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Magari and others on Gödel’s ontological proof

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... Some of these experiments were reconstructed in the proof assistant Coq [23], see [19]. Additional follow-up work contributed similar studies, see e.g., [16,37,39,5], for a range of variants of the ontological argument proposed by other authors, such as Anderson [2,3], Hajek [42,43,44], Fitting [35], and Lowe [48]. Moreover, in ongoing work [10], ultrafilters are used to study the distinction between extensional and intensional positive properties in the variants of Scott, Anderson and Fitting. ...
... One might conclude, therefore that the premises of Gödel's argument imply that everything is determined (we may even say: that there is no free will). Further variants of Gödel's argument, in which his premises were weakened to address the above issues, were proposed by Anderson, Hájek, Fitting, and Bjørdal [2,3,42,43,44,35,25]. These variants have also been formally analysed. ...
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Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.
... Probably the best known is Andersons's given in [2] and modified in [3]. Andersonian systems were further explored and critically discussed and modified, for instance by Hájek [21] [23] and Szatkowski (e.g., [38] and [39]). Sobel's proposal is to exclude modal collapse by deleting Axiom 5 ( [34] and [35], preventing the provability of Theorem 1 .4, ...
... too). Another approach was proposed by Hájek [21], consisting in the weakening of the ontological system to a belief system with the KD45 propositional base, where Theorem 1 .4 is not provable. Fitting proposed a change from intensional to extensional types of variables, preserving the validity of Theorem 1 .4. ...
... Yet Gö del's derivation of the Necessity Claim also relies on some of the implausible axiological principles we have rejected. Há jek (1996Há jek ( : 128, 2002, however, proposes some emended proofs of the Necessity Claim, which 16 Consider the following negative analogues of (16) and (17) These principles also seem plausible. Note that, here, 'negative' should be understood as negative in 'the moral aesthetic sense'. ...
Article
Kurt Gödel’s version of the Ontological Proof derives rather than assumes the crucial (yet controversial) Possibility Claim, that is, the claim that it is possible that something God-like exists. Gödel’s derivation starts off with a proof of the Possible Instantiation of the Positive, that is, the principle that, if a property is positive, it is possible that there exists something that has that property. I argue that Gödel’s proof of this principle relies on some implausible axiological assumptions. Nevertheless, I present a proof of the Possible Instantiation of the Positive, which only relies on plausible axiological principles. Nonetheless, Gödel’s derivation of the Possibility Claim also needs a substantial axiological assumption, which is still open to doubt.
... Some experiments were also conducted with the proof assistant Coq [10]. Further work (see the references in [24,4]) contributed a range of similar studies on variants of the modal ontological argument that have been proposed by Anderson [1], Anderson and Gettings [2], Hájek [21,22,23], Fitting [16], and Lowe [26]. Particularly relevant for this article is some prior formalization work by the authors that has been presented in [18,17]. ...
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Full-text available
Three variants of Kurt Gödel's ontological argument, proposed by Dana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory reading they are in fact closely related. This has been revealed in the computer-supported formal analysis presented in this article. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.
... Some of these experiments were reconstructed in the proof assistant Coq [10]. Additional follow-up work contributed a range of similar studies (see the references in [24]) and includes a range of variants of the ontological argument proposed by other authors, such as Anderson, Hájek, Fitting, and Lowe [1,2,21,22,23,16,26]; particularly relevant for this article is also some prior work of the authors [18,17]. The use of ultrafilters to study the distinction between extensional and intensional positive properties in the variants of Scott, Anderson and Fitting has first been mentioned in an invited paper presented in 2018 at AISSQ conference in Bhubaneswar, India [6], and subsequently at FMSPh-2019 conference in Dubrovnik, Croatia [3]. ...
Preprint
Full-text available
Three variants of Kurt Gödel's ontological argument, as proposed byDana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument, the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory reading, they are in fact closely related, as our computer-supported formal analysis (conducted in the proof assistant system Isabelle/HOL) reveals. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.
... Some of these experiments were reconstructed in the proof assistant Coq [10]. Additional follow-up work contributed a range of similar studies (see the references in [24]) and includes a range of variants of the ontological argument proposed by other authors, such as Anderson, Hájek, Fitting, and Lowe [1,2,21,22,23,16,26]; particularly relevant for this article is also some prior work of the authors [18,17]. The use of ultrafilters to study the distinction between extensional and intensional positive properties in the variants of Scott, Anderson and Fitting has first been mentioned in an invited paper presented in 2018 at AISSQ conference in Bhubaneswar, India [6], and subsequently at FMSPh-2019 conference in Dubrovnik, Croatia [3]. ...
Preprint
Full-text available
Three variants of Kurt Gödel's ontological argument, as proposed byDana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer. In contrast to Scott's version of Gödel's argument, the two variants contributed by Anderson and Fitting avoid modal collapse. Although they appear quite different on a cursory reading, they are in fact closely related, as our computer-supported formal analysis (conducted in the proof assistant system Isabelle/HOL) reveals. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.
... 6]. Subsequent development has taken various paths (1) emendations of the axioms (such as [2] and [7], or Hájek's [14] emendation that is based on Gödel's philosophical notes), (2) cautious comprehension principles [13], or (3) a distinction between intensional and extensional types where essence and positivity are interpreted extensionally (which implies rigidity) [9, Ch. 11, sec. 9] and [4]. ...
Article
Gödel’s ontological proof is by now well known based on the 1970 version, written in Gödel’s own hand, and Scott’s version of the proof. In this article new manuscript sources found in Gödel’s Nachlass are presented. Three versions of Gödel’s ontological proof have been transcribed, and completed from context as true to Gödel’s notes as possible. The discussion in this article is based on these new sources and reveals Gödel’s early intentions of a liberal comprehension principle for the higher order modal logic, an explicit use of second-order Barcan schemas, as well as seemingly defining a rigidity condition for the system. None of these aspects occurs explicitly in the later 1970 version, and therefore they have long been in focus of the debate on Gödel’s ontological proof.
... They have utilized the proof assistant Isabelle/HOL together with the external automated higher-order provers Leo-II and Satallax for computer-assisted analysis of the Anderson-Hájek's Ontological Controversy. C.A. Anderson [1], Hájek [10] and Bjørdal [5] have proposed emendations of Gödel's axioms and definitions. They require neither comprehension restrictions nor more complex semantics and are therefore technically simpler to analyze with computer support. ...
... C.A. Anderson [1], Hájek [10] and Bjørdal [5] have proposed emendations of Gödel's axioms and definitions. They require neither comprehension restrictions nor more complex semantics and are therefore technically simpler to analyze with computer support. ...
... To prove ModalCollapsLem, we will need the following lemma: C.A. Anderson [1], Hájek [11] and Bjørdal [6] have proposed emendations of Gödel's axioms and definitions. They require neither comprehension restrictions nor more complex semantics and are therefore technically simpler to analyze with computer support. ...
... include the separation of relevant from irrelevant axioms, the determination of mandatory properties of modalities, and undesired side-implications of the axioms such as the modal collapse. 5 Further variants of Gödel's axioms were proposed by Anderson, Hájek and Bjørdal [57,58,59,60,61,62]. These variants have meanwhile also been formally analysed, and ATPs have even contributed to the clarification of an unsettled philosophical dispute between Anderson and Hájek [63]. ...
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Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed.
... include the separation of relevant from irrelevant axioms, the determination of mandatory properties of modalities, and undesired side-implications of the axioms such as the modal collapse. 5 Further variants of Gödel's axioms were proposed by Anderson, Hájek and Bjørdal [57,58,59,60,61,62]. These variants have meanwhile also been formally analysed, and ATPs have even contributed to the clarification of an unsettled philosophical dispute between Anderson and Hájek [63]. ...
Preprint
Full-text available
Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed. DOI: 10.1016/j.scico.2018.10.008
... Further variants of Gödel's axioms were proposed by An- derson, Bjordal and Hájek (Anderson, 1990;Anderson and Gettings, 1996;Hájek, 1996, 2001Hájek, 2002;Bjørdal, 1999). These variants have also been formally analysed, and, in the course of this work, theorem provers have even contributed to the clarification of an unsettled philosophi- cal dispute between Anderson and Hájek ( . ...
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Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is well suited for realising a universal logic reasoning approach. Universal logic reasoning in turn, as envisioned already by Leibniz, may support the rigorous formalisation and deep logical analysis of rational arguments within machines. A respective universal logic reasoning framework is described and a range of exemplary applications are discussed. In the future, universal logic reasoning in combination with appropriate, controlled forms of rational argumentation may serve as a communication layer between humans and intelligent machines.
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Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.
Preprint
Full-text available
Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.
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