We describe a KB Gödelian ontological system, and some other weak systems, in a fully formal way using theory of types and natural deduction, and present a completeness proof in its main and specific parts. We technically and philosophically analyze and comment on the systems (mainly with respect to the relativism of values) and include a sketch of some connected aspects of Gödel's relation to Kant.

Kurt Gödel’s version of the Ontological Proof derives rather than assumes the crucial Possibility Claim: 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: 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 but it can be patched so that it only (...) relies on plausible principles. But Gödel’s derivation of the Possibility Claim also needs a substantial axiological assumption, which is still open to doubt. (shrink)

Caramuels’ proof of non-existence of God is compared with Gödel’s proof of existence.

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. (shrink)

A shallow semantic embedding of an intensional higher-order modal logic in Isabelle/HOL is presented. IHOML draws on Montague/Gallin intensional logics and has been introduced by Melvin Fitting in his textbook Types, Tableaus and Gödel’s God in order to discuss his emendation of Gödel’s ontological argument for the existence of God. Utilizing IHOML, the most interesting parts of Fitting’s textbook are formalized, automated and verified in the Isabelle/HOL proof assistant. A particular focus thereby is on three variants of the ontological argument (...) which avoid the modal collapse, which is a strongly criticized side-effect in Gödel’s resp. Scott’s original work. (shrink)

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 is based on implausible axiological principles, but can be appropriately reformulated in terms of plausible axiological principles. 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, does not work against Gödel's Ontological Argument. (shrink)

Classical higher-order logic, when utilized as a meta-logic in which various other 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. (shrink)

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted (...) in larger scale by philosophers. (shrink)

Стаття присвячена онтологічному аргументу Курта Ґьоделя, його місцю в історії філософії і сучасних дебатах стосовно валідності онтологічного доказу. По-перше, ми аргументуємо що Ґьоделів аргумент є необхідним кроком в історії розвитку онтологічного аргументу. По-друге, ми показуємо, що Ґьоделів аргумент засновується на незадовільних аксіологічних принципах, проте може бути переформульований в термінах задовільних аксіологічних принципів. Також ми розглядаємо дебати щодо валідності Ґьоделевого аргументу між сучасним нео-ґаунілістом Ґремом Опі та захисником онтологічного аргументу Майклом Ґетінґсом. Ми доходимо висновку, що онтологічний аргумент Ґьоделя не вразливий до (...) ґаунілізму Опі. Нарешті, враховуючи той факт, що пародія Опі є, імовірно, найбільш досконалим аргументом ґаунілістського ґатунку в історії філософії, ми висновуємо, що ґаунілістська лінія аргументації, навіть якщо є успішною проти аргументу Ансельма, не працює проти аргументу Ґьоделя. (shrink)

Anderson-like ontological proofs, studied in this paper, employ contingent identity, free principles of quantification of the 1st order variables and classical principles of quantification of the 2nd order variables. All these theories are strongly complete wrt. classes of modal structures containing families of world-varying objectual domains of the 1st order and constant conceptual domains of the 2nd order. In such structures, terms of the 1st order receive only rigid extensions, which are elements of the union of all 1st order domains. (...) Terms of the 2nd order receive extensions and intensions. Given a family of preselected world-varying objectual domains of the 2nd order, non-rigid extensions of the 2nd order terms belong always to a preselected domain connected with a given world. Rigid intensions of the 2nd order terms are chosen from among members of a conceptual domain of the 2nd order, which is the set of all functions from the set of worlds to the union of all 2nd order preselected domains such that values of these functions at a given world belong to a preselected domain connected with this world. (shrink)

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. (shrink)

The aim of this paper is to prove strong completeness theorems for several Anderson-like variants of Gödels theory wrt. classes of modal structures, in which: (i). 1st order terms order receive only rigid extensions in the constant objectual 1st order domain; (ii). 2nd order terms receive non-rigid extensions in preselected world-relative objectual domains of 2nd order and rigid intensions in the constant conceptual 2nd order domain.

In the paper we introduce a wide range of Anderson-like variants of Gödel's theory and prove for each of them strong completeness theorem wrt. corresponding class of modal structures.These theories — all formulated in the 2nd order modal language with a 2nd order unary predicate of positiveness — differ among themselves with respect of: properties of the necessity operator and of the predicate of positiveness, axioms characterizing identity between 1st sort terms, definitions of identity between 2nd sort terms, the treatment (...) of Godlikeness as a term of the 2nd order, the treatment of necessary existence as a term of the 2nd order if there exists the necessity of its presence, the treatment of Ix (Ix(y) is read: terms x, y are identical), for every 1st sort term x, as a term of the 2nd order if there exists the necessity of its presence.Concerning to semantical commitments, the following are essential: (i) 1st order terms order receive only rigid extensions in the constant objectual 1st order domain; (ii) 2nd order terms receive non-rigid extensions in the constant objectual domain of 2nd order and rigid intensions in the constant conceptual 2nd order domain; (iii) The behavior of the identity between 1st sort terms shifts from possible worlds to possible worlds. (shrink)