contraposition)
◊~◊~p → ~◊~p (Double negation)
◊□p → □p (Definition of □)

Right. Thanks.

If, in S5, if god is possible then god is necessary, Gödel's ontological proof shows that god is not possible in S5.

Not what the Op wanted. :wink:

2. ◇∃x□(Fx ∧ Ax) ∴ ∃x□(Fx ∧ Ax)
3. ◇∃x□(Fx ∧ ¬Ax) ∴ ∃x□(Fx ∧ ¬Ax) — Michael

both 2 and 3 are valid under S5 — Michael

EDITED post:

I think I see how you got :

pEx(nQ) -> Ex(nQ)

(I'm using 'Q' instead of e.g. the more specific 'Fx & Ax'.)

I don't know the deductive system, but I guess this is a validity:

pEx(nQ) -> Ex(pnQ)

And we have:

pnQ <-> nQ

So we have:

pEx(nQ) -> Ex(nQ)

But you say that is in S5. But, as far as I know, S5 is merely a modal propositional logic.

If, in S5, if god is possible then god is necessary — Banno

S5 does not say that pQ -> nQ.

Or am I missing something in your context?

S5 does not say that pQ -> nQ. — TonesInDeepFreeze

It does say that ◊□p → □p. Hence if ~□p, it follows that ~◊□p.

If god is not necessary, then god is not possible. If god is not necessary, then god is not god.

While the coffee here is not strong enough, it does seem to me that if the ontological argument fails then there is something contradictory in the notion of god. God cannot be just possible. A contingent god is not god.

No, you are not correctly applying the formulas.

This is correct:

If it is not necessary that Q, then it is not possible that is necessary that Q.

That is not equivalent with your incorrect application:

If it is not necessary that Q, then it is not possible that Q.

If it is not necessary that Q, then it is not possible that is necessary that Q. — TonesInDeepFreeze

I bet you are fun at parties :wink:

Note that god is by all accounts necessary. Hence, a contingent god is not god. If it is not necessary that there is a god, then, as you say, it is not possible that it is necessary that there is a god...

Hence, if it is not necessary that there is a god, then there is no god.

This by way of setting out what is at stake for the theist - it's all or nothing.

(edit: hence, where Q is god, if it is not necessary that Q, then it is not possible that Q).

I bet you are fun at parties — Banno

I don't go to parties to talk about modal logic. Have your party hearty fun about the ontological argument. I'm not stopping you. I merely pointed out that the modal theorem you cited is not correctly applied as you did.

Cheers.

one or both of these is true:

4. ¬◇∃x□(Fx ∧ Ax)
5. ¬◇∃x□(Fx ∧ ¬Ax) — Michael

I think I'm with you that far. But I'm not sure what the following quotes mean or how they follow from the above quote:

Therefore we cannot assume that ◇∃x□Px is true for any logically consistent Px. — Michael

(What do you mean by 'logically consistent' rather than plain 'consistent'?)

Am I correct that by "we cannot assume pEx(nPx) is true for any logically consistent Px" you mean "For all consistent Px, we have that pEx(nPx) is not logically true"?

(I would think that to say "we cannot assume Q" means "We don't have sufficient basis to assume Q since Q is not logically true".)

or do you mean

"It is not the case that for all consistent Px we have pEx(nPx)"?

I surmise you mean the former, since:

we cannot assume that a necessary unicorn [...] is possible. — Michael

I take it that by a "A necessary unicorn is possible" you mean "It is possible that there is an x such that necessarily x is a unicorn". I.e. pEx(nUx).

Are you saying: If Ux is consistent, then pEx(nUx) is not logically true?

If I'm not mistaken, pEx(nUx) is not logically false:

Let Ux be Dx <-> Dx. So nUx. So Ex(nUx). So pEx(nUx).

If I understand correctly, you're saying that the first part of your argument (up to 5.) shows that if Ux is consistent then pEx(nUx) is not logically true? What is your argument for that?

If I understand correctly, you are saying that

(ExFx -> E!xFx) -> (~pEx(n(Fx & Ax)) v (~pEx(n(Fx & ~Ax))) (which seemed correct to me when I glanced over it)

implies

If Ux is consistent, then pEx(nUx) is not logically true

If that is what you're saying, then what is your argument?

/

P.S. I'm assuming we have "If Q is consistent then Q is not logically false".

The irony of Modal Logic is that there are so many alternatives to choose from, corresponding to the fact that Logic and a forteriori modal logic, has no predictive value per se. But modal theologicians aren't using Modal Logic to derive or express predictions, rather they are using Modal Logic to construct a Kripke frame with theologically desired properties. So ontological arguments aren't necessarily invalid for achieving their psychological and theological purposes, provided they aren't construed as claims to knowledge.

In fact, i'm tempted to consider Anselm's argument to be both valid and sound a priori, and yet unsound a posteriori. This is due to the fact that although our minds readily distinguish reality from fiction, I don't think that this distinction is derivable from a priori thought experiments.

I'll translate it into English for ease.

Neither of these are contradictions:

1. There exists a unique creator god who performs miracles
2. There exists a unique creator god who does not perform miracles

But they cannot both be true. Therefore, under S5, at least one of these is false:

3. It is possibly necessary that there exists a unique creator god who performs miracles
4. It is possibly necessary that there exists a unique creator god who does not perform miracles

Therefore, we cannot just assume that because some X is not a contradiction that it is possibly necessary. We need actual evidence or reasoning to support such a claim.

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. — Tarskian

It's not a mathematically unobjectionable proof.

In its simplest form it is:

◊p
p ≔ □q
∴ ◊□q
∴ q

But given the second line, this is equivalent to:

◊□q
∴ q

Which begs the question.

Hey, calling cranks 'the crank' is my schtick. Please don't steal my act! — TonesInDeepFreeze

I would usually use "chauvinist" instead but the subject-individual doesn't qualify as such.

A quick look through his profile will show you are wasting your time.

What do you think of this rendition in English (1-6) of the argument https://thephilosophyforum.com/discussion/comment/913745 to show where the circularity is?

If, in S5, if god is possible then god is necessary, Gödel's ontological proof shows that god is not possible in S5. — Banno

Well, that is the contention over the argument, innit. Some folks will insist that it proves God is necessary in S5.

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

Understood.

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. — Michael

To me, this is circumvented by D1, defining God as having all positive properties. That way, performing miracles is a positive property (or not, whatever our choice is). All positive properties are possibly exemplified (T1). So, if performing miracles is a positive property:

"Is it possible that there necessarily exists a God who is unique and performs miracles?"

Yes.

"Is it possible that there necessarily exists a God who is unique and does not perform miracles?"

Since this "God" does not have the positive property of performing miracles, let's call it entity instead.

"Is it possible that there necessarily exists an entity who is unique and does not perform miracles?"

We don't know. The question meaningfully boils down to "is there a being that necessarily exists?". Now:

"Is it possible that there necessarily exists a entity who is unique and has every positive property except performing miracles?"

The answer to that seems to be yes, because necessary existence is a positive property. So, there would an infinite amount of lesser gods each having all positive properties except one, except two, and so on.

This seems to be the reply that Sobel gives (source #7):

Sobel (1987), playing Gaunilo to Godel's Anselm, showed the argument could be applied to prove more than one would want.

So under these axioms, in S5, every possible positive property is exemplified in at least one being, meaning that necessarily there are innumerably many beings — every possible being with a certain set of positive properties necessarily exists. If there are n many positive properties (necessary existence being one of them), there necessarily are (n-1)! many beings; if n is infinite, there are infinitely many beings. This reminds me of modal collapse, which is implied by the argument put on the OP, and verified by computers that it does collapse.

To discuss the argument that does not imply modal collapse, we would have to discuss Anderson's and Fitting's, which I found be, at a first glance, impenetrable, especially when Fitting uses extensional properties rather than intensional (I don't know what the implication of that are and neither does Fitting by his own admission in his book).

To me, this is circumvented by D1, defining God as having all positive properties. — Lionino

Here are three different claims:

1. If X is God then X has all positive properties
2. If X has all positive properties then X is God
3. X is God if and only if X has all positive properties

Which of these is meant by "God is defined as having all positive properties"?

Hm, good question!

D1 uses ≡, so I will say 3: "if and only if".

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 — sime

I don't see where that is implied in the argument.

P(ψ)≡¬N(ψ) — sime

If N is supposed to mean necessary existence, that is a rejection of axiom 5.

Well, that is the contention over the argument, innit. Some folks will insist that it proves God is necessary in S5. — Lionino

I guess that is what is meant by

Hence, if it is not necessary that there is a god, then there is no god. This by way of setting out what is at stake for the theist - it's all or nothing. — Banno

So, X is God if and only if X has all positive properties.
Necessary existence is a positive property.
Being all powerful is a positive property.
Being all knowing is a positive property.
Therefore, X is God if and only if X necessarily exists, is all powerful, is all knowing, etc.

Now, what does "God possibly exists" mean? In modal logic we would say ◊∃xG(x) which translates to "it is possible that there exists an X such that X is God."

Using the definition above, this means:

It is possible that there exists an X such that X necessarily exists, is all powerful, is all knowing, etc.

But what does this mean? In modal logic we would say ◊□∃x(P(x) ∧ K(x) ∧ ...) which translates to "it is possibly necessary that there exists an X such that X is all powerful, is all knowing, etc."

Notice how "it is possible that there exists an X such that X necessarily exists ..." becomes "it is possibly necessary that there exists an X such that X ...". This step is required to make use of S5's axiom that ◊□p ⊢ p. But it also removes necessary existence as a predicate.

All we are left with is the claim that it is possibly necessary that there exists an X such that X is all powerful, is all knowing, etc. This is a claim that needs to be justified; it isn't true by definition.

Hence, if it is not necessary that there is a god, then there is no god. — Banno

Both this claim and the claim that God is necessary amuse/confuse me.

Imagine that some intelligent, all powerful, all knowing, creator of the universe actually does exist, but that because it doesn't necessarily exist then we refuse to call it God, as if the name we give it is what matters.

- It seems that @Banno understands better than you what the word "God" means.

Now, what does "God possibly exists" mean? In modal logic we would say ◊∃xG(x) which translates to "it is possible that there exists an X such that X is God."

Using the definition above, this means:

It is possible that there exists an X such that X necessarily exists, is all powerful, is all knowing, etc.

But what does this mean? In modal logic we would say ◊□∃x(P(x) ∧ K(x) ∧ ...) which translates to "it is possibly necessary that there exists an X such that X is all powerful, is all knowing, etc."

Notice how "it is possible that there exists an X such that X necessarily exists ..." becomes "it is possibly necessary that there exists an X such that X ...".

...

All we are left with is the claim that it is possibly necessary that there exists an X such that X is all powerful, is all knowing, etc. This is a claim that needs to be justified; it isn't true by definition. — Michael

Then the modal logic fails to translate, because <it is possible that there exists a necessary being> does not mean <it is possibly necessary that there is a being>. The former is an epistemic claim, and in my opinion the ◊ operator of modal logic does not capture this (others might argue that it is not epistemic, but I would still say that it is not represented by ◊). Logical possibility and epistemic possibility do not seem to me to be the same thing. When most people say, "It is possible that there exists a necessary being," what they mean is that there may exist a necessary being that they do not have knowledge of, for the necessity of some being does not guarantee knowledge of it (i.e. necessity does not preclude epistemic possibility).

Necessity opposes possibility on any given plane (logical, epistemic, theoretical, actual...). But epistemic possibility does not oppose logical necessity, or actual necessity, etc. Thus, supposing God exists, He is actually necessary (i.e. he is a necessary being), but it does not follow that he is epistemically necessary (i.e. that everyone knows He exists and is a necessary being). Thus someone who does not know that God exists is perfectly coherent in saying, "It is possible that God exists."

's point is well put but I would phrase it somewhat differently. Suppose there were a modal argument that proved God's existence. What would the hardened atheist say? "Why put so much faith in modal logic?" This is not wrong. Modal logic is derivative on natural language, and therefore to assent to an argument in modal logic that cannot be persuasively translated into natural language is to let the tail wag the dog. What I find is that most who dabble in modal logic really have no precise idea what the operators are supposed to mean ('◊' and '□'), and as soon as they try to nail them down other logicians will disagree. Is the nuance and flexibility of natural language a bug, or is it a feature?

So the English language claim that "God is defined as necessarily existing" is a deception. — Michael

You are letting the tail wag the dog. The problem isn't the English, it's the modal logic. Everyone who speaks English knows that things cannot be defined into existence. @Banno both understands the definition of God as necessarily existing and nevertheless denies his existence, and this does not make Banno incoherent.

-

Here is Aquinas:

Objection 2. Further, those things are said to be self-evident which are known as soon as the terms are known, which the Philosopher (1 Poster. iii) says is true of the first principles of demonstration. Thus, when the nature of a whole and of a part is known, it is at once recognized that every whole is greater than its part. But as soon as the signification of the word "God" is understood, it is at once seen that God exists. For by this word is signified that thing than which nothing greater can be conceived. But that which exists actually and mentally is greater than that which exists only mentally. Therefore, since as soon as the word "God" is understood it exists mentally, it also follows that it exists actually. Therefore the proposition "God exists" is self-evident.

Objection 3. Further, the existence of truth is self-evident. For whoever denies the existence of truth grants that truth does not exist: and, if truth does not exist, then the proposition "Truth does not exist" is true: and if there is anything true, there must be truth. But God is truth itself: "I am the way, the truth, and the life" (John 14:6) Therefore "God exists" is self-evident.

On the contrary, No one can mentally admit the opposite of what is self-evident; as the Philosopher (Metaph. iv, lect. vi) states concerning the first principles of demonstration. But the opposite of the proposition "God is" can be mentally admitted: "The fool said in his heart, There is no God" (Psalm 53:2). Therefore, that God exists is not self-evident.

I answer that, A thing can be self-evident in either of two ways: on the one hand, self-evident in itself, though not to us; on the other, self-evident in itself, and to us. A proposition is self-evident because the predicate is included in the essence of the subject, as "Man is an animal," for animal is contained in the essence of man. If, therefore the essence of the predicate and subject be known to all, the proposition will be self-evident to all; as is clear with regard to the first principles of demonstration, the terms of which are common things that no one is ignorant of, such as being and non-being, whole and part, and such like. If, however, there are some to whom the essence of the predicate and subject is unknown, the proposition will be self-evident in itself, but not to those who do not know the meaning of the predicate and subject of the proposition. Therefore, it happens, as Boethius says (Hebdom., the title of which is: "Whether all that is, is good"), "that there are some mental concepts self-evident only to the learned, as that incorporeal substances are not in space." Therefore I say that this proposition, "God exists," of itself is self-evident, for the predicate is the same as the subject, because God is His own existence as will be hereafter shown (I:3:4). Now because we do not know the essence of God, the proposition is not self-evident to us; but needs to be demonstrated by things that are more known to us, though less known in their nature — namely, by effects.

Reply to Objection 2. Perhaps not everyone who hears this word "God" understands it to signify something than which nothing greater can be thought, seeing that some have believed God to be a body. Yet, granted that everyone understands that by this word "God" is signified something than which nothing greater can be thought, nevertheless, it does not therefore follow that he understands that what the word signifies exists actually, but only that it exists mentally. Nor can it be argued that it actually exists, unless it be admitted that there actually exists something than which nothing greater can be thought; and this precisely is not admitted by those who hold that God does not exist.

Reply to Objection 3. The existence of truth in general is self-evident but the existence of a Primal Truth is not self-evident to us. — Aquinas, ST I.2.1 - Is the proposition that God exists self-evident? (NB: objection 1 and its reply omitted)

Note in particular, "it does not therefore follow that he understands that what the word signifies exists actually, but only that it exists mentally."

Then the modal logic fails to translate, because <it is possible that there exists a necessary being> does not mean <it is possibly necessary that there is a being>. — Leontiskos

Modal ontological arguments try to use modal logic to prove the existence of God. In particular, they use S5's axiom that ◊□p ⊢ □p.

At their most fundamental, their premises take the following form:

1. X is God if and only if X necessarily exists and has properties A, B, and C¹.
2. It is possible that God exists.

To prevent equivocation, we must use (1) to unpack (2), reformulating the argument as such:

1. X is God if and only if X necessarily exists and has properties A, B, and C.
3. It is possible that there exists some X such that X necessarily exists and has properties A, B, and C.

The phrase "it is possible that there exists some X such that X necessarily exists" is somewhat ambiguous. To address this ambiguity, we should perhaps reformulate the argument as such:

1. X is God if and only if X necessarily exists and has properties A, B, and C.
4. It is possibly necessary that there exists some X such that X has properties A, B, and C.

We can then use S5's axiom that ◊□p ⊢ □p to present the following modal ontological argument:

1. X is God if and only if X necessarily exists and has properties A, B, and C.
4. It is possibly necessary that there exists some X such that X has properties A, B, and C.
5. Therefore, there necessarily exists some X such that X has properties A, B, and C.

This argument is valid under S5. However, (4) needs to be justified; it is not true a priori.

If, as you claim, (3) and (4) are not equivalent, then prima facie one cannot derive (5) from (3), and so something other than S5 is required.

_{¹ The particular properties differ across arguments; we need not make them explicit here.}

Modal ontological arguments try to use modal logic to prove the existence of God... — Michael

You asked:

Now, what does "God possibly exists" mean? — Michael

You responded:

In modal logic we would say ◊∃xG(x) which translates to "it is possible that there exists an X such that X is God." — Michael

And I pointed out, among other things, that:

Then the modal logic fails to translate, because <it is possible that there exists a necessary being> does not mean <it is possibly necessary that there is a being>. — Leontiskos

The implications of the natural English propositions and the implications of the modal logic propositions diverge drastically, and it would be silly to prefer the modal logic to the natural English. That would be to let the tail wag the dog, as I argued (). Presumably Godel is making the same sort of error, equivocating on "possibility."

a. It is possibly necessary that there exists some X such that X created the universe — Michael

No one thinks creation was necessary. It seems that you have gotten your theology from Richard Dawkins.

The implications of the natural English propositions and the implications of the modal logic propositions diverge drastically, and it would be silly to prefer the modal logic to the natural English. That would be to let the tail wag the dog, as I argued — Leontiskos

I'm addressing modal ontological arguments. These arguments try to use modal logic to prove the existence of God.

No one thinks creation was necessary. It seems that you have gotten your theology from Richard Dawkins. — Leontiskos

It was just an example. Replace with "omnipotence", "omniscience", or whatever you want.

I'm addressing modal ontological arguments. These arguments try to use modal logic to prove the existence of God. — Michael

You literally said:

Now, what does "God possibly exists" mean? In modal logic we would say ◊∃xG(x) which translates to "it is possible that there exists an X such that X is God." — Michael

You asked what an English sentence means, and then you tried (and failed) to translate it into modal logic.

◊∃xG(x) is false given the fact that it denies what is true of God by definition. "God possibly exists" is not false, and it is not false precisely because it is an epistemic claim. Therefore your translation into modal logic fails. Modal logic is not capable of distinguishing the notion of necessity from the actuality of necessity, and that is precisely what is required in order to translate, "God possibly exists." Modal logic is not sophisticated enough to represent the claim, "A necessary being possibly exists." I explained why above ().

See the opening post, where Gödel's argument is presented. See line C:



These are the kinds of modal ontological arguments that I am addressing.