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” 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.
'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.
unenlightened: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.
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.
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."
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.
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.
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.
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
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.
If it is not necessary that Q, then it is not possible that is necessary 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.
Therefore we cannot assume that ??x?Px is true for any logically consistent Px.
(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)"?
we cannot assume that a necessary unicorn [...] is possible.
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.
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.
To me, this is circumvented by D1, defining God as having all positive properties.
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"?
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.
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.
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.
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."
Reply to unenlightened'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.
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.
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.
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>.
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]
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 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 (Reply to Leontiskos). Presumably Godel is making the same sort of error, equivocating on "possibility."
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
I'm addressing modal ontological arguments. These arguments try to use modal logic to prove the existence of God.
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."
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 (Reply to Leontiskos).
Reply to Michael
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.
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.
I do see now that to show that (3) is not the case, we need rely only on pnQ -> nQ and the fact that it is not the case that pQ |- nQ.
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.
I think the most remarkable and amusing part of Godel's argument, is in the beginning before the use of modal logic, in which he argues for the existence of a 'god term' by turning the principle of explosion on its head.
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.
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"?
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.
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"?
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 don't see where that is implied in the argument.
P(?)?¬N(?) — sime
If N is supposed to mean necessary existence, that is a rejection of axiom 5.
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.
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.
Isn’t all of this just begging the question? I mean, are we not allowed to challenge the assertion that “necessary existence” is a “positive quality?” Isn’t it possible for necessary existence to be a negative quality? After all, human beings exist, and they are imperfect and mortal; they make mistakes, they sin, etc. So maybe “non-existence” (as opposed to “necessary existence”) is the (more) positive quality.
Consider this: “non-existent” beings don’t age, suffer and die. And because they transcend time and space, non-existent beings aren’t 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 I’ve always found troubling about Pascal’s “Wager.” Pascal argued that belief in God will get you into heaven after you die if God does exist. And yet, Pascal continued, you won’t be worse off by believing in God if God doesn’t 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 God’s characteristics is rewarding believers after death. But what if God rewards those who don’t believe? Maybe God prefers critical thinkers over those who dogmatically follow religious tenets. Why does Pascal’s assumption of “God-rewards-those-who-believe-God-exists” take precedence over someone else’s assumption of “God-rewards-those-who-don’t-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 it’s cracked up to be. (?)
"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.
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.
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” wouldn’t entail “necessary”.
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.
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.
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.
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.
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"?
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 <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?
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.
-
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 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?
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.
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.
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.
Consider this: “non-existent” beings don’t age, suffer and die. And because they transcend time and space, non-existent beings aren’t 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 I’ve always found troubling about Pascal’s “Wager.” Pascal argued that belief in God will get you into heaven after you die if God does exist. And yet, Pascal continued, you won’t be worse off by believing in God if God doesn’t 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 God’s characteristics is rewarding believers after death. But what if God rewards those who don’t believe? Maybe God prefers critical thinkers over those who dogmatically follow religious tenets. Why does Pascal’s assumption of “God-rewards-those-who-believe-God-exists” take precedence over someone else’s assumption of “God-rewards-those-who-don’t-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 it’s 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