What Is Philosophy: An Ontological Proof of the Existence of Philosophy, or: Gödel on the Existence of God (Column 160)
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With God’s help
What Is Philosophy: An Ontological Argument for the Existence of Philosophy,
or: Gödel on the Existence of God
Dedicated to my dear sons Nachman and Shlomo
My son Nachman sent me the following link, which presents the proof proposed by the logician and mathematician Kurt Gödel for the existence of God. This is a good opportunity to apply a bit of what we have seen, and at the same time to analyze a work by a mathematical genius, which is sometimes taken as a formal joke,[1] but it seems to me that he meant it in complete seriousness. And when Gödel says something serious about matters of logic, all of us would do well to take him with the utmost seriousness.
Our discussion here provides a fitting conclusion to the series of columns dealing with philosophy. In column 155 we saw that mathematics is a branch of philosophy, and here we shall see what that means, both its lights and its shadows.
This column is also an elementary lesson in logical formalism. I know that quite a few people are put off by it, but I nevertheless suggest not giving up in advance. I am trying to explain things so that every reader can understand them. Whoever arms himself with patience and is willing to go through the material systematically can expect a fascinating surprise about my entire line of thought in the last few columns. Believe me, I am not just pestering you. It will be genuinely interesting!
Kurt Gödel was one of the greatest mathematicians and logicians of all time. His principal achievement (though not his only one) is his completeness and incompleteness theorems in logic, which have aroused much interest in the philosophical community as well (which itself has provoked much criticism in the mathematical community, since many claim that some philosophers do this without understanding what they are talking about.[2] Many mathematicians argue that Gödel’s theorems do not have especially important philosophical implications beyond the philosophy of mathematics. I, in all my smallness, disagree with them, but who knows—perhaps I too understand nothing…).
They say that the only friend Gödel had in his life was Einstein, whom he met after arriving in the United States at the Institute for Advanced Study in Princeton (both had fled there from the Nazis). Gödel was not in the habit of expressing views for which he had no proof, and apparently he had a proof for his views about God; that is what we shall discuss here.
To understand what sort of person we are dealing with, let me tell you that two of his friends, Oskar Morgenstern (the well-known mathematician from game theory) and Einstein, took him to his interview with the American immigration authorities, for which he had to study the Constitution and American history. Gödel, of course, studied those subjects thoroughly in preparation for the interview (poor interviewer), and excitedly told Morgenstern that he had discovered a contradiction in the American Constitution (one of his concerns in logic was the consistency of systems). He insisted that it opened the door to dictatorship. Fortunately, his two friends, who were somewhat more grounded than he was, persuaded him to spare the consul his fascinating discovery, and thus Gödel received American citizenship.
Try tangling with someone like that. I must say that your faithful servant experiences a similar frustration whenever a police officer stops him (it has happened before, to my shame) for speeding or for not wearing a seatbelt, and lectures him on how he is not sparing his own life and the lives of others. As a grounded person, I know there is no chance that I could explain to the honorable officer the simple philosophical fact that endangering life is an arbitrary normative line that can be placed wherever one wishes, and above all that the legislator and the police understand this no better than I do (see here). And no, this is not merely a matter of lacking time.☺ Ask any philosopher, and he will immediately tell you that police officers and judges (and apparently American consuls too) are two communities among whom you will find a disgraceful lack of interest in philosophy (unlike taxi drivers and barbers/hairdressers, for example).
Grounded or not, if I had to wager, I would bet a hundred to one that at least on the logical level there really is a contradiction in the American Constitution. Now that we know whom we are dealing with, we can turn to Gödel’s proof of the existence of God.
The ontological argument and ontological arguments in general
The ontological argument is one of the oldest arguments for the existence of God. Its first formulation is attributed to Anselm of Canterbury in the twelfth century, and after him it received quite a few further formulations, refutations, and lively philosophical discussions. You can find an extended discussion of the argument in the first booklet here on the site (and at the beginning a short account of its history, which does not include some of its formal formulations, one of which we shall discuss here).
The essence of Anselm’s argument is that if we define God as the most perfect being that can be conceived (possessing all positive properties, those that constitute perfection), then his existence can be proved by reductio. If we assume that he does not exist, that assumption will lead us to a contradiction. If he does not exist, then one can conceive of a being more perfect than he is (that same being with those same properties, except that it also exists), which contradicts its definition as the most perfect being that can be conceived. In simple terms, the claim is that a perfect being must also exist.
If this sounds to you like a foolish argument, you are in good company. Many excellent thinkers think so, but as I showed in the booklet, they are mistaken. This argument is very far from foolish, and as Bertrand Russell said (one of the most prominent atheists of the twentieth century), it is much easier to shout that it is foolish than to put one’s finger on what is wrong with it.
As Kant explained, one of the fundamental problems with this argument is its ontological character. Anselm claimed that this was nothing but conceptual analysis, that is, an argument without premises. One starts from a definition of a concept, and logical analysis of the definition yields the conclusion that God exists. But philosophers are unwilling to accept an ontological argument, that is, an argument with no premises, and in particular no factual premises, whose conclusion is a factual claim (that God exists). In their view, such a thing cannot happen.
The problem with this objection is that it does not point to the place where the argument fails, but merely declares that it fails. If the argument is in fact logically valid, then we have before us a living example that an ontological argument is indeed possible. Here you have a logically valid argument that proceeds from definitions of concepts, without any factual premise, and derives from them a factual conclusion. Such criticism is a declaration, not an argument, and it ignores a fact that stands before it. It simply assumes that an ontological argument cannot exist, and when such an argument is presented to it, it merely declares that this cannot be so (in the spirit of: do not confuse me with facts). Serious philosophical-logical criticism must also point to a failure in the logic of the argument. Only after a failure is found, if one is found at all, can one claim that in principle an ontological argument is impossible.
As stated, Kurt Gödel presented a formal version of this proof. Those familiar with the material will immediately understand that this is not a formalization of Anselm’s argument, but a different ontological argument for the necessary existence of God. In any case, discussion of the formal version helps us sharpen the discussion of the proof and of ontological arguments in general. At the end of this column we shall see that this also sheds surprising light on the question of philosophy in general (which has occupied us in the last few columns). But first, a brief introduction to the formalization of arguments.
On formalizing philosophical arguments
Formalizing a philosophical argument means translating it into a formal logical language. Such a translation has advantages and disadvantages. On the one hand, it conceptualizes and sharpens the argument, exposes all its premises, and helps us focus on what is essential while removing what is secondary. On the other hand, a formal formulation of a philosophical argument is dangerous, because it appears necessary and persuasive with mathematical certainty, whereas that is not necessarily the case. First, at the base of the argument there are premises, and in order to form a view of its conclusion one must examine one’s attitude to those premises. Second, people tend to forget that the process of formalization (the translation of the argument into formal logical language) is itself based on premises and on various interpretations of the argument, and there there may be problems and refutations that do not arise in the discussion of the validity of the formal argument itself. In many cases mathematical formalism simply hides part of the philosophy, and very often precisely its problematic part.[3]
Gödel’s proof
The formulation proposed by Gödel is the following:[4]
Ax. 1. {P(φ)∧□∀x[φ(x)→ψ(x)]}→P(ψ)
Ax. 2. P(¬φ)↔¬P(φ)
Th. 1. P(φ)→◊∃x[φ(x)]
Df. 1. G(x)⟺∀φ[P(φ)→φ(x)]
Ax. 3. P(G)
Th. 2. ◊∃xG(x)
Df. 2. φ ess x⟺φ(x)∧∀ψ{ψ(x)→□∀y[φ(y)→ψ(y)]}
Ax. 4. P(φ)→□P(φ)
Th. 3. G(x)→G ess x
Df. 3. E(x)⟺∀φ[φ ess x→□∃yφ(y)]
Ax. 5. P(E)
Th. 4. □∃xG(x)
Convinced? Forward this by email to Amnon Yitzhak.
Note the difference in length between my first booklet and this little box (although it is not so short compared to the two chapters of the Proslogion cited in the first booklet). This is the advantage of formalism. But as I explained, and as we shall also see below, if you think this makes philosophical discussion unnecessary, you are simply mistaken.
I will now offer a more detailed explanation of this formulation, including an explanation of the logical symbols for those who are less familiar with this language. I shall begin with the notation, and then go through the proof step by step. In the course of explaining the proof, the use of the symbols and their meaning will also become clearer. And again, I strongly recommend following along, because at the end there is a surprise for everyone who has stayed with me until this point.
Explanation of the notation
- This proof contains three kinds of lines (axiom, theorem, and definition):
Ax – this is an axiom (Ax. 1. is axiom no. 1).
Th – this is a theorem.
Df – this is a definition.
- The basic statement used by this formalization has the form of a function: φ(x). x is some object (or concept; see below), and φ is a property (a predicate, in the standard terminology of logic). This functional notation means that object x has the property φ (it is endowed with the property φ). Likewise, ψ(x) means that x has the property ψ.
- There is another relation here between a property and an object, defined here (Definition 2): ess, meaning essential. When we say φ ess x, we mean to say that the property φ is essential to x (in a certain sense defined there. We shall see this in the explanation of the argument).
- We need here four logical operators (and, implication, negation, and equivalence):
W∧T – W and T.
W→T – W implies T (that is, it is impossible for W to be true and T false).
¬φ – the negation of φ.
W⟺T – equivalence between W and T, that is, if W is true then T is true, and vice versa.[5]
- There are two quantifiers, that is, symbols of quantity (the universal quantifier and the existential quantifier):
Universal quantifier ∀ – meaning ‘for every’. [∀x[φ(x) means that φ(x) holds for every x. In other words: all x’s have the property φ. The square brackets are not necessary.
Existential quantifier ∃ – meaning ‘there exists’. ∃x[φ(x)] means that there exists an x with the property φ.
There is a logical relation between these two quantifiers (dealt with by Boethius’s square of opposition, which also appears in __Words of Logic__ by Maimonides): if every x has the property φ, then there is no x that does not have that property. And if there exists an x that has that property, then it is not true that every x lacks that property.
Formally, such a relation can be denoted as follows: ¬∀x[φ(x)] ⟺ ∃x[¬φ(x)]
Meaning: the statement ‘It is not true that every x has the property φ’ is equivalent to the statement ‘There exists an x that has the property not-φ’ (that is, that does not have the property φ).
- There are two modal operators (necessary and possible):
□T – it is necessary that T.
◊T – it is possible (contingent) that T.
There are also logical relations between these two: ¬◊φ(x) ⟺ □[¬φ(x)]
Meaning: ‘It is not possible that φ’ is equivalent to ‘It is necessary that not-φ’.
One must understand that the claim ‘There is light outside now’ is true, but not necessarily true. Therefore, from the standpoint of modal logic it is only possible (there are worlds one can imagine in which this claim would not be true), but not necessary. By contrast, the claim ‘It is impossible for both X and not-X to be true’ (the law of non-contradiction), or the claim ‘Every bachelor is unmarried,’ are necessarily true (that is, they hold in every world one can conceive).
- The use of parentheses is intended to determine the order of the logical operations, and is familiar to all of us.
- In this argument there are three logical predicates (divinity, necessary existence, and positivity):
G – having divine properties. Thus G(x) means that x is an object with divine properties (that is, it is God).
E – exists necessarily (it is impossible that it not exist). Thus E(x) means that x exists necessarily.
P – positive (in the sense of good, perfect; positive). P is a second-order predicate, that is, it is a property of predicates (and not of objects, x). Thus P(φ) means that the property φ is positive (that is, a kind of perfection).
Explanation of the argument
We shall now go through the argument step by step and explain it.
The first axiom:
{P(φ)∧□∀x[φ(x)→ψ(x)]}→P(ψ)
There is here a structure of logical implication whose antecedent is a conjunction of two claims (one of which is itself an implication). The meaning of the axiom is that if it is the case both that the property φ is a perfection, and that necessarily, whenever this property characterizes the object x, the property ψ also characterizes it, then the property ψ is also a perfection.
Notice that this is a property of the perfection-character of properties (this is an interpretation of predicate P). The assumption is that if there is a necessary implication (and not merely an implication) between two properties for every entity we choose, then if the first is a property of perfection, so is the second. A kind-hearted person may have two ears (there is no kind-hearted person who does not have two ears), but that does not mean that having two ears is a perfection (God need not have ears). Why not? Because there is no necessity here. Even if this happens to obtain in reality, and even if it obtains for all kind-hearted human beings, it need not obtain. Put differently: if the implication between the properties is necessary, this means that the second property follows from the goodness of the first, that is, that the second is good as well.
The second axiom:
P(¬φ)↔¬P(φ)
This is a relation of exclusivity between two opposite properties. If ¬φ is a property of perfection (a positive property), then φ is not such a property, and vice versa. If helping another person is a good property, then refraining from doing so is not a good property. This may sound trivial to you, but among mathematicians nothing is trivial. Anything that is assumed must appear as an assumption. Otherwise the derivation of the conclusion cannot be carried out. This is an advantage of the formal formulation, in which all the implicit premises that in an ordinary verbal formulation we might not have noticed must enter explicitly.
The first theorem:
P(φ)→◊∃x[φ(x)]
This is an interesting theorem, which says that if φ is a positive property, then it is possible that there exists an object x endowed with it. Theoretical properties that cannot be instantiated by any object are not defined as positive properties.
There is a difference between an axiom and a theorem. The former is assumed, and each person must decide whether it seems reasonable or not. The latter, by contrast, must be proved on the basis of the axioms. What is the proof of this theorem? It presumably proceeds by reductio. Let us assume that there could be a good property φ such that no object can possibly instantiate it (not merely that in fact no object does). In that case, for every property ψ one can write the implication: □∀x[φ(x)→ψ(x)], and it will be necessarily true for every x (for there will never be a case in which, for some x, the antecedent is true and the consequent false, because the antecedent is always false). In particular, of course, we may substitute for ψ the property ¬φ. But according to Axiom 1, that means that the property ¬φ is also good. And that, lo and behold, contradicts Axiom 2. From the negation of this theorem we have arrived at a contradiction, and that proves by reductio that the theorem is indeed correct.
The first definition:
G(x)⟺∀φ[P(φ)→φ(x)]
Here we introduce the subject of the discussion, God. The definition does not deal with him himself, but with the property ‘having the properties of God,’ or the property of ‘godlikeness.’ One must understand that this is not a definition of an object, but of a property (which is really a collection of its properties). A definition of an entity is the presentation of its properties; therefore the subject of the definition is not an entity but properties. Thus, when we define a human being, we say that he is a rational animal, that is, we specify the properties that characterize him.
Here we define a divine being as an object endowed with every good (positive) property whatsoever. If it is positive, then it has it.
The third axiom:
P(G)
Godlikeness is a perfection; that is, godlikeness is a positive property (a property of perfection).
Again, there is something here that in philosophical wording seems trivial, but the formal formulation forces us to put it on the table as an axiom.
The second theorem:
. ◊∃xG(x)
It is possible that there exists an x endowed with the property of godlikeness (that is, the existence of the object God is possible). This is already beginning to look like a bite taken out of the atheistic thesis (some atheists say that this is not possible at all). Therefore this is indeed not an axiom but a theorem, that is, a proved claim (on the basis of the axioms). It is easy to see that its proof is based on an application of Theorem 1 and Axiom 3.
The second definition:
φ ess x⟺φ(x)∧∀ψ{ψ(x)→□∀y[φ(y)→ψ(y)]}
Here the essentiality of a property φ for the object x is defined. It is defined as follows: object x satisfies the property φ (first of all, it must be one of its properties), and it is also true that for every other property ψ possessed by object x, there is a relation between it and the property φ: with respect to every entity whatsoever, the existence of φ necessarily implies ψ. In simple language, this means that all of x’s other properties follow from its property φ. This is the definition here of essentiality (or perhaps, more accurately: fundamentality, basicness).
The fourth axiom:
P(φ)→□P(φ)
If φ is a positive property, then it is necessarily such (there is no world we can imagine in which it would not be positive).
The third theorem:
G(x)→G ess x
The property of godlikeness of some object (= God) is essential to it. Again, this is a theorem and not an axiom or definition, and therefore we must prove it. This seems quite self-evident, because by its very definition (see Definition 1), godlikeness includes all the positive properties and only them (that is, God has no properties that are not positive—from Axiom 2). If so, every property of God is positive, and therefore it is necessarily positive (from Axiom 4). If so, God necessarily has it.
The third definition:
E(x)⟺∀φ[φ ess x→□∃yφ(y)]
Here we define the predicate of existing necessarily, that is, the necessity of the existence of object x. The definition is that x exists necessarily if, for every property φ that is essential to it, there necessarily exists an object endowed with it.
One should note that the claim that the property φ is essential to x does not mean that such an object exists. The subject of that claim (x) is the concept, not the object. φ is a property of a concept and is essential to it, and the definition requires that in such a case the concept necessarily be realized in reality (that is, there is in our world an object of which this is the concept).[6] Thus, for example, the horse is essentially four-legged, but that does not mean that a horse exists in reality. Here the horse is a concept, not an object. The definition is that the real horse is a necessary existent if, for every property essential to it, there necessarily exists an object that instantiates it.
It is worth noting that the property of being four-legged is not essential to the horse. It is necessarily included in its Platonic idea (horseness), but it is not true that all its other properties necessarily derive from it (which is what essentiality means according to Definition 2). In the sense of Definition 2, horseness (the collection of properties that creates the concept of horseness) is the essential property of the horse.
The fifth axiom:
P(E)
Being a necessary existent is positive (it is a property that expresses perfection). This is an assumption that is certainly accepted in philosophical-theological thought. Precisely for this reason, in the central theological traditions God is defined as a necessary existent (this follows from his perfection).
The fourth theorem (conclusion):
□∃xG(x)
The proof proceeds as follows: from Theorem 2 it follows that it is possible that there exists a being with the property of godlikeness. That is, the assumption of its existence is not contradictory, and therefore one can speak of it. From Definition 1 it follows that something with divine properties has every perfect property. In particular, it has the property E (for this too is a perfection, from Axiom 5). According to Theorem 3, for every divine object the property of godlikeness is essential. From Definition 3 of the property E (when one of the properties φ is G), it follows that such a being must exist.
[By the way, from Definition 2 it follows that all the other properties φ of the divine being derive from its property of godlikeness (for that property is essential and positive), and therefore the other properties in the implication in Definition 3 are not important.]
Conclusion: necessarily there exists an object x endowed with the property of godlikeness. In other words: God exists!!!
Q.E.D. (= which was to be shown)
Three remarks
It is worth noticing that Gödel proves here not only that God exists, but that he exists necessarily. As I showed in the first booklet, these are in fact the two arguments that Anselm presents in chapters 2 and 3 of the __Proslogion__.
It is also worth noting that he does not prove here that God is unique, but only that at least one God exists (at least one being with the property of godlikeness).
The atheist, of course, need not be convinced. He can give up one of the five premises of the argument (the axioms), and then he will not be committed to its conclusion either. That is why I wrote above that formalization does not make philosophical discussion superfluous, but only focuses it. It is now easier to focus the philosophical debate on the axioms, and that will determine whether we ought to adopt the conclusion that God exists or not. But do not go anywhere; the real surprise still lies ahead.
Back to our line of thought: the ontological argument for the existence of philosophy
Now we come to the punchline. My main remark concerns the ontological character of this argument. Anselm pretended to present his argument as the product of pure conceptual analysis (that is, an argument not based on premises at all, but only on definitions of concepts and their analysis). But in Gödel’s formal formulation one sees that the argument (which admittedly is not identical with Anselm’s) is indeed based on premises (and that is also what I showed in the booklet regarding Anselm’s own argument). But it is important to notice that nowhere along the way did we assume factual premises (go through the five axioms and you will see this). These are all philosophical-conceptual premises about ideas and the relations among them. Thus Gödel’s argument is indeed ontological in a softer sense: it yields a factual conclusion from non-factual premises that are not the product of observation.
It is important to understand that this description does not depend at all on whether you agree with Gödel’s conclusion. Even a complete atheist, convinced that there is no God, must agree that from this collection of five axioms his existence can be proved.[7] He simply thinks that at least one of them is probably not correct, and therefore the conclusion is not binding for him either. But even he must admit that if one accepts these premises, the conclusion follows from them, and of course he must admit that this is a factual claim about the world. In other words, even he agrees that there exists a valid ontological argument that necessarily yields a factual conclusion from non-observational premises. He disputes the premises and therefore the conclusion as well, but that is not our concern here.
But now the question arises: where do these premises come from? If they are not the product of observation (science), Ron Aharoni will say that they are necessarily hallucinations, that is, arbitrary definitions of concepts that we have invented out of our own minds. But then this is indeed strange and unacceptable. It is not plausible that from such premises one can derive conclusions that are factual claims about the world (that God exists).[8] Moreover, one way or another Aharoni will have to admit that in principle it is possible to arrive at conclusions about the world by non-observational means, and so in either case his thesis has fallen (which, as we saw, is based on empiricism).
As we saw in previous columns, if philosophy indeed exists, then necessarily there is a third category, beyond subjective definitions (hallucinations) and scientific observations. We saw that this must be intuitive observation of ideas, that is, of parts of the world that are inaccessible to scientific-sensory observation. And here we have a wonderful demonstration of this. Gödel has presented us with a logical argument based on non-scientific premises (intuitive observations), and yet it yields a conclusion that is a factual claim about the world. Perforce, at the foundation of these premises there lie observations of the world of some kind. True, we saw that these premises contain no scientific observation whatsoever, but necessarily there is here observation of another sort. What leads us to these premises is apparently eidetic vision, that is, intuitive contemplation.
In this sense, Gödel’s argument may be seen as a proof of my approach to philosophy. I proposed a definition of philosophy, and argued that it deals with intuitive observations of the world. Aharoni denied the existence of such a category and therefore the existence of the field of philosophy at all. But notice that here I have not only demonstrated my definition of philosophy; I have also proved a claim: there is such a field. There is such a kind of observation, and in principle one can reach conclusions with it (that are inaccessible to science).
It seems to me that what I have done here is itself an ontological argument for the existence of philosophy.[9] In effect, I have proved here that there is another kind of observation beyond sensory-scientific observations (for if not, the premises would necessarily be mere definitions, and then we could not derive from them a factual conclusion). Consequently, the definition I proposed for philosophy is not only possible (the concept of philosophy is coherent), but we have proved that it is also actually instantiated. Not only can one define a field in a third category (besides hallucinations and scientific observation), one can show that it actually exists, realized. If so, in Gödel’s language we have proved.
∃xPH(x)
There exists an argument x that has a philosophical character.
Alternatively, we have proved:
Or: ∃xIn(x)
There exists a claim x that is neither a hallucination nor a scientific claim, and this is what we called a product of intuitive observation (and therefore it can yield a factual claim about the world).
As stated, this itself is an ontological argument for the existence of philosophy.[10] Q.E.D.
1.
Footnotes
- See here, in section 5, on formal parodies of ontological arguments.
- See, for example, here and here.
- For a similar phenomenon, see column 108, in the discussion of the topology of convexity and concavity.
- A similar version of the argument can be found in the Stanford Encyclopedia of Philosophy SEP, here in section 6. And also in this article.
- In the language we are using here, equivalence also serves for definitions. In Definitions 1-3, an equivalence appears, meaning that its right-hand side is the definition of the left-hand side.
- In the first booklet I elaborated on this distinction, which underlies the idea of ontological arguments.
- Unless he can point to a flaw in the logic of the argument. But there is no chance of that, friends. Do not forget that this is Gödel’s logical argument—sorry for the ad hominem.
- See booklet 0 for why, categorically speaking, the claim ‘God exists’ is a factual claim, even though it cannot be scientifically verified (that is, confirmed or refuted by scientific observation).
- See here in the discussion of ‘modal collapse’ in Gödel’s argument. It should be remembered that even if we correct some interpretation of one of the predicates and the argument thereby becomes invalid, that changes nothing whatsoever with respect to my claim here. For my purposes it is enough that there exists at least one particular interpretation of the predicates according to which a fact can be derived from non-factual premises.
- I leave it to the reader to ask whether this ontological argument is purely conceptual (that is, whether it follows from conceptual analysis alone, without recourse to premises), or whether it is based on several premises (though admittedly not sensory-scientific facts).
Discussion
Couldn’t you have proved philosophy in a simpler and shorter way? The very fact that it is so long and complicated makes me suspect that some trick is hidden here.
Thanks, I liked it.
I have a problem with the predicate “necessarily exists.” The analytic criticism of the ontological argument is that in predicate calculus existence is a quantifier and not a predicate. It seems that here he is trying to get around that obstacle by means of a kind of “existence” (E) that is in fact a predicate, and thus move from relations between predicates to a claim of existence.
Indeed, I also think this is one of the focal points here.
Even without predicate calculus—it is philosophically quite clear that existence is not a predicate. In objections to the proof, many have already pointed out this (and I among them, in the first booklet): existence is not an attribute (it does not relate to form but to essence). The assumption in predicate calculus only reflects this philosophical statement.
As for the matter itself, one should note that E is not existence but necessary existence. In one of my books (I no longer remember which) I argued that necessary existence is not different from ordinary existence on the ontic plane (contrary to Yuval Steinitz’s assumption). Both things exist, and that’s it. But necessary existence can indeed be considered a predicate, unlike ordinary existence. The dimension of necessity in it has the character of a predicate.
Be that as it may, Gödel defined this predicate completely by means of a legitimate definition. If so, whatever its interpretation may be, the proof is valid. The question is whether this proof proves something about the world (there is a God) or about myself (I arrive at the conclusion that there is a God). In my booklet I discussed this at length and argued that even a proof of the second type is enough, since if I have proved that I must think there is a God, then if I am not a skeptic and I do not doubt my conclusions unless there is a good reason to do so, then from my perspective this is like a proof about something in the world.
By that lazy logic, you should not accept Fermat’s theorem. Its proof is much longer and more complicated. If for you simple = true, then we have a deep disagreement.
Suspicions are good, and definitely worth listening to (that is the power of intuition that I discussed here). But they are supposed to be a motor for examination, not a decisive consideration for drawing a conclusion.
I didn’t understand what is unclear. I proved something for every property (predicate) psi. In particular, “not phi” (what you called minus theta) is also a predicate, and therefore it can be substituted in place of psi. So it is relevant to it as well. I think that if you follow the sentences I wrote, everything is there clearly.
The proof of Theorem 1 is seemingly incorrect. It is based on a ‘contradiction’ in universal statements, but contradiction exists only in existential statements. For example, for every X that is an element of the empty set, X5 holds—this is not a contradiction.
The statement I wrote that is not a contradiction is that for every X that is an element of the empty set, X is greater than 5 and also less than 5. The greater-than and less-than notation apparently gets interpreted as HTML code.
But why does the dimension of necessity in it have the character of a predicate if “necessity” is also a quantifier in the calculus?
Seemingly, what we have here is a predicate, necessarily exists, built from two quantifiers that Gödel uses in the proof. On the face of it, that is strange.
Hello,
A completely technical comment. Thank you very much for uploading it in this format, so it can be printed out (I am aware that one can always print it out)
and read. It is much more convenient this way. And one more small suggestion for the benefit of the public: years ago I wanted to read the book “Two Carts and a Hot-Air Balloon,” but it looked heavy and tiring, and then I came across Dr. Nadav Shnayerb’s essay, “The Emptiness of the Analytic,” which in my view serves as a kind of excellent introduction to the book (by way of an amusing anecdote, he too admits that when writing a critique one should read the book, and here, exceptionally, he tried to peek at the book as little as possible). After that it was easier for me to read the book and follow the rest of your articles.
I think that if possible, this essay ought to appear on the site; it provides a good starting point before reading your books and articles.
Thanks. I have no way to put it in here. If I put it among my articles it will get swallowed up there, and then it will have no effect whatsoever. I don’t think it is right to put a general recommendation for everyone to read it before reading every book of mine.
I don’t understand. Are you claiming that one cannot define a predicate if quantifiers appear in its definition? That sounds very strange to me. When one quantifies over predicates, it is still a predicate.
This is an interesting question in logic, and I’m not sure you’re right. I don’t have time right now to delve into it, but at first glance the expression “both greater than 5 and less than 5” is itself contradictory. Therefore inserting it into any expression drains it of meaning, even if it is applied to elements of the empty set.
It seems to me that you are assuming that this expression is a kind of implication: if X exists then X5 (greater and less than 5), and since it does not exist, the antecedent is always false and therefore the implication is always true. But that is not at all a necessary interpretation.
I am really not a logician, but what I am saying is what they teach in every basic course (which I did not take) in logic (so says the local mathematician).
As I said, I doubt it. But perhaps I’ll try to look into it when I have time. In any case, the claim itself is proved according to Gödel, even if not in the way I suggested (which also appears in the link I referred you to).
With God’s help, Wednesday, for the portion “Show me Your greatness,” 5778
From a glance at the Wikipedia entry on Gödel, it seems that he really did grasp that God rules His world absolutely and that there is no chance in the world, but that this insight was bound up with constant fear and distrust that anyone seeks his good. What Gödel lacked was the insight that the greatness of the Holy One, blessed be He, is the measure of His goodness (as explained by Rashi on Deuteronomy 3:24, s.v. “Your greatness”). The God of Abraham is first and foremost: “great,” beneficent and abundant in kindness, and from that He is also “mighty and awesome,” whose attribute of judgment comes to keep His creatures from straying from the measure of goodness and kindness.
Regards, S.Z. Levinger
According to the current formulation, the proof is completely valid. The problem is with Axiom 1, which also applies to the case where the first property does not hold of any x whatsoever (one can still accept it, but it greatly lowers the level of the argument.
No, I am claiming that the term “necessary existence” is composed of two terms: exists, necessary. Both already appear in our calculus, but not as predicates, rather as quantifiers (notice that in the conclusion we go back to expressing “necessarily exists” by combining the quantifiers).
On the face of it this is possible; after all, if we have the predicate “table” and the predicate “small,” we can also define a third predicate—“small table” (even though we don’t really need it). By contrast, it is clear that Gödel had to express the term “necessary existence” not by combining the quantifiers, in order for it to be a predicate.
It seems to me that what Gödel lacked was mainly mental health, and less foundations of a worldview.
A.H.,
This was important for Gödel, but from my point of view, as someone who is now dealing with the question of what philosophy is and whether there is such a thing, it does not matter. There is a valid argument here that yields a fact from non-factual premises. Even if the premises are unreasonable, that does not change this fact. It only means that it is less reasonable to adopt the conclusion.
Incidentally, even Theorem 1 itself does not really require the existence of a good being but only the possibility of its existence. That is, the goodness of a phi property need not be actualized; only the capacity to be actualized is required. So one way or another, Gödel is talking about good properties only in that sense.
I still don’t understand. You keep returning again and again to semantics, while I am dealing with syntax. The predicate E is defined in some way. Do you think that definition is problematic (contradictory)? A definition is a definition, and if there is no contradiction in it, it is legal and can be used. What meaning you give to E is another question, but it is not important to our discussion because E does not appear in the conclusion.
Agreed, I don’t find any syntactic problems here.
I was talking about Gödel’s hairsplitting and not yours (but apparently I was mistaken, because from what I checked this does not ruin the continuation of the proof).
What is the difference between the ontological argument in the first booklet and this one? Seemingly one can give up Axioms 1-2 and Theorems 1-2, and we are left with exactly Anselm (albeit in a somewhat different formulation of necessary existence) = necessary existence is positive, therefore God exists, QED.
I didn’t say that the conclusion should not be accepted; I wondered whether there is not a simpler way to prove the existence of philosophy. Can’t one isolate the assumption (or assumptions) that prove it and build an argument only from them?
But if there are no syntactic problems, then there are no problems at all. After all, your semantic problems concern E, and it does not appear in the conclusion. So why should I care what E means? As long as the semantics of the conclusion are agreed upon, of course.
A.H., in any case what is relevant to me is only the validity of the hairsplitting, not its conclusion.
Can you formalize a valid argument without Axioms 1-2 and Theorems 1-2 and arrive at the same conclusion?
Well, you didn’t read, so it’s hard to talk. There is nothing here to isolate. My proof is from the argument as a whole. In the future, it is advisable to read before criticizing.
You need Axiom 2 in order to prove Theorem 3, but apart from that yes. Delete those lines from the proof and nothing will change, if I’m not mistaken.
In light of your comment I saw that I had omitted the proof of the final conclusion (Theorem 4). It was brought without proof. I will ask Oren to insert an updated version that includes it, and now you can check again whether there is anything superfluous (I would be very surprised if so. After all—Gödel).
That’s it, corrected.
Hello,
You did not prove that Gödel’s system of axioms is non-contradictory. As is well known, from an axiomatic system that contains a contradiction one can prove anything (including an ontological claim). Therefore this is still not an ontological proof for the existence of philosophy…
Incidentally, in the link you brought, there is a link to another article showing that Gödel’s axioms are indeed too strong, and that something implausible follows from them—that every true proposition is necessary. In any case, I did not see a proof that the system contains no contradiction.
Well, I’m out of my depth. I’ll need a healthy ad hominem here. Even what you quoted here is still not a contradiction.
Rabbi Michi, with your permission, a few questions/thoughts.
Axiom 1: Suppose “all-knowing” is a perfection; the question is whether everything necessarily entailed by it is a perfection. For example, in every possible world, if I am “all-knowing” then I have no possibility of learning anything additional. Is the inability to learn anything additional a perfection-property when it stands on its own?
Axiom 3: You write that this is self-evident, but it is not really clear to me. If God is *defined* as possessing all perfections (Definition 1), how can He Himself be a perfection? For then He would have to be part of the definition, and that would create circularity.
Axiom 4: Is it really the case that a perfection-property is such in every possible world? Suppose “eternal existence” is a perfection-property. There is a possible world in which the world is full of suffering beyond the bearable limit, an eternal Auschwitz—would eternal existence still be a perfection then?
Hello Guy.
Axiom 1: That is a good question. There is room to say yes, since my inability to learn anything additional is because there is nothing additional. So it can be formulated differently: there is nothing additional for me to learn. And that is already a perfection. The question is how you regard the additional property when it stands on its own: is it the property “it is impossible to learn,” or the property “there is nothing more to learn”?
I have now thought of a better formulation: it is not true that I cannot learn, for I certainly can. There is simply nothing on which to apply this ability. Therefore the second formulation above is the correct one.
In any case, my main concern here is of course not to prove that there is a God but that there is philosophy, and for that matter it is not really important whether this assumption is true or not. It is enough for me to show that from a non-factual assumption one can derive a factual conclusion.
Axiom 3: I wrote that in the philosophical formulation this sounds trivial, because Anselm and Gödel define God this way (as possessing all perfections). Possessing all perfections (being perfect) is itself a perfection. There is indeed a proposition here that also refers to itself, but circularity is not necessarily a problem. See my critique of the cat in column 157 (regarding type theory). But now I think that as a definition it is indeed problematic, since the concept is defined by means of itself. So perhaps there is no sufficient definition here, but there is no contradiction here. The loop enters not by virtue of the definition itself, but only in light of Axiom 3 that is added afterward (perhaps that itself is problematic, since when defining divinity as including all perfections we should already be clear what all the perfections are). Be that as it may, I do not think there is a contradiction here. It is a loop but not a contradiction.
And regarding the definition, one should now consider whether this loop really creates some problem in the definition. After all, any claim I want to examine about G I can examine by means of this definition. Its circularity does not interfere. But that is certainly a remark worth further thought.
I have now thought that regarding any object one can ask a similar question. It has a set of properties, but it also possesses the set of properties. The set too is one of its properties. Is there the same loop here? On second thought, no, because it does not really possess the set of properties, but each one of them separately. The set as such is not a property. And one can discuss and analyze whether the property that is the conjunction of all the properties (which is certainly a property) is identical to the set of all the properties (which is not a property).
Wow, there is a genuine topic for philosophical research here.
Axiom 4: Eternal existence is a perfection, except that in a world full of suffering that perfection brings suffering and problematic consequences. But in itself it is a perfection. It may be that by virtue of being a perfection it cannot appear in a world that is entirely suffering. Incidentally, it is more correct to raise the objection from a world in which entities inflict suffering and do not experience suffering. If someone suffers, that does not mean he is not perfect (in the sense of positive). Among Christians it is exactly the opposite (“surely he has borne their sins”).
Bottom line, the first and third remarks challenge Gödel’s argument (that there is a God), and I think one can manage with them. But the second remark may undermine my argument as well (about philosophy), because if the concept under discussion is not well defined, the argument is probably invalid. I am wondering about this.
What about non-intuitionist philosophy (Kant, Russell)—is that too proved here according to your method? Or are they too intuitionists?
I think every philosopher is an intuitionist, even if he does not admit it or is not aware of it. Without that there is no philosophy, as has been proved in my columns here. And similarly, anyone who is moral is an implicit believer, even if he does not admit it or is not aware of it.
Thanks for the response.
The discussion of Axiom 1 reminded me of the question of the divine will, which the Scholastic philosophers struggled with, because on the one hand it testifies to a deficiency in the one who wills, and on the other hand a complete negation of will itself seems like a serious deficiency. Which may be a particular case of the more general problem of the ‘stasis of the perfect,’ its inability to progress and perfect itself, which in its case is perceived as a perfection in itself (in the example I gave, the ability to learn exists in him, but he has nothing to learn—which is tragic).
Which raises the question whether the concept “perfect” is even a coherent concept. It seems to me that Rav Kook spoke of our being, כביכול, the contractors carrying out the property of self-perfection, and therefore in a certain sense completing God; but we ourselves are not God, so it is unclear how that solves the problem.
I’ve strayed a bit from the original text.
Indeed, this is the problem of perfection and self-perfection in volume 2 of Orot HaKodesh. I wrote about it briefly in my article on Zeno’s arrow. See there the mechanism that shows that inability to perfect oneself in actuality is no deficiency at all (because perfection is not the movement but the potential for movement, and that it has in a perfect way. Rav Kook argues that in our case the potential goes from potentiality to actuality). It really reminds me that this touches on our discussion.
https://mikyab.net/%D7%9B%D7%AA%D7%91%D7%99%D7%9D/%D7%9E%D7%90%D7%9E%D7%A8%D7%99%D7%9D/%D7%97%D7%99%D7%A6%D7%95-%D7%A9%D7%9C-%D7%96%D7%99%D7%A0%D7%95%D7%9F-%D7%95%D7%94%D7%A4%D7%99%D7%A1%D7%99%D7%A7%D7%94-%D7%94%D7%9E%D7%95%D7%93%D7%A8%D7%A0%D7%99%D7%AA1/
Therefore there is nothing tragic here at all.
For the time being, before I read the article you attached, it seems to me that a potential that *cannot* be actualized (that is, there is no possible world in which it is actualized in fact) is an oxymoron.
Besides that, in my view, the very actualization of the potential contains an element of perfection (and not only in the potential itself or in the final result of its actualization).
But I’ll read the article and see whether I have anything to add, thanks.
With God’s help, eve of the 15th of Av, 5778
It should be noted in Gödel’s praise that, despite his excessive suspiciousness toward every person—he placed full trust in his wife, to the point that he agreed to eat only what she prepared for him. It seems that faith and trust do not depend only on intellectual clarification, but to a large extent on emotion, and where there is love—there is faith!
Regards, S.Z. Levinger
Regarding Anselm’s argument:
“The gist of Anselm’s argument is that if we define God as the most perfect being that can be conceived
(possessing all the positive properties, those that constitute perfection), then one can prove by reductio ad absurdum
his existence. If we assume that he does not exist, that assumption will lead us to a contradiction. If he does not exist, then one can conceive
of a being more perfect than him (that same being with the same properties, except that it also exists), which contradicts his definition
as the most perfect being that can be conceived.”
It is not clear to me where the contradiction is; that is, in what way is the second God more perfect than the first? If by his existence—who said he exists? If one assumes that, that is begging the question; and if I can conceive him as existing even though in truth he does not exist, then it is also possible with the first God to conceive him as existing even though he does not really exist. So where is the God that is more perfect than the first?!
(That is aside from the problem of who said that the concept ‘most perfect that can be conceived’ is well defined; perhaps the set of things that can be conceived has no maximum/supremum. It is not clear to me whether “most perfect” is the maximum or the supremum of the set—in any case, there is an assumption here about the set, that it has one of them. Exactly as if we define the divine number as the greatest number that can be conceived, then we will prove that it exists, because if it does not exist then one can conceive a greater number than it… exactly the same proof, and it is not true.)
You surely don’t mean to drag me here into the ontological proof. I summarized it here briefly for the benefit of the readers. If you are interested, go and study the first booklet.
Could you please explain the difference between: exists (the quantifier), necessarily exists (the existential quantifier plus the box symbol), and necessarily existent (the predicate)?
Hello Avner.
As for the first two, it is not clear to me what here requires explanation. Do you understand the difference between a true proposition and a necessarily true proposition? A necessarily true proposition is true in every possible world (that is the modal meaning). True means true here, but not in every possible world. For example, arithmetic is necessarily true, but the laws of physics are true in our world, though there could be a world in which the physics is different. That is, it is not necessarily true.
That is also the difference between exists and necessarily exists. To say that X exists is a proposition. There is a difference between saying that this proposition is true and saying that it is necessarily true.
As for the predicate (I understand that you mean E), I have already explained here to one of the questioners that there is no need at all to give it semantics (I brought that in only to clarify the meaning of the move). Since it is explicitly defined in the proof, and the definition apparently is not contradictory, there is nothing preventing us from using it as a legitimate predicate that forms part of the proof. After all, I can define whatever I want so long as I used the definition and the defined predicate in a legal and consistent way. It does not appear in the conclusion, and therefore even if you do not describe E in the form of a predicate of necessary existence, that does not really matter. It seems to me that it is well defined and therefore a legitimate predicate.
True, your attitude toward Axiom 5 depends on the question of what E is, since there it is assumed to be a perfection (a positive property). But that may have mattered to Gödel. For my purposes here, it does not matter, because I only wanted to show that one can infer a factual conclusion from non-factual premises. For that purpose it really does not matter whether you agree with Axiom 5 or not, so long as the argument is valid.
One can of course continue and prove that he is unique if one adds an axiom saying that exclusivity is positive (an intuitive axiom, since if divinity is not exclusive to him, that detracts from his power and perfection).
Indeed. Here again the question arises that someone here already raised, whether the axioms do not contain a contradiction. If we add this axiom as well, the question may perhaps broaden.
I did not understand why you say that the definitions and axioms contain no assumptions about the world.
In fact, Axiom 1 includes that if a property is positive and applies to no entity, then its opposite is also positive. The second axiom says that it cannot be that a property and its opposite are both positive. From this Theorem 1 follows. Is that not clearly begging the question?
Likewise regarding the final conclusion: in Definition 3 we define the property of existing necessarily, define it as positive, and from this it follows that it applies to an entity to which the property G applies. I do not understand where the proof of the existence of philosophy is here; the talk about existence in reality is definitely present in the assumptions.
This looks more like an indictment of philosophy, which deals in empty hairsplitting.
The claim that one cannot infer conclusions from logic about reality without adding premises about reality seems to me as valid as ever.
Actually, in Steinitz’s book there is a nicer version of the ontological proof. He says that something one of whose properties is necessary existence can either necessarily exist or necessarily not exist, but it cannot exist and not exist contingently. On the assumption that the only possible status of necessary non-existence is a logical contradiction, then as long as we have not found a logical contradiction in the definition of God, we should think that he necessarily exists.
Another question—what exactly is the definition of necessary existence? When one says in every world that can be conceived, whose mind are we talking about? Not all minds are equal.
There is a misunderstanding here.
1. You surely know the fact that every logical argument presupposes what it seeks to prove. I have elaborated on this in several places. Is the conclusion that Socrates is mortal not contained in the premises that all human beings are mortal and Socrates is a human being? Therefore one can broaden even further the claim about the factual emptiness of logic: not only can a logical argument not yield a fact from non-factual premises, it cannot yield anything at all (not even something non-factual) that is not already contained in those premises themselves.
2. If the claim that no factual conclusion can be derived from non-factual premises means this, then it is trivial and says nothing. It simply says that from premises there cannot emerge what is not in them. By the same token you could challenge the need for and value of plane geometry. After all, all the conclusions are contained in the axioms, so what is the point of engaging in it?
Therefore, the discussion of the possibility of an ontological argument must assume the broader and non-trivial interpretation of the factual emptiness of logic. If each premise by itself contains no factual information, then their conjunction will not yield such information either.
3. Neither of the two axioms you mentioned asserts a factual claim about reality. And yet from the combination of the two there emerges a factual conclusion. Therefore the short argument you wrote here shows that there are ontological proofs.
4. And if you insist, the very conclusion you drew here from Axiom 1 (this is not the axiom itself) is itself such a logical argument. In fact, in that passage you summarized the ontological move of this argument.
5. And from here, a demonstration of the value of philosophy. It succeeds in showing you that from some non-factual premises one can derive factual conclusions. Of course, this concerns only the “non-factuality” of the premises in the broad sense I described above. Exactly like the value there is in geometry, even though all its conclusions are in some way contained in the set of axioms. If philosophy has no value, then geometry has none either; and if geometry has value—then philosophy has value too. QED.
1. As is well known, with the help of logic one can draw conclusions that are not clearly included in the premises. That is, they are included in the premises, but not clearly. One can say that this is what is done in geometry and in mathematical arguments.
According to your line of argument, why didn’t you prove the existence of philosophy from geometry? If what you are trying to prove is that there is value in logical and systematic thinking, that seems like a trivial claim. Even materialists who do not like philosophy very much like logic and scientific and mathematical thinking.
2. The move from geometry to reality is made by means of the assumption that geometry constitutes a model of reality. That seems a reasonable assumption.
The problem in the ontological argument is that it starts from definitions of concepts and arrives at a conclusion about reality. Is this done in a reasonable attempt to model reality? If it is so trivial that one cannot prove something not contained in the premises, why is that not enough to reject the ontological argument?
3. I have no problem at all with philosophy; I even like philosophy (philo-philosophy). Only this “proof” seems forced and unconvincing to me. Gödel’s line of argument seems like something forced that was built in order to reach a certain conclusion, and that of course is no problem to do.
If I were to assume two axioms: (a) A entails the necessary existence of God, (b) “not A” entails “A,” (c) conclusion A, (d) conclusion God exists—here too in the two axioms I have not assumed the existence of anything, and in the conclusion I derived the existence of God. Does that constitute proof of the existence of philosophy?!
4. One of the main claims against philosophy is that there is no progress in it. Open scientific and mathematical questions are generally solved (or at least often). All the great philosophical questions are still open, and there is no consensus about them. What does that mean? Perhaps that we have no good way to address those questions? Or others would say that we have not really defined (and cannot define) the questions properly at all.
If one wants to prove the importance of philosophy, one must prove that although the questions are still open, philosophical thinking advances us in thinking about those problems.
1. This is a misunderstanding. One cannot prove the existence of philosophy from geometry. Geometry proves that there is value to systematic and logical thinking, but the proof I am looking for is that non-factual premises can lead to a factual conclusion. The greater includes the lesser. Only after my proof from Gödel’s argument can I write to you that if geometry has value, so does philosophy. Without this, the comparison is incorrect.
2. I explained that the ontological argument does not make do with definitions, but has premises. However, they are non-factual premises.
3. Here too there is a misunderstanding. I explained that it really is not important whether Gödel’s argument is forced. It still shows that one can derive a factual conclusion from premises that are not factual.
4. There is definitely progress in it. First, because it stands at the basis of science (even if scientists are not aware of that). Second, it certainly teaches us new things. Contrary to what is commonly accepted, in my opinion the disagreements in philosophy are minor. True, it also teaches us to ask the questions, but not only that.
And for the benefit of everyone who does not accept the proof, here is an alternative proof, the onto-ontological proof: only God could have created someone stupid enough to think of the ontological proof and assume that it is valid.
Just one small correction:
Only God could have created someone stupid enough to think that it is stupidity to think that this proof is valid. See the first book in the trilogy at length.
I did not understand what is meant by perfection or positivity. If it means what I understand as positive, that depends very much on who you are, what your worldview is, and what case you are discussing. And to the best of my knowledge it is very hard to define objective positivity.
I thought perhaps the intention was omnipotence, meaning any property that adds ability to him. But that is simply not what is written.
I would be happy for an explanation.
I am not sure you are right. For the purpose of the discussion, you can think about it according to your own worldview. Everyone checks whether the argument is valid from his own perspective. In particular, you have to check whether existence is one of the perfections, which is the only thing important for this argument.
It follows that everyone has a different God, one that has no truly correct objective property. That is a difficult claim.
Seemingly, you are returning to Anselm’s argument; there is a matter of cognition here and not pure logic. There are premises here that are not built on reason alone but also on assumptions dependent on emotion and personal perspective. That greatly weakens this argument.
I am not trying to weaken or strengthen. The proof proves the existence of God if you include existence among the perfections. If in your view existence is not part of perfection, then from your perspective this is not a proof (although there may be other proofs). That’s all.
Thank you for the quick response.
“So I praised the dead who have already died more than the living who are still alive. But better than both is he who has not yet been, who has not seen the evil work that is done under the sun.”
So God does not exist. QED.
Most axioms that try to speak about the real world are agreed upon by almost everyone or by everyone, and that is where their force comes from. Here it is something that by definition everyone thinks differently about. How can you project from that onto the world? [Unlike Kant, I am asking: how can an axiom that changes from person to person, and even within me from time to time if you are Descartes, project onto reality?]
But you gave an interesting answer. I need to think about it.
Gödel’s argument came up; see also here: https://aish.com/how-the-greatest-mathematical-mind-of-the-20th-century-proved-the-existence-of-god/
It seems to me that the proof falls at the axiom (p(x- is equivalent to (p(-x, there is no reason to assume that if white has some perfection, blue will not have some perfection.
I read the rabbi’s column 160 on Gödel’s ontological issue.
At the end, when the rabbi proved that our intuition speaks about the world, I could not understand the rabbi’s proof.
Why can’t one say that the way we assumed premises in the ontological argument is just a matter of fantasizing fantasies?
One can claim anything. Any premise of any argument can stem from a hallucination. I did not prove that intuition speaks about the world; I argued it.
So why connect it to a formal logical proof?!
Everything was explained there. The proof does not come to prove its premises, and not even its conclusion.
A fascinating and unique column. Just two questions (or comments) on the content:
a) I did not understand what there is in the ontological argument in Gödel’s formulation (which I had known in a more simplistic form) that is not in the classical ontological argument. (Gödel’s argument is indeed different from Anselm’s argument, but I do not see that it differs essentially from the familiar argument of Descartes and Spinoza and Leibniz.) The classical argument is that God’s essence is that he necessarily exists, and therefore one cannot claim that he does not exist, since to say that a being that necessarily exists does not exist is a contradiction. In modal logic this is formulated in terms of possible worlds: since God exists in one possible world (for the concept of God includes no logical contradiction), and in that possible world he exists necessarily—then God exists in all possible worlds, since there cannot be a being that necessarily exists in one possible world and not in the other possible worlds. (This is Alvin Plantinga’s formulation, for example.) The problematic aspect of this argument is that it mixes different concepts of necessity. The fact that God exists necessarily is a metaphysical property of his existence—that his existence is necessary and not contingent. But even if there is a possible world in which God necessarily exists—there is no reason to assume that he also exists in our world; in one possible world he exists necessarily and in our world he does not exist at all. What point is there in Gödel’s argument that neutralizes this difficulty? Does he too merely assume that if God as a necessarily existing being is possible—then that means that he also exists? Is the uniqueness of his argument only the proof that God is a possible being?
b) I did not understand Gödel’s proof of the first theorem, that if a certain property is positive—then that means that it is at least possible. Why can’t one say that love is a positive property, but it necessarily is not realized in the world; yet if it were to be realized (which cannot happen), then it would necessarily also entail giving? There are severe intuitive problems in this statement. It is not reasonable to assume that there is a property that necessarily is not realized in the world; there is simply no reason to assume that, nor is it clear how we know this property if it is not realized. But what logical problem is there in the statement?
Then what is fascinating and unique about the column? The column does not deal with Gödel’s argument but with its principled significance. There is an argument here that does not assume factual premises and reaches a conclusion about the world. I showed that by analyzing its premises and structure. And indeed, I showed a similar phenomenon in the first conversation of The First Existing about Anselm. Although there I showed that not only are there premises, but the premises are factual (about what can be conceived).
As far as I remember, I also dealt here in the past with Plantinga’s argument, but I do not remember its details now, and this is not the place to enter into comparisons among the arguments.
See column 580, in which I distinguished between these concepts of necessity. And likewise in other places.
My question was: what does the proof prove?!
If the novelty is that intuition asserts things about the world, then great, I discovered that I can “know.”
But can intuition still not be an illusion?
Can the concept “perfection” still not be an illusion that grants non-necessary properties to the concept?
I would be glad for an answer, because apparently I took the rabbi’s proof as too far-reaching…
I explained. Gödel’s proof proves the existence of God. I was not dealing with that, but rather with the question of whether in principle one can prove claims about the world from non-factual premises (the question of rationalism, which in my view defines philosophy, as distinct from empiricism). Premises are always products of intuition, and still there are factual premises and non-factual ones.
My question was that from the wording in the column it seems that the rabbi accepts the basic premise of Gödel’s ontological proof, (that if God necessarily exists in one world—he necessarily exists in every possible world). If not—then there is no ontological proof here at all for the existence of philosophy, since in truth it is impossible to derive a factual claim about the existence of God from the fact that he exists in some way in some possible world, just as in the classical empiricist-Kantian argument (or more precisely: claim), that one cannot derive factual claims from a purely logical argument. That argument only expands and explains why this is impossible even when one is dealing with a claim about necessary existence.
Incidentally, my difficulty was among other things because I remembered that the rabbi does not essentially accept the ontological argument, and now I saw that in column 580 (to which you referred) the rabbi indeed rebuts Plantinga’s argument precisely for this reason, that one cannot derive the existence of God in our world from his necessary existence in some possible world, since the necessity is metaphysical and not logical. (I remembered that the rabbi argued this somewhere, but I did not have the heart to go looking.) So this is really a frontal contradiction: there the rabbi declares that he does not believe an ontological argument is possible, and that is the motivation to seek and find the flaw in Plantinga’s argument; whereas here the rabbi claims that an entirely identical argument (with respect to this specific flaw) is valid, and even constitutes an ontological proof for the existence of philosophy. How could such a villainy have been done in Israel? Perhaps our rabbi has retracted?
P.S. Does the rabbi really not understand what is fascinating about the column? My question was about the uniqueness of Gödel’s argument from the standpoint of its ontologicality. I did not understand what there is in his argument that is not exposed to the attacks directed at all the other ontological arguments, but there is no doubt that this is another interesting formulation worthy of examination, especially since in addition there is also the ontological proof (the invalid one) for the existence of philosophy.
I was not dealing with the question of whether I accept the argument or not. I used it in order to show that whoever accepts the premises (the non-factual ones) will arrive at a factual conclusion. That is enough for me. In column 580 I did indeed detail my opinion on the link between these two kinds of necessity.
I do not understand what you want.
My claim is precisely about that, namely that it is not true that whoever accepts the premises arrives at a factual conclusion. One can accept all the premises of Gödel’s argument and still remain with the conclusion that there is no God, because even if God necessarily exists in one possible world—it can still be that he does not exist at all in our world. So there is no ontological proof (for the existence of God, and) for the existence of philosophy.
In light of what the rabbi wrote in column 580 about Plantinga’s ontological argument, has the rabbi retracted the conclusion of this column? I mean the conclusion that Gödel’s ontological argument is logically valid, but that it assumes premises whose source is intuition, and therefore we have no way to verify them.
I no longer remember the details. As far as I remember, the distinction between the two kinds of necessity is nothing but a challenge to one of the premises of the argument.
That is a mistake. Gödel does not explicitly assume that premise, so this is a challenge to the argument and not a dispute with one of its basic premises.
Incidentally, even if it is one of the premises—that is not relevant, since clearly one can assume unjustified premises and deduce from them the existence of God. If we assume that the existence of the table entails the existence of God and that the table exists, we will reach the conclusion that God exists. Does that constitute an ontological proof of the existence of philosophy? True, if someone happens not to agree with one of the premises and still admits that in principle an ontological argument that yields a factual claim is possible—then there is here proof of the existence of philosophy. But here the challenge concerns the very possibility of inferring a factual conclusion from a philosophical argument. The argument of the challenge is that defining a concept as necessary does not mean that it is actualized in fact as necessary, but only that if it exists—then that is the character of its existence. So if Gödel’s ontological argument for God’s existence is invalid—then the ontological argument for the existence of philosophy is also invalid.
Why did Gödel need to assume all the premises he assumed?
Is it not enough to define God as a necessary being, and from that derive his existence?
Is there an essential need to define necessity as a positive property, and God as perfect?
I am not currently immersed in the details of the argument. Offhand, he wants to prove the existence of God and not just of some being. In addition, perhaps the assumption that he is a necessary being is meaningless—for example, that the property of necessity does not apply to any object. And finally, what you are proposing is just trivial begging the question. If I assume x, I can prove x from it. So what?
Why is this a premise?
In the column here the rabbi wrote at the end that the argument is logically valid, but that its problem is that it assumes intuitive premises, whereas in column 580 the rabbi further added and showed that the argument does not prove what we thought it proved.
What I mean by my question is to clarify whether I did not understand the rabbi’s words, or whether the rabbi retracted.
I would be happy for the rabbi’s answers.
As I wrote, I am no longer in this topic, and unfortunately I do not have time right now to get into it.
After several attempts to understand (the problem is with me, ad hominem), can you clarify even more than you already did the proof that the first theorem is proved by reductio ad absurdum?
That is, how did you manage to push minus theta into the first axiom?