Gödel's ontological proof of God
It seems there is no thread around Gödel's ontological proof of God, so, prompted by another thread, I will make one for such as it is high time.
If you don't know the history around the argument, here it is:
Ontological arguments in their modern sense go back to Sant'Anselmo d'Aosta. Monk Gaunillon of Tours makes a rebuttal often deemed as succcessful, parodying the argument to imply "the perfect island exists". Gaunillon argues we cannot go from idea to reality. Ontological arguments evolved through times.
Descartes writes his own argument, where necessary existence is part of the definition of God. When Descartes's argument is formalised, we find out that, if God's existence is possible, it is necessary; however the argument falls short of proving that God's existence is possible [7].
Leibniz somewhat accepts Descartes' argument, but identifies a critical issue it assumes the being is conceivable or possible [8]. Leibniz attempts to patch the argument by proving God's existence is possible.
Gödel writes his ontological argument, based on Leibniz's, which comes from Descartes' [7]. A manuscript can be seen here:
Anticipating an imminent death, Gödel shared his proof with Dana Scott. Although Gödel lived for 8 more years, he never published the proof. The details of the proof became known through a seminar led by Scott, whose slightly modified version was eventually made public.
As logic was not well-developed in Gödel's time as it is today, it is unsure what kind of logic Gödel had in mind when writing the proof, so others took it upon themselves to put the proof on firm grounds. Sobel plays Gaunillon to Gödel's Anselmo, wanting to show the argument could be applied to prove more than one would want [7].
Sobel also proves that the argument ends in modal collapse:
[hide]For the proof of modal collapse, let Q be some arbitrary truth. We will show that ?Q. We know, from Gödels theorem 3, that a Godlike being exists: call it j again. So, we know G(j). We also know, from theorem 2, that G is the essence of j. This means that G necessitates all of js actual properties. Since Q is true, j has the property of being such that Q (i.e., from (Q&j = j), we can deduce that j has the property x[Q&x = x]). Thus, being G must necessitate being such that Q. Since G is instantiated in every world, it follows that something is such that Q is true in every world. Hence, ?Q.[/hide]
Scott raises that Gödel's proof really amounts to an elaborate begging of the question. The proof itself is correct (if God is possible, it is necessary), but it breaks down at the same point as Descartes: proving that God is possible [7].
Software is developed that is able to verify arguments in higher-order logic. Gödel's ontological argument fails a consistency test.
Scott's is confirmed consistent for including a conjunct in D2, but it implies modal collapse (everything that is possible is also necesssary, and it has been argued this implies no free will) [1][9]. Here is Dana Scott's version of the argument:
Gödel's original axioms (without the conjunct) are proven to be inconsistent [4][5][6].
C. Anthony Anderson and Melvin Fitting then provide different versions of the consistent proof, to avoid modal collapse [7][10]. Fitting's version works with extensional properties, rather than intensional properties as is believed that Gödel had in mind [7].
Anderson along with Michael Gettings argues that his own emendation may be refuted by the same objection as Gaunilo raised against St. Anselmo [3].
Both Anderson's and Fitting's proofs are computer-verified to avoid modal collapse. Both proofs are verified as consistent [9].
Note: Gödel's original "positive properties" is to be interpreted in a moral-aesthetic sense only [2], it is possibly related to the Cartesian terminology of "perfections".
[1]
"Automating Gödel's Ontological Proof of God's Existence with Higher-order Automated Theorem Provers", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2014).
[2] K. Gödel, Appx.A: Notes in Kurt Gödels Hand, 144145
[3] "Gödel's Ontological Proof Revisited", C. Anthony Anderson and Michael Gettings (1996)
[4] "Experiments in Computational Metaphysics: Gödels Proof of Gods Existence", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2015)
[5] "An Object-Logic Explanation for the Inconsistency in Gödels Ontological Theory (Extended Abstract)", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2016)
[6] "The Inconsistency in Gödel's Ontological Argument: A Success Story for AI in Metaphysics", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2016)
[7] "Types, Tableaus, and Gödel's God", Melvin Fitting (Trends in Logic, Volume 12)
[8] "Two Notations for Discussion with Spinoza", Gottfried Leibniz (1676)
[9] "Types, Tableaus and Gödels God in Isabelle/HOL", David Fuenmayor and Christoph Benzmüller (2017)
[10] ""Some Emendations of Gödel's Ontological Proof", C. Anthony Anderson (1990)
[hide="Reveal"]The post will be updated eventually to include objections from scholars and replies.[/hide]
If you don't know the history around the argument, here it is:
Ontological arguments in their modern sense go back to Sant'Anselmo d'Aosta. Monk Gaunillon of Tours makes a rebuttal often deemed as succcessful, parodying the argument to imply "the perfect island exists". Gaunillon argues we cannot go from idea to reality. Ontological arguments evolved through times.
Descartes writes his own argument, where necessary existence is part of the definition of God. When Descartes's argument is formalised, we find out that, if God's existence is possible, it is necessary; however the argument falls short of proving that God's existence is possible [7].
Leibniz somewhat accepts Descartes' argument, but identifies a critical issue it assumes the being is conceivable or possible [8]. Leibniz attempts to patch the argument by proving God's existence is possible.
Gödel writes his ontological argument, based on Leibniz's, which comes from Descartes' [7]. A manuscript can be seen here:
Anticipating an imminent death, Gödel shared his proof with Dana Scott. Although Gödel lived for 8 more years, he never published the proof. The details of the proof became known through a seminar led by Scott, whose slightly modified version was eventually made public.
As logic was not well-developed in Gödel's time as it is today, it is unsure what kind of logic Gödel had in mind when writing the proof, so others took it upon themselves to put the proof on firm grounds. Sobel plays Gaunillon to Gödel's Anselmo, wanting to show the argument could be applied to prove more than one would want [7].
Sobel also proves that the argument ends in modal collapse:
[hide]For the proof of modal collapse, let Q be some arbitrary truth. We will show that ?Q. We know, from Gödels theorem 3, that a Godlike being exists: call it j again. So, we know G(j). We also know, from theorem 2, that G is the essence of j. This means that G necessitates all of js actual properties. Since Q is true, j has the property of being such that Q (i.e., from (Q&j = j), we can deduce that j has the property x[Q&x = x]). Thus, being G must necessitate being such that Q. Since G is instantiated in every world, it follows that something is such that Q is true in every world. Hence, ?Q.[/hide]
Scott raises that Gödel's proof really amounts to an elaborate begging of the question. The proof itself is correct (if God is possible, it is necessary), but it breaks down at the same point as Descartes: proving that God is possible [7].
Software is developed that is able to verify arguments in higher-order logic. Gödel's ontological argument fails a consistency test.
Scott's is confirmed consistent for including a conjunct in D2, but it implies modal collapse (everything that is possible is also necesssary, and it has been argued this implies no free will) [1][9]. Here is Dana Scott's version of the argument:
Gödel's original axioms (without the conjunct) are proven to be inconsistent [4][5][6].
C. Anthony Anderson and Melvin Fitting then provide different versions of the consistent proof, to avoid modal collapse [7][10]. Fitting's version works with extensional properties, rather than intensional properties as is believed that Gödel had in mind [7].
Anderson along with Michael Gettings argues that his own emendation may be refuted by the same objection as Gaunilo raised against St. Anselmo [3].
Both Anderson's and Fitting's proofs are computer-verified to avoid modal collapse. Both proofs are verified as consistent [9].
Note: Gödel's original "positive properties" is to be interpreted in a moral-aesthetic sense only [2], it is possibly related to the Cartesian terminology of "perfections".
[1]
"Automating Gödel's Ontological Proof of God's Existence with Higher-order Automated Theorem Provers", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2014).
[2] K. Gödel, Appx.A: Notes in Kurt Gödels Hand, 144145
[3] "Gödel's Ontological Proof Revisited", C. Anthony Anderson and Michael Gettings (1996)
[4] "Experiments in Computational Metaphysics: Gödels Proof of Gods Existence", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2015)
[5] "An Object-Logic Explanation for the Inconsistency in Gödels Ontological Theory (Extended Abstract)", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2016)
[6] "The Inconsistency in Gödel's Ontological Argument: A Success Story for AI in Metaphysics", Christoph Benzmüller and Bruno Woltzenlogel Paleo (2016)
[7] "Types, Tableaus, and Gödel's God", Melvin Fitting (Trends in Logic, Volume 12)
[8] "Two Notations for Discussion with Spinoza", Gottfried Leibniz (1676)
[9] "Types, Tableaus and Gödels God in Isabelle/HOL", David Fuenmayor and Christoph Benzmüller (2017)
[10] ""Some Emendations of Gödel's Ontological Proof", C. Anthony Anderson (1990)
[hide="Reveal"]The post will be updated eventually to include objections from scholars and replies.[/hide]
Comments (98)
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 God[sub]1[/sub] exists, where necessary existence is one of God[sub]1[/sub]'s properties, is to claim that it is possibly necessary that God[sub]2[/sub] exists, where necessary existence is not one of God[sub]2[/sub]'s properties.
The claim that it is possibly necessary that God[sub]2[/sub] exists isn't true a priori, and so the claim that it is possible that God[sub]1[/sub] 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 wouldnt entail necessary.
1 Being god-like is the essence of a god-like being
2 A god-like being has all positive properties
3 Being god-like implies having all positive properties
4 Necessary existence is a positive property
5 Being god-like implies having necessary existence
6 God exists because he has the property of necessary existence
We give a property to God (existence) before his existence is established. It reminds me of Descartes, simply defining God into existence. The reports on automated results however don't bring that up.
Besides, "essence" here is used strangely. It does not mean "the thing that defines X" or "the thing without which X is not itself", but seems to be "the thing from which all other features of X sprawl". Perhaps it is due to being a translation from German.
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.
:fire: :up:
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 ?x
where p is the proposition, x is a (bound) variable, S a set, and ? a predicate.
Note that x is bound by S, and ? and ? aren't quite interchangeable.
Less confusion invited.
See note. If being all-knowing is positive, being not-all-knowing is not positive. Beautiful, not beautiful.
Quoting tim wood
Everything that necessary follows from a positive property is also a positive property.
On A3, there is this from [3]:
Quoting jorndoe
Anselmo did reply to Gaunillon by basically saying the latter misses the point. The perfect island may be thought of as non-existant, while God, which is exactly the greatest being, may not. On the other hand, if God is complex, we may think of him as having all the attributes he does, besides existence. Then however, Anselmo affirmed the doctrine of divine simplicity, so on that point the counter does not work.
On divine simplicity:
Though I am not convinced by C.T. that divine simplicity has been out of favour, especially because of the sources given.
That is not a property, is it?
http://www.ptta.pl/pef/haslaen/a/anselm.pdf
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.
Ignore the schizophrenic above. He has been shown to be an ill-informed sophist in another thread several times. The argument as shown in the OP is verified as valid. You can't easily convert to truth tables.
Quoting EricH
The definition is given in D1, and see the note in OP.
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.
Oh, really?
Go ahead and solve the following operation:
[math] \bigtriangledown \times (2 x y, 2 y z, 2 x z) [/math]
Give the answer with the unit vectors specified, no parenthesis in the notation. On the left is the nabla operator, so we are clear.
Edit: The crank, despite online and active, hides when pressed to give an answer to an extremely basic vector calculus (an undergraduate subject for everyone in science) that one could do in one's head. That is all it takes to show the cranks claiming to know "mathematics" do not have surface knowledge of what they are babbling about.
The (p ? ¬q) ? ¬(p ? q) is invalid by the way, the left side is not the same as the right side.
Quoting tim wood
Yes, moral-aesthetic sense. What you quoted is me translating A2.
Correct.
Quoting tim wood
I did. The logical rendering of A1 is as it is written in the image. Your rendering is invalid because it can entail contradictions.
P. Q. ¬(p ? q):
0. 0. 0
0. 1. 0
1. 0. 1
1. 1. 0
P. Q. (p ? ¬q)
0. 0. 1
0. 1. 1
1. 0. 1
1. 1. 0
Therefore (p ? ¬q) ? ¬(p ? q) is false
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.
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
[math] P(\psi) := \Box \forall x, \psi (x) [/math]
and if we replace axiom A1 above with
[math] P(\psi) \equiv \neg N(\psi)[/math]
where
[math] N(\psi) := \Diamond \exists x, \neg \psi (x) [/math]
Then i expect that the resulting argument reduces to a trivial tautology of S5 in which all individuals are infected by the godliness virus.
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
?
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.
Not what the Op wanted. :wink:
Quoting 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.
S5 does not say that pQ -> nQ.
Or am I missing something in your context?
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.
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 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.
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:
Quoting 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:
Quoting 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".
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.
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.
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?
Well, that is the contention over the argument, innit. Some folks will insist that it proves God is necessary in S5.
Quoting Michael
Understood.
Quoting 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):
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).
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"?
D1 uses ?, so I will say 3: "if and only if".
I don't see where that is implied in the argument.
Quoting sime
If N is supposed to mean necessary existence, that is a rejection of axiom 5.
I guess that is what is meant by Quoting 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.
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.
Quoting Michael
Then the modal logic fails to translate, because
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?
Quoting 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.
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Here is Aquinas:
Quoting 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."
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 C[sup]1[/sup].
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.
[sub][sup]1[/sup] The particular properties differ across arguments; we need not make them explicit here.[/sub]
You asked:
Quoting Michael
You responded:
Quoting Michael
And I pointed out, among other things, that:
Quoting 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."
Quoting Michael
No one thinks creation was necessary. It seems that you have gotten your theology from Richard Dawkins.
I'm addressing modal ontological arguments. These arguments try to use modal logic to prove the existence of God.
Quoting Leontiskos
It was just an example. Replace with "omnipotence", "omniscience", or whatever you want.
You literally said:
Quoting 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.
Quoting Leontiskos
So we both agree that modal ontological arguments like Gödel's fail to prove the existence of God.
You asked readers to consider a formal argument you started. Since that was interesting to me, I considered it in detail as far as I could. The argument involves uniqueness, inferences in S5 and inferences with both quantification and modal operators. I asked questions whose answers might allow me to understand your locutions about the argument and to see that your argument would be completed. But then your answer is to just drop that formal buildup; moreover, to give an English argument that does't come close to the specifics of your previous formal argument. So I don't understand your point in your formalisms if you don't follow through with them; I don't see why I should have spent my time on them if you're just going to ditch them anyway.
But regarding your answer (I'm using 'Q' rather than 'G' or 'U' to steer clear of theological or fictive connotations):
If I understand (I've not read subsequent posts to your answer to me), your argument starts with: Q is consistent and ~Q is consistent, so S5 proves ~pnQ v ~pn~Q.
I can see that argument if these are theorems of S5:
Q -> ~pn~Q
~Q -> ~pnQ
Are they? If not, then what is the argument that "Q is consistent and ~Q is consistent" implies that S5 proves ~pnQ v ~pn~Q?
Then you say, "Therefore, we cannot just assume that because some X is not a contradiction that it is possibly necessary."
I take that to mean: "Q is consistent" does not imply S5 |- pnQ.
S5 has as an axiom that ??p ? ?p.
Therefore, under S5, these cannot both be true:
1. ??q
2. ??¬q
Therefore, under S5, this is not true:
3. ¬?¬p ? ??p
This then relates to the post above.
Assuming that (a) means (b), (b) needs to be justified. Given that (3) is false, this is false:
4. ¬?¬?xC(x) ? ???xC(x)
So ???xC(x) must be justified some other way for a modal ontological argument to work.
"Q"?
But It is difficult to follow you as you jump around among very different formal formulations and among different English formulations and different kinds of examples. I started out trying to sort out your original argument as originally formulated but now you've twice jumped to different, though related, formulations. I'm giving up for now. It would help if you would give one self-contained argument with transparent inferences from start to finish.
The claim that ???xP(x) where P is every positive property besides necessary existence.
The claim comes T1 and D1. A God possesses all positive properties. Positive properties are possibly exemplified. Then we have C. Being that necessary existence is a positive property, and it is possibly exemplified, we end up with ???xG(x). Your argument seems to be that, to take advantage of S5, for ??p ? p, we must pull the predicate NE out of the variable, so we end up with ???xP(x) where P is every positive property except necessary existence, and since necessary existence has been pulled out, we don't know whether ???xP(x) is true. Is that right?
I have seen elsewhere that some think that D1 is basically Zermelo-Russell's paradox. I would say it doesnt imply the paradox. The set of all sets argument is paradoxical precisely because its universal. D1 however is restrictive, all positive properties isnt the same as all properties. Having all of something isnt an issue, an all-colourful being can be defined as having all colours, such a being is not only possible but also exist in real life. Stars emit all frequencies in the visible light spectrum.
Then that, for A3 to be valid, D1 cannot yield paradoxes (violation of LNC). I would say it doesnt end up in paradox because of A1 it makes sure that no positive properties conflict with each other. For example, being colourless is the denial of being colourful, only one of those is positive. Being transparent ?? being colourless, therefore being transparent is also the denial of being colourful (this example ties in with A2).
The explanation of the argument here presents the problem more clearly.
I think the previous argument did that? Perhaps you could let me know which line(s) you'd like me to explain further?
The proof we are discussing is Gödel's proof. From what I've read, epistemic modality has nothing to do with the proof, the proof uses S5. On the contrary, Gödel is trying to prove exactly something it is clear we have no clue about.
Quoting Leontiskos
Many-worlds semantics.
But then:
Quoting Leontiskos
Such is the life of the philosophuck.
Quoting Leontiskos
That would be out of scope just like discussing whether S5 obtains or whether existence is a predicate. We can make a thread for ontological arguments in general.
Your transition to from 3 to 4 is basically ??x(P(x)?NE(x)) ? ???xP(x).
Your reply to Leontiskos fights with the definition of God including necessary existence or not. I don't think it has to include it. As you said:
Quoting Michael
I don't see any issues because, differently from others, for me your 3 and 4 are equivalent. Once we establish ??x(P(x)?NE(x)), we can then claim ???xP(x). The issue with this transition, as I pointed, is that it invites a Gaunillon counter-argument: Quoting Lionino
The first table by the way is a NOR gate with an inverted P, and the second table a NAND gate. Physically different.
Constructively speaking, an existential proposition is proved by constructing a term that exemplifies the proposition, as per the Curry Howard Isomorphism. Classically speaking, an existential proposition can also be derived by proving that it's negation entails contradiction, as per the law of double negation.
In Godel's proof however, he defines a so-called Godliness predicate G, where as usual ~G(x) corresponds to the principle of explosion
G(x) --> B(x)
G(x) --> ~B(x)
where B is any predicate.
But in Godel's case, he defines G as only implying properties that satisfy a second-order predicate he calls "Positivity", which is a predicate decreeing that G(x) --> B(x) and G(x) --> ~B(x) cannot both be true.
So in effect, Godel crafted a non-constructive proof-by-absurdity that implies the existence of a god term on the basis that non-existence otherwise causes an explosion! this is in stark contrast to the normal constructive situation of proofs-by-absurdity in which a term exemplifying a negated existential proposition is constructed in terms of a function that sends counterexamples to explosions.
The rest of Godel's proof is unremarkable, since he defined G as implying it's own necessity, meaning that if G is said to be true in some world, then by definition it is said to be true of adjacent worlds, which under S5 automatically implies every world.
My questions were here:
https://thephilosophyforum.com/discussion/comment/914470
Your response was to switch to a different description of your idea.
We could start with the first question:
Quoting TonesInDeepFreeze
We can assume anything. So I take it "cannot assume" is colloquial for something more logically definite. Thus my question above.
Also, you have a modal operator after a quantifier. I don't think S5 can do anything more with that than to regard the quantified formula as just a sentence letter, so S5 sees pEx(nPx) as just pQ.
about what?
Yes, good catch. I should have used ???xP(x).
Quoting TonesInDeepFreeze
What I am saying is that ??xP(x) ? ???xP(x), i.e "it is possible that X exists" does not entail "it is possibly necessary that X exists".
I hope it won't be too long that I'll have time to resume going over your argument with the emendations.
N was supposed to mean the possibility modality (N standing for Negative Properties, in order to stand for the opposite of Positive Properties). The question here I was interested in, is how to give a syntactical definition of Positive Properties such that the resulting argument follows as a valid tautology in some modal logic. This was partly in order to help clarify the the definitions Godel provided, even his assumptions need to be altered slightly and the resulting argument and its conclusion aren't quite the same.
For example, taking Positive properties to refer to what is necessarily true of all individuals in every possible world, turns Axiom A2 into the definition of a functor, which is rather tempting. It also makes the possibility of god follow as a matter of tautology.
Also, Godel's definition of essences seems close to the definition of the Categorical Product. So why not take the essence of an individual to be the conjunction of his properties?
One thing I overlooked was that God was defined as referring to the exact set of positive properties, which would mean that according to my definition of P, all individuals would be identical. But then supposing we weaken the definition of "Godliness" to refer to a set that contains all the positive properties and possibly some of the negative (i.e contigent) ones?
I think there is quite a few pedagogically useful questions here.
If it is worth something, ??x(?Px) ? ??x(Px) is valid in S5.
I don't understand that proof.
Where can I see a specification of S5 extended to a deduction calculus with quantifiers?
I don't know what deduction in S5 permits:
inferring line 4 from line 3. (~nQ does not imply ~Q)
inferring line 5 from line 2. (pQ does not imply Q)
line 6 from line 5 is existential instantiation applied to a modal formula, but S5 is only a modal propostional logic.
I don't understand. You said a certain formula is valid in S5. The proof generator shows a deduction of the formula. But I can't make sense of the deduction at the lines I mentioned. The proof generator makes no mention of exemplification and positive. Bringing in exemplification and positive does not address my points. And I'm not even talking about Godel. I'm just looking at certain claims about what is derivable in S5, as those claims don't invoke exemplification or positive.
Are you waiting on me for something else or are you saying that you're currently too busy to examine what I've said?
I'm saying that I'll take your latest note and incorporate it as I go over your argument again. Not waiting on you.
Thanks.
For the record, I don't think the criticism, in the post you referred to, is successful. I developed why in my post last page.
I don't know which of my posts or comments you are commenting on.
In a recent post, I said that I don't understand the proof at the proof generator.
I'm not stating a criticism of Michael's posts. I'm just trying to figure them out.
Trying to put this in logical terms, I think it would be ??[??x?(x)]. From source [3]:
Consider this: non-existent beings dont age, suffer and die. And because they transcend time and space, non-existent beings arent restricted by the laws of physics. In fact, non-existent beings are not adversely affected by anything in the universe including hatred, discrimination, war, ignorance and greed. Taken one step further, if God exists (or even if only the idea or concept of God exists), then perhaps God (or the concept of God) values non-existence over (necessary) existence. Why does Anselmo or Descartes or Gödel get to decide what God (or the concept of God) values most?
This is what Ive always found troubling about Pascals Wager. Pascal argued that belief in God will get you into heaven after you die if God does exist. And yet, Pascal continued, you wont be worse off by believing in God if God doesnt exist after all; your death will be met with the same fate whether you believe in God or not. So you might as well believe in God.
But Pascal is assuming (begging the question) that one of Gods characteristics is rewarding believers after death. But what if God rewards those who dont believe? Maybe God prefers critical thinkers over those who dogmatically follow religious tenets. Why does Pascals assumption of God-rewards-those-who-believe-God-exists take precedence over someone elses assumption of God-rewards-those-who-dont-believe-God-exists?
And so it is with any ontological proof of God whether it be valid or not, sound or not, or well-argued or not. Maybe existence is not the positive quality its cracked up to be. (?)
If I haven't made any mistakes here:
At least for me, this is more exact and clear:
(1) E!xFx ... premise
(2) pExAx ... premise
(3) pEx~Ax ... premise
(4) {(1), (2), (3)} is consistent
(5) pE!x(Fx & Ax) ... (1),(2)
(6) pE!x(Fx & ~Ax) ... (1),(3)
(7) pnEx(Fx & Ax) -> nEx(Fx & Ax) ... theorem
(8) pnEx(Fx & ~Ax) -> nEx(Fx & ~Ax) ... theorem
(9) {(1), (2), (3)} |/- pnEx(Fx & Ax) ... (1),(4),(6),(7)
(10) {(1), (2), (3)} |/- pnEx(Fx & ~Ax) ... (1),(4),(5),(8)
* But the inferences at (5) and (6) are invalid (according to the validity checker).
https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1~9~7xAx))~5~9(~7x(~6y((Fy~1Ay)~4x=y)))||universality
https://www.umsu.de/trees/#(~7x~6y(Fy~4x=y)~1~9~7xAx)~5~9~7x~6y((Fy~1Ay)~4x=y)
"There exists exactly one falcon, and it possible that there exists a non-falcon" doesn't entail "It is possible that there exists exactly one falcon that's a non-falcon".
and
"There exists a falcon, and it possible that there exists a non-falcon" doesn't entail "It's possible that there exists a falcon that's a non-falcon".
But
{(1), (2), (3)} |/- pnEx(Fx & Ax)
and
{(1), (2), (3)} |/- pnEx(Fx & ~Ax)
are correct anyway (according to the validity checker). Just not by your argument.
https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1(~9~7xAx~1~9~7x~3Ax)))~5~9~8~7x(Fx~1Ax)||universality
* I don't see the relevance of this to your specific argument:
https://www.umsu.de/trees/#~9~7xP(x)~5~9~8~7xP(x)||universality
What you want to prove is not just that that formula is invalid but to prove:
{(1), (2), (3)} |/- pnEx(Fx & Ax)
and
{(1), (2), (3)} |/- pnEx(Fx & ~Ax)
Those are correct (according to the validity checker). Just not by your argument.
(1) E!xFx ... premise
(2) pEx(Fx & Ax) ... premise
(3) pEx(Fx & ~Ax) ... premise
(4) {(1), (2), (3)} is consistent
(5) pnEx(Fx & Ax) -> nEx(Fx & Ax) ... theorem
(6) pnEx(Fx & ~Ax) -> nEx(Fx & ~Ax) ... theorem
(7) {(1), (2), (3)} |/- pnEx(Fx & Ax) ... (1), (3), (4), (5)
(8) {(1), (2), (3)} |/- pnEx(Fx & ~Ax) ... (1), (2), (4), (6)
https://www.umsu.de/trees/#((~7x~6y(Fy~4x=y)~1(~9~7x(Fx~1Ax)~1~9~7x(Fx~1~3Ax))))~5~9~8~7x(Fx~1Ax)||universality
I think you've misunderstood these:
1. ??x(F(x) ? A(x))
2. ??x(F(x) ? ¬A(x))
They say:
1. It is possible that there exists some X such that X is the only unicorn and is male
2. It is possible that there exists some X such that X is the only unicorn and is not male
They are not inferences but independent premises and might both be true.
My argument is that we cannot then infer these:
3. ???x(F(x) ? A(x))
4. ???x(F(x) ? ¬A(x))
Which say:
3. It is possibly necessary that there exists some X such that X is the only unicorn and is male
4. It is possibly necessary that there exists some X such that X is the only unicorn and is not male
Under S5 they cannot both be true.
This matters to modal ontological arguments because (3) and (4) are equivalent to the below:
3. It is possible that there exists some X such that X is the only unicorn and is male and necessarily exists
4. It is possible that there exists some X such that X is the only unicorn and is not male and necessarily exists
The switch from "possibly necessary that there exists some X" to "possible that there exists some X such that X necessarily exists" is a sleight of hand. It is used to disguise the fact that asserting the possible existence of God where necessary existence is a property of God begs the question.
You wrote in the argument:
Quoting Michael [emphasis added]
So in my first post I captured that implication.
And in my second post I gave a version in which instead they are premises:
Quoting TonesInDeepFreeze