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If someone (no matter who) designates something (for example, the property of God-likeness) as “positive”, then this does not mean that it is actually positive. Moreover, it does not mean that the property of God-likeness generally correlates with reality in view of the extreme doubtfulness of the existence of God as the initial object of likeness. In addition, there is no comprehensive definition of the concept of “positivity” due to the extreme ambiguity of this term (for example, positive and negative electric charges). These factors turn the specified text into an empty chatterbox.
Let me also remind you that if the “axiomatics” of a certain boltology contains elements that are not confirmed by empirical science, then this turns it into a boltology due to its isolation from reality (precisely because of the presence of elements that are not confirmed by empirical science). And how well the elements of boltology fit together does not correct the essence of the situation at all. In extreme cases, it can be kept in reserve, as one of the many similar tools, in case there is an empirical justification for one of them. And before the appearance of such trust in each of them, as “the only true”, of course, there is no
Does Godel's ontological argument make sense? This question contains three sub-questions: 1) Is Godel's proof logically correct? 2) is this proof ontological? 3) is it a proof of the existence of God (i.e. Gottesbeweise, as it is called in Godel's native language)?
1). So, does Godel's proof have a purely logical meaning? As you know, a logical formula does not make sense if it is not a well-constructed formula. But this is a trivial case, where the presence or absence of meaning is determined by a simple comparison with the syntax. The logical meaning of the proof is somewhat more complicated, although the main principle is the same: it must be correctly constructed, i.e. logically correct. A proof of a theorem has logical meaning if it is a nonempty finite sequence of formulas, where each formula is either an axiom or a theorem of the logical system used, or a formula obtained from the previous formulas of the sequence according to any of the inference rules, and the last formula coincides with the theorem being proved. Accordingly, if assumptions are used in addition to axioms, then the specified sequence of formulas is a conclusion, but in any case, the set of axioms, assumptions, and rules of inference should not lead to contradictions (in the sense that inconsistency is defined in this logical system). In Godel's own sketches, only the key points of the proof were indicated. But this makes it quite possible to present it in the above-mentioned “canonical” form: as, for example, in the reconstruction of V. V. Gorbatov (see fig. https://pp.vk.me/c633927/v633927371/16aad/dOKIzdjY0TU.jpg). Of great importance is the research carried out in 2013-2014 by Christoph Bentzmuller of the Free University of Berlin and Bruno Paleo of the Vienna University of Technology, using the theoretical developments of Paulson of the University of Cambridge (http://ebooks.iospress.nl/publication/36922). And as Christoph Bentzmuller pointed out in an interview, “now we can say with great confidence: the logical chain of argument in this proof of God is provable correctly” (http://www.heise.de/tp/artikel/39/39766/1.html). There is now little doubt that Godel's proof makes sense in this respect.
2). Does Godel's proof make sense as an ontological proof? Now there is also no doubt that Godel, at least, set himself the task of constructing a proof of this type: according to Oskar Morgenstern, referring to Godel's own words, he first of all “sought to show that such a proof with classical statements (completeness, etc.) correctly axiomatized is possible” (see Sobel J. H. Logic and Theism. Arguments For and Against Beliefs in God. NY. Cambridge University Press. 2004. p. 115-116). In its most general form, the essence of ontological proof consists in deducing the existence of an object from the concept of this object. Therefore, we are also talking about what kind of existence is meant. The simplest example of ontological proofs is deducing the existence of mathematical objects from their definitions. Although here, too, different criteria of existence are possible, and, accordingly, a different understanding of the ontological status: from nominalism to Platonic realism (it is enough to recall formalism, intuitionism, logicism, etc.). However, in any case, it concerns the ideal existence within the framework of a particular mathematical theory, and the question of the reality of such ideal objects is not in itself part of the subject of mathematics. Ontological proofs of a metaphysical nature, on the contrary, have in mind precisely the real existence (no matter how it is understood), where the logical truth (“tautology”) of the proved existential formula would coincide with its actual truth. Elements of this kind of evidence are found in modern physics, which is forced to operate, among other things, with fundamentally unobservable objects (such as virtual particles, the existence of which is proved purely theoretically, and empirical verification is possible only by indirect signs). In this case, Godel's proof is a kind of explication of Leibniz's ontological argument. If Descartes assumed that the Most Perfect Being is conceivable, and that the negation of existence would be a contradiction in the concept itself (i.e., a contradiction in the concept of existence), then it would be a contradiction in the concept itself. If he was thinking entirely within the framework of classical logic), then Leibniz noted: thinkability means a possibility that must itself be proved by establishing the consistency of the combination of all perfections. And from such a proven possibility, the necessity of existence immediately follows, because if necessity is possible, then it is necessary (now this is a theorem of the modal system S5, following from the axiom “everything possible is necessarily possible”). Leibniz based his proof on the postulate that perfection is “a simple quality that is positive and absolute, that is, it is not a simple quality.” without any restrictions, it expresses what it expresses” – therefore, each perfection does not contain anything that could contradict others, which implies the consistency of the totality of all perfections. However, the main counter-argument to this option was the objection that the existence of such “simple positive qualities” is by no means self-evident and must itself be proved. Godel eliminates this difficulty by implying that the term “positive property” becomes implicit in the logical system itself, receiving its definition through it (in particular, through one of the main propositions that the necessary logical consequences of positive properties are always positive). Another serious objection to ontological proofs was Kant's thesis that existence is not a real (“real”) predicate-a thesis based on the identity of the content of the concept of an object, regardless of the ontological status of this object. In Godel's proof (as, we note, in systems of free logics) this objection is removed by the fact that here existence is a second-order predicate. In short, on the whole, it is legitimate to conclude that Godel really succeeded in demonstrating the logical possibility of constructing ontological proofs of a metaphysical type. So in this respect, too, his “ontological argument” makes sense. It is interesting to note, by the way, that the question of the possibility of constructing ontological proofs without involving the apparatus of modal logic is not removed from the agenda. Thus, the outstanding Russian logician V. A. Bocharov attempted to demonstrate this possibility by means of first-order logic of predicates with equality in the system of natural inference (see Bocharov V. A., Yuraskina T. I. Divine Attributes, Moscow: MSU Publishing House, 2003, pp. 111-147). And although he failed to achieve a specific goal (proof of the chosen thesis), the attempt itself deserves special attention.
3) Does Godel's proof make sense as proof of the existence of God? This, of course, is the most controversial point. But mostly for non-logical reasons. However, leaving aside all sorts of ideological aspects, we can say that even the main “shortcomings” of Godel's ontological argument push for a positive answer to the question posed. First, the proof itself is based on such a model structure <G, K, R>, where the relation R ensures the reachability of all possible worlds K relative to the selected (real) world G. This is a necessary condition if we are talking about the existence of God, and not just some metaphysical entity with a relative need for existence. But this, as it was believed until recently, implies the mandatory use of the S5 modal system, the attitude to which is ambiguous among logicians. Now, after the study of Benzmuller and Paleo, the criticism of Godel's proof from this side is no longer relevant. Secondly, one of the essential conclusions of this study is the reference to the” monotheism ” of Godel's ontological argument. This removes criticism about the possible ” multiplicity “(expressed, for example, http://plato.stanford.edu/entries/ontological-arguments/#GodOntArg)by . However, third, it was confirmed that “modal collapse” (in the form of “everything that is valid is necessary”) is a peculiar price. This is a very serious counterargument. But the point is that such a circumstance is by no means fatal for Godel's proof, but rather the opposite. For, as Robert C. Koons points out, there is “a way to avoid modal collapse. The key question concerns the range of properties over which second-order quantifiers should run”, while the range of values of variables could only serve as “internal properties” (http://goo.gl/ZIiZPe). Moreover, it is precisely the danger of “modal collapse” that requires appropriate clarification. Apparently, this is what Godel meant when he meaningfully characterized ” positivity in the moral aesthetic sense (regardless of the random structure of the world). Only then are the axioms true” (https://pp.vk.me/c633927/v633927371/16ab6/N0jMnN32pjM.jpg). And such positive properties exist: for example, “good” or “beautiful” – it is in relation to them that the actual presence implies the necessity of having without the occurrence of a” modal collapse ” (if something is good or beautiful, then it is necessarily good or beautiful, by the definition of these concepts themselves). This approach requires a generalization of the concept of empirical experience, extending not only to sensory experience, but also to religious, moral, and aesthetic experience with appropriate logical explications. It is no coincidence that the same Benzmuller and Paleo qualified their research on the “automation” of Godel's ontological proof as “a glimpse of what can be achieved by combining computer science, philosophy and theology.”
It seems that yes-contrary to popular suspicions, it still makes sense. But this very meaning is still completely uncertain for us.
To begin with, let's recall the essence of this argument (GOA – God's Ontological Argument). As a basis, we will take the original formulation of Godel himself (given in: Manin Yu. I. Mathematics as a metaphor, Moscow: ICNMO, 2008, p. 88; the formal record can be viewed in the same place).
Definitions:
D1. to be a God (God-like being) means to have as essential properties all the positive properties
D2. For the property F to be essential in relation to the object x means that any other property inherent in the given object necessarily implies the property F; in other words, if the object has any properties at all, then there is also the property F.
D3. Existence is inherent in the object x, when all the essential properties of x imply that it is necessary to find an object that has these properties
Axioms:
A1. the conjunction of positive properties is a positive property
A2. A property is not positive if and only if its negation is positive
A3. a positive property is necessarily positive (and a non-positive property is necessarily not positive)
A4. existence is a positive property
A5. everything that necessarily follows from a positive property is a positive property
The theorem: God must exist
(For a detailed discussion of both the argument itself and its relation to other versions of the ontological proof, see Graham Oppie's article “Ontological Arguments” for the Stanford Encyclopedia of Philosophy: stanford.edu )
So, does the above set of sentences make any sense? What does it even mean for formal reasoning like GOA to “make sense”? It should be understood that we are dealing with a rather sophisticated argument formulated in a language much richer than the language of first-order logic. Both predicate quantifiers and modal operators are used here. Such arguments cannot be clearly divided into correct and incorrect ones: quite often the question of their correctness depends on a number of meta-theoretical assumptions. And the very concept of “correctness” is not very suitable here – it is better to talk about “rational acceptability”, which is divided into several independent criteria: simplicity, economy, consistency, non-triviality, etc.
The argument under consideration is doing quite well with this: it can be subjected to syntactic and semantic analysis, check the consistency of axioms and definitions, build models, and generally apply various methods of working with proofs (including computer ones). So, in 2014, German researchers Paleo and Benzmuller published the results of a thorough verification of GOA by means of “computational theoretical philosophy” (including several programs for automatic proof of higher-order theorems). The results were very encouraging: the stated statement, meaningfully interpreted as “God necessarily exists”, really follows from the accepted axioms and definitions.
These axioms and definitions do not contain internal contradictions, as many suspected.
In addition, they turned out to be quite economical (the KB system is sufficient to justify the modal part of the proof, and not the stronger S5, as some claimed).
The authors confirmed Sobel's previously published result on “modal collapse”: Godel's axioms are so strong that the concepts of “necessary truth” and “truth”, in fact, stick together (this, frankly, is a big minus!)
Good news for supporters of monotheism: it turns out that these axioms can be deduced not only the existence, but also the uniqueness of God.
Can we say that Godel's version of the ontological argument is somehow significantly better than the previous ones? In my opinion, yes. It is constructed on the basis of much more sophisticated and sophisticated logical means than those available to Godel's great predecessors, from Anselm to Hegel. Even compared to the versions of Hartshorn, Malcolm, and Plantinga (who also used the conceptual framework of modern modal logic), it looks much more rigorous, general, and detailed. The level of detail and formalization is unprecedented.
So does Godel's ontological argument make sense? Definitely, yes. What exactly is the point? This question remains open. As you can see, the key concept of “positive property” is not accompanied by any meaningful definition. It is defined only implicitly – through axioms. Godel himself warned against too direct an interpretation of this concept, emphasizing that “positive means positive in the moral and aesthetic sense (regardless of the accidental structure of the world).”
Is the property of “being red” positive, for example? Unlikely. After all, this property implies the negative property of “not being green”. You can't be red and green at the same time – at least not in our world.
And is the positive property of “being alive”, for example? Well, it seems like if a person is kind – then he is alive, and if he is evil – then he is still alive, and if he is dead,then he is no longer at all. But even here everything is not so simple: we can imagine a world with such an arrangement that the property of “being dead” in it turned out to be ontologically more fundamental: all other properties significant for that world would follow from it, and living beings would turn out to be” nothing”.
But” equal to yourself ” (x=x), according to Godel, is a positive property. After all, it does not follow any negative properties, and it itself follows from any property (including all positive properties). I wonder what Godel would say about bosons – subatomic particles that don't have this property (thequestion.ru)? It seems that “positivity” even for the property “equal to yourself” is somewhat questionable.
Thus, Godel's argument may be quite rationally acceptable. It may indeed follow that ” God necessarily exists.” And this “God” must represent the totality of all positive properties. Only, we are not able to name any of the supposed properties of such a “God” with confidence. Unfortunately, we don't even have the faintest idea what such properties might actually consist of.