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Godelian ontological arguments
Abstract
Ontological arguments are arguments, for the conclusion that God exists, from premises which are supposed to derive from some source other than observation of the world — e.g., from reason alone. In other words, ontological arguments are arguments from nothing but analytic, a priori and necessary premises to the conclusion that God exists. The first, and best-known, ontological argument was proposed by St. Anselm of Canterbury in the 11th. century A.D. In his Proslogion, St. Anselm claims to derive the existence of God from the concept of a being than which no greater can be conceived. St. Anselm reasoned that, if such a being fails to exist, then a greater being — namely, a being than which no greater can be conceived, and which exists — can be conceived. But this would be absurd: nothing can be greater than a being than which no greater can be conceived. So a being than which no greater can be conceived — i.e., God — exists. In the seventeenth century, René Descartes defended a family of similar arguments. For instance, in the Fifth Meditation, Descartes claims to provide a proof demonstrating the existence of God from the idea of a supremely perfect being. Descartes argues that there is no less contradiction in conceiving a supremely perfect being who lacks existence than there is in conceiving a triangle whose interior angles do not sum to 180 degrees. Hence, he supposes, since we do conceive a supremely perfect being — we do have the idea of a supremely perfect being — we must conclude that a supremely perfect being exists.
... Gödel's ontological argument, in its different variants, is amongst the most discussed formal proofs in modern literature, and so most of its premises and inferential steps have been subject to criticism in some way or another (see e.g. [41], [42] and [50]). We can therefore conceive of this argument as a network of (abstracted) nodes, some of them standing for some argument supporting the respective premise and others standing for attacking arguments (cf. ...
... In a logic such as S4, where symmetric accessibility relations between worlds are not enforced, the argument fails. This is confirmed again in lines 44-45, where the countermodel is computed directly for the stated validity conjecture (excluding the possibility that there might be an alternative proof in S4 to the one attempted in lines [32][33][34][35][36][37][38][39][40][41][42]. For logic S5 the Scott argument is valid again, which is not a surprise given that logic S5 entails logic KB. ...
We motivate and illustrate the utilization of (higher-order) automated deduction technologies for natural language understanding and, in particular, for tackling the problem of finding the most adequate logical formalisation of a natural language argument. Our approach, which we have called computational hermeneutics, is grounded on recent progress in the area of automated theorem proving for classical and non-classical higher-order logics, and it integrates techniques from argumentation theory. It has been inspired by ideas in the philosophy of language , especially semantic inferentialism and Donald Davidson's radical interpretation; a systematic approach to interpretation that does justice to the inherent circularity of understanding: the whole is understood (compositionally) on the basis of its parts, while each part is understood only in the context of the whole (hermeneutic circle). Computational hermeneutics is a holistic, iterative, trial-and-error approach, where we evaluate the adequacy of some candidate formalisation of a sentence by computing the logical validity of (i) the whole argument it appears in, and (ii) the dialectic role the argument plays in some piece of discourse.
... Perhaps, Patterson wants to hide the best clues for the scholars of the future generation who will challenge the most fundamental problem of logic and philosophy of logic still left unsolved. 142 8 Patterson on Tarski's Definition of Logical Consequence many interesting responses including Oppy (1996), where a parody of the Gödelian proof reminiscent of Gaunilo's objection to Anselm's proof is presented. As one might expect, such a parody has invited friends of ontological proofs to follow in the footsteps of Anselm. ...
Frege frequently complains that others are ignorant of the distinction between “falling under” and “subordination”. This criticism is not only directed against the philosophers who are under the influence of Aristotelian logic but also against the mathematicians of his time. I shall show that this distinction must be the vantage point for understanding Frege in both historical and philosophical contexts. Strangely, this distinction is not studied extensively nowadays. There are some good reasons for this. First, ironically, it is so well established as to become a triviality. Secondly, some people think that Frege’s criticism of the aggregate view of sets is outdated. Consequently, we cannot understand why this distinction was so important to Frege. In what problem situation did Frege formulate this distinction? Were there any rival theories of predication? Was this distinction an ad hoc device for Frege in order to establish other important theses? What would happen if we lack this distinction? This chapter aims at a partial answer to these questions.
... Perhaps, Patterson wants to hide the best clues for the scholars of the future generation who will challenge the most fundamental problem of logic and philosophy of logic still left unsolved. 142 8 Patterson on Tarski's Definition of Logical Consequence many interesting responses including Oppy (1996), where a parody of the Gödelian proof reminiscent of Gaunilo's objection to Anselm's proof is presented. As one might expect, such a parody has invited friends of ontological proofs to follow in the footsteps of Anselm. ...
Michael Friedman’s project both historically and systematically testifies to the importance of the relativized a priori. The importance of implicit definitions clearly emerges from Schlick’s General Theory of Knowledge (Schlick 1918). The main aim of this paper is to show the relationship between both and the relativized a priori through a detailed discussion of Friedman’s work. Succeeding with this will amount to a contribution to recent scholarship showing the importance of Hilbert for Logical Empiricism.
... Perhaps, Patterson wants to hide the best clues for the scholars of the future generation who will challenge the most fundamental problem of logic and philosophy of logic still left unsolved. 142 8 Patterson on Tarski's Definition of Logical Consequence many interesting responses including Oppy (1996), where a parody of the Gödelian proof reminiscent of Gaunilo's objection to Anselm's proof is presented. As one might expect, such a parody has invited friends of ontological proofs to follow in the footsteps of Anselm. ...
I attempt to get further insights from Cocchiarella’s history and philosophy of logic in understanding the contrast of Aristotelian and Fregean logic. Recently Cocchiarella proposed a conceptual theory of the referential and predicable concepts used in basic speech and mental acts (Cocchiarella in Synthese 114:169–202, 1998). This theory is interesting in itself in that singular and general, complex and simple, and pronominal and nonpronominal, referential concepts are claimed to be given a uniform account. Further, as a fundamental goal of this theory is to generate logical forms that represent the cognitive structure of our speech and mental acts, as well as logical forms that represent only the truth conditions of those acts, it is an indispensable part of Cocchiarella’s conceptual realism as a formal ontology for general framework of knowledge representation. In view of the recent surge of interest in his formal ontology by cognitive scientists and AI people, at least, Cocchiarella’s theory of reference deserves careful examination. Above all, however, the utmost value of Cocchiarella’s theory of reference must be found in its challenge against what he calls “the paradigm of reducing general reference to singular reference of logically proper names” that pervades the 20th century (Cocchiarella in Synthese 114:169–202, 1998, 170). The aim of the present chapter is to provide an impressionistic sketch of Cocchiarella’s challenge.
اعتمدت المدارس الكلامية الإسلامية قديما على المنطق الصوري الأرسطي (formal logic) بشكل رئيس في الاستدلال على القضايا الإيمانية، وصياغة حجج كانت آنذاك قادرة على معالجة الإشكالات العقدية الطارئة، أما وقد تغير المجال التداولي المعرفي في القرنين العشرين والحادي والعشرين (لغة الرموز Symbols language)، حيث أصبح المنطق الأرسطي متجاوزا، كما تغيرت الإشكالات العقدية المطروحة، فقد أصبح لزاما تغيير العدة المنطقية الـمُستعملة في الاستدلالات الكلامية (Kalam inferences)، بالانفتاح على المنطق المعاصر -كمنطق الموجهات (Model Logic)، ومنطق الرتبة الأولى (First-order logic FOL)...- حتى يتسنى لمتكلمي المسلمين عموما والأشاعرة على وجه الخصوص بناء مقدمات منطقية سليمة، تقود إلى استنتاجات عقلانية، ومناسبة من الناحية الفكرية لمستوى الاستشكال العقدي المعاصر. من هذا المنطلق حاولت في هذه الورقة البحثية عرض نماذج تطبيقية لاستعمال العدة المنطقية المعاصرة في بناء الحجج الإيمانية، من خلال ثلاث أنواع من الحجج: الحجة الوجودية (Ontological argument)، والحجة الكونية (Cosmological argument)، والحجة الغائية (Teleological argument). وهي حجج اخترتها بسبب مركزيتها في النسق الميتافيزيقي القديم والمعاصر، وبسبب قدرتها على بناء صورة واضحة عما وصل إليه الاستدلال العقدي المعاصر، وعما يجب أن يكون عليه التجديد في الدرس الكلامي الإسلامي المعاصر.
Varios argumentos han sido formulados para defender la existencia de Dios. Entre todos estos podemos reconocer distintos tipos de pruebas. Sin embargo, la prueba más atractiva ha sido el conocido argumento ontológico el cual ha sido analizado por varios filósofos a lo largo de la historia del pensamiento occidental. Anselmo fue el primero en exponerla; pero no fue el único. También, Descartes y Leibniz han intentado elaborar la demostración más simple y definitiva con respecto a este tópico. No obstante, recién Kurt Gödel en el s. XX logró establecer una inferencia bien construida en base a la lógica modal de segundo grado. Es este último argumento el que pretendemos exponer en este trabajo.
The article is devoted to Gödel’s ontological argument, its place in the history of philosophy, and the current debate over the validity of ontological proof. First, we argue that Gödel's argument is a necessary step in the history of the development of ontological proof. Second, we show that Gödel’s argument (namely, its core concept of “positive property”) is based on implausible axiological principles (this fact raises many objections like Hajek’s counter-argument), but can be appropriately reformulated in terms of plausible axiological principles (Gustafsson’s argument). Also, we consider the debate over the validity of Gödel’s argument between contemporary neo-Gaunilist Graham Oppy and the advocate of Gödel’s ontological proof Michael Gettings. We conclude that Gödel’s ontological argument is immune to Oppy’s neo-Gaunilism. Finally, given the fact that Oppy’s parody is arguably the most fine-grained Gaunilo-style argument in the history of philosophy, we conclude that Gaunilist line of argumentation, even if successful in refuting Anselm’s ontological proof of God’s existence, does not work against Gödel’s ontological argument (what is evidenced by the results of the debate between Oppy and Gettings).
In this paper, I present what I call the symmetry conception of God within 1st order, extensional, non-well-founded set theory. The symmetry conception comes in two versions. According to the first, God is that unique being that is universally symmetrical with respect to set membership. According to the second, God is the universally symmetrical set of all sets that are universally symmetrical with respect to set membership. I present a number of theorems, most importantly that any universally symmetrical set is identical to its essence, that show that the two symmetry conceptions intersect with some dominant theological conceptions of God. The theorems also show that both of the symmetry conceptions of God entail that God has a fractal like structure.
In recent years there has been a surge of interest in Gödel’s ontological proof of the existence of God. In spite of all this extensive concern, it is not certain whether there is any improvement in understanding the motivations of Gödel’s ontological proof. Why was Gödel so preoccupied with completing his own ontological proof? To the best of my knowledge, no one has dealt with this basic question seriously enough to answer it. In this chapter, I propose to examine Gödel’s ideas against a somewhat larger background in order to understand his motivation for establishing the ontological proof. I shall point out that the value of Gödel’s proof is to be found in the possible role of his proof of the existence of God in his philosophy as a whole as well as in its relative merit as an ontological proof. Hopefully, my guiding question as to Gödel’s motivation will turn out to be extremely fruitful by enabling us to fathom his mind regarding God and mathematics.
“I began to ask myself whether there might be found a single argument which would require no other for its proof than itself alone, and would suffice to demonstrate that God truly exists.”—St. Anselm