Adams, Robert Merrihew. 1995. "Introductory Note to *1970." In Kurt Gödel - Collected Works Vol. Iii: Unpublished Essays and Lectures, edited by Feferman, Solomon, 388-402.
New York: Oxford University Press.
Anderson, Anthony C. 1990. "Some Emendations of Gödel's Ontological Proof." Faith and Philosophy no. 7:291-303.
"Kurt Gödel's version of the ontological argument was shown by J. Howard Sobel to be defective, but some plausible modifications in the argument result in a version which is immune
to Sobel's objection. A definition is suggested which permits the proof of some of Gödel's axioms."
Anderson, Anthony C., and Gettings, Michael. 1996. "Gödel Ontological Proof Revisited." In Gödel '96, edited by Hájek, Petr, 167-172. Berlin, Heidelberg: Springer.
"Gödel's version of the modal ontological argument for the existence of God has been criticized by J. Howard Sobel (5) and modified by C. Anthony Anderson (1). In the present paper
we consider the extent to which Anderson's emendation is defeated by the type of objection first offered hy the Monk Gaunilo to St. Anselm's original Ontological Argument. And we try to push the
analysis of this Gödelian argument a bit further to bring it into closer agreement with the details of Gödel's own formulation. Finally, we indicate what seems to be the rnain weakness of this
emendation of Gödel's attempted proof."
Bjørdal, Frode. 1999. "Understanding Gödel's Ontological Argument." In The Logica Yearbook 1998, edited by Childers, Timothy, 214-217. Praha: Filosofia.
Cook, Roy T. 2004. "God, the Devil, and Gödel's Other Proof." In The Logica Yearbook 2003, edited by Behounek, Libor, 97-109. Prague: Philosophia.
"Gödel's 1970 proof of the existence of a god-like being (i.e., a being having all the 'perfective' properties) is investigated. The proof is streamlined and reformulated within the
weakest logic possible - and intuitionistic version of second-order logic with a modal operator no stronger than the system K. It is shown that even in this weak context one can derive that it is
necessary that a god-like being is at least possible. Finally, the prospects for a similar proof of the existence of the Devil (or, more carefully, a being who lacks all perfective properties) is
investigated. Technical reasons are given for why such a parallel proof of the existence of an evil being is not forthcoming."
Czermak, Johannes. 2002. "Abriss Des Ontologischen Argumentes." In Kompendium Zum Werk, edited by Köhler, Eckehart, Buldt, Bernd, De-Pauli Schimanovich-Göttig, Werner,
Klein, Carsten, Stöltzner, Michael and Weibel, Peter, 309-324. Wien: Öbv & Hpt.
Kurt Gödel. Wahrheit & Beweisbarkeit. Vol.1: Dokumente und historische analysen; Vol. 2: Kompendium zum Werk.
Fitting, Melvin. 2002. Types, Tableaux and Gödel's God. Dordrecht: Kluver Academic Press.
See Part III. Ontological arguments. Chapter 10. Gödel argument, background 133; Chapter 11. Gödel argument, formally 145-172.
"Finally, Part III is devoted to ontological proofs. Chapter 10 gives a brief history and analysis of arguments of Anselm, Descartes, and Leibniz. This is followed by a longer,
still informal, presentation of the Gödel argument itself. Formal methods are applied in Chapter 11, where Gödel's proof is examined in great detail. While Gödel's argument is formally correct, some
fundamental flaws are pointed out. One, noted by Sobel, is that it is too strong -- the modal system collapses. This could be seen as showing that free will is incompatible with Gödel's assumptions.
Some ways out of this are explored. Another flaw is equally serious: Gödel assumes as an axiom something directly equivalent to a key conclusion of his argument. The problematic axiom is related to a
principle Leibniz proposed as a way of dealing with a hole he found in an ontological proof of Descartes. Descartes, Leibniz, and Gödel (and also Anselm) all have proofs that stick at the same point:
showing that the existence of God is possible.
If the Gödel argument is what you are interested in, start with Part III, and pick up earlier material as needed. Many of the uses of the formalism are relatively intuitive. Indeed,
in Gödel's notes on his ontological argument, formal machinery is never discussed, yet it is possible to get a sense of what it is about anyway." (pp. XIII-XIV).
Fuhrmann, André. 2005. "Existenz Und Notwendigkeit - Kurt Gödels Axiomatische Theologie." In Logik in Der Philosophie, edited by Spohn, Wolfgang, Schroeder-Heister, Peter
and Olsson, Erik J., 349-374. Heidelberg: Synchron Publishers.
Gödels ontologischer Beweis pp. 354-360.
Goldman, Randolph Rubens. 2000. Gödel's Ontological Argument, University of California at Berkeley.
Available at UMI Dissertation Express. Order number: 9979637.
Gonçalves, Gomes Nelson. 2006. "Summum Bonum." Analytica.Revista de Filosofia no. 10:43-105.
(The article is in Portuguese).
"This article contains a presentation of Gödel's ontological proof, from both the intuitive and the formal points of view. Two other contemporary variations of it are also
presented. The article
discusses the philosophical and the theological criticisms of the proof. The conclusion to be drawn is that the soundness of Gödel's reasoning is an important logical result, which
nevertheless demands further work of analysis. The metaphysical idea of a positive property (or positive set) for instance needs clarification."
Hájek, Petr. 1996. "Magari and Others on Gõdel's Ontological Proof." In Logic and Algebra, edited by Ursini, Aldo and Agliani, Paolo, 125-136. New York: Dekker.
"Gödel's proof of the necessary existence of God is analyzed from the point of view of modal logic together with related papers by Magari and Anderson. ln particular, Magari's claim
on redundance in axioms is analyzed and shown to be only true for some extension of Godel's system (but true for Anderson's modification, as it as shown elsewhere). Completeness of the underlying
modal logic is proved; and it is shown that the "ontological" proof may use only the logic KD45 (logic of belief) instead of S5 (logic of knowledge)."
———. 2002. "A New Small Emendation of Gödel Ontological Proof." Studia Logica no. 71:149-164.
"Two variants of monadic fuzzy predicate logic are analyzed and compared with the full fuzzy predicate logic with respect to finite model property (properties) and arithmetical
complexity of sets of tautologies, satisfiable formulas and of analogous notion restricted to finite models."
———. 2002. "Der Mathematiker Und Die Frage Der Existenz Gottes." In Kompendium Zum Werk, edited by Köhler, Eckehart, Buldt, Bernd, De-Pauli Schimanovich-Göttig, Werner,
Klein, Carsten, Stöltzner, Michael and Weibel, Peter, 325-336. Wien: Öbv & Hpt.
Kurt Gödel. Wahrheit & Beweisbarkeit. Vol.1: Dokumente und historische analysen; Vol. 2: Kompendium zum Werk.
Hazen, Allen P. 1998. "On Gödel's Ontological Proof." Australasian Journal of Philosophy no. 76:361-377.
"Gödel, in a cryptic note given to Dana Scott in 1970, introduces the notion of a positive property. (Thus the formalized version uses a Third-Order constant, the predicate
P(F), expressing the positivity of the property F). The plausibility of his 'axioms' and the theological relevance of its conclusion depend on the interpretation of this notion; Gödel says that it
means 'positive in the moral aesthetic sense (independently of the accidental structure of the world)', but also allows that it may mean 'pure "attribution" as opposed to
"privation".' The evidence available from his notebooks suggests that he never found an interpretation of this notion that fully satisfied him, and it is perhaps best to assume that he
thought of his ontological argument not as a conclusive proof of the existence of God, but as an attempt at a reconstruction of Leibniz's argument. In any event, he laid down certain axioms
concerning the notion. Any property entailed by a positive property is positive, and the conjunction of two positive properties (the property, that is, that an individual has if and only if it has
both of the given properties) is positive. Together, these amount to saying that the positive properties form a filter on the Boolean algebra of properties. (Gödel adds in a footnote that
the positivity of conjunctions of positive properties holds for arbitrary numbers of conjuncts, not just for two: as we shell see, this includes infinite numbers.) Further, positivity is a
non-contingent feature of a property: any property which is positive is necessarily positive, and no property which is not positive could be.
Another axiom said that, of any pair of properties consisting of a property and its negation (the property necessarily holding of all and only the individuals not possessing the
first property), precisely one is positive. (Positive properties form an ultrafilter.) We are now In a position to prove our first theorem: positive properties are at least possibly
instantiated, Proof: the contradictory property (self-non-identity) entails all properties, including Its own negation (self-Identity), so if it were positive both properties of such a pair would be
positive. (As has been noted by commentators, the proof uses only part of the strength of the latest axiom: of a pair of a property and its negation at most one is positive.)
Gödel's remaining axiom is formulated in terms of a defined notion. First define a property to be an essence of an individual if it is a property, possessed by the
individual (Godel left this clause out in (1), but this appears to have been an oversight - -it is Included In related manuscripts), which entails every property the individual has. The terminology
is somewhat unfortunate: the notion defined is close to Leibniz's notion of the complete concept of an object, but is not at all what current philosophical usage calls an essence. 'Essence'
and 'essentialism' were bad words for most analytic philosophers of the 1960s -- many found the doctrines of Kripke downright shocking -- but, with the flowering of modal metaphysics since, they have
come to have well-understood and generally agreed meanings, defined in terms of de re modal locutions. An individual. x, has a property, F, essentially iff x has F and it is
necessarily the case that x, provided it exists, has F. An essence of x, in this post-1970 sense, is a property which entails all and only the properties x has essentially. To avoid
confusion, therefore, let us re-christen Gödel's
notion: the property an individual has that entails every property it has is its character." pp. 364-365.
Kegler, Jeffrey. 2008. The God Proof: CreateSpace.
A novel on the ontological proof, useful as a non-technical, but reliable, introduction to Gödel's argument.
Koons, Robert C. 2006. "Sobel on G¨Odel’s Ontological Proof." Philosophia Christi no. 8:235-247.
Kovac, Srecko. 2003. "Some Weakened Gödelian Ontological Systems." Journal of Philosophical Logic no. 32:565-588.
"We describe a K B 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 aspect of
Gödel's relation to Kant."
Magari, Roberto. 1988. "Logica E Teofilia." Notizie di logica no. 7:11-20.
Ristampato in: Kurt Gödel - La prova matematica dell'esistenza di Dio - Trino, Bollati Boringhieri, 2006, pp. 99-120.
Muck, Otto. 1992. "Eigenschaften Gottes Im Licht Des Godelschen Arguments." Theologie und Philosophie no. 67:60-85.
"In his sketch of an Ontological Proof (dated February 10th 1970) Kurt Gödel introduces the concept of a positive property and proves the necessary existence of exact one being
which instantiates all positive properties -- he calls it 'summun bonum'. Special emphasis in the discussion of this argument is put on the logical structure of positive property and the
comparison with the concept of (pure) perfection as it is used in traditional philosophy of God for dealing with divine attributes."
———. 1992. "Religioser Glaube Und Gödels Ontologischer Gottesbeweis." Theologie und Philosophie no. 67:263-267.
"In discussing the rationality of religious belief, Franz von Kutschera (Vernuft und Glaube, 1990) criticises attempts to clarify divine attributes and to demonstrate the
existence of God, including Gödel's Ontological Proof. The article argues that the criticism proposed neglects the concept of pure perfection and that the logical structure of this concept can be
clarified in further developing Gödel's concept of positive property."
Oppy, Graham. 1996. "Gödelian Ontological Arguments." Analysis no. 56:226-230.
"The main thesis of this paper is that Gödel's ontological argument is subject to a kind of objection which has hitherto been overlooked, but which has often been levelled at other
ontological arguments, viz. that it can be paralleled by apparently equally persuasive proofs of the existence of beings in which no one should wish to believe. (Compare Gaunilo's objection to St.
Anselm: No one should wish to believe in the existence of an island than which no greater island can be conceived.) "
Park, Woosuk. 2003. "On the Motivations of Gödel's Ontological Proof." Modern Schoolman:144-153.
"In recent years there has been a surge of interest in Gödel's ontological proof of the existence of God. Gödel showed his proof (Gödel *1970) to Scott, and Scott made a note of the
proof and presented it in his seminar at Princeton University in the fall of 1970. From then on, Gödel's proof has become widely circulated. It was finally published in Sobel 1987 as an appendix and
later included in volume three of Gödel's Collected Works. Recent discussions of Gödel's proof mostly start from Sobel's criticisms. As is well known, the most influential criticism of Sobel
is that Gödel's proof leads to a consequence unacceptable to most philosophers, i.e. that all truths are necessary truths. Anderson 1990 viewed this as the modal collapse of Gödel's assumptions, and
tried to save Gödel's proof by some plausible modifications.
Anderson's emendation secured 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.
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 article, 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."
Perzanowski, Jerzy. 1991. "Ontological Arguments Ii - Cartesian and Leibnizian." In Handbook of Metaphysics and Ontology, edited by Burkhardt, Hans and Smith, Barry,
625-633. München: Philosophia Verlag.
Roetti, Jorge Alfredo. 2004. "El Argumento Ontológico: La Variante De Gödel De La Versión De Leibniz." Dialogos.Revista del Departmento de Filosofia Universidad de Puerto
Rico no. 39:77-105.
"The paper contains a full version of the well-known Gödel's variant of the ontological argument. It is based on the Leibniz's version of the proof, the axiom systems of Gödel 1970
and of Anderson 1990, with some weakenings. We discriminate between several types of existence; the form of existence of God we assert is, in a constructive fashion, a weaker one. Later we deal with
Kant's criticism against the predicate of existence and offer several commentaries on the proof, and on topics such as perfection and existence."
Scott, Dana. 1987. "Gödel's Ontological Proof." In On Being and Saying. Essays for Richard Cartwright, edited by Thomson, Judith Jarvis, 257-258. Cambridge: MIT Press.
Sobel, Jordan Howard. 2006. "To My Critics with Appreciation: Responses to My Critics with Appreciation: To Taliaferro, Swinburne, and Koons." Philosophia Christi no.
Szatkowski, Miroslaw. 2005. "Semantic Analysis of Some Variants of Anderson-Like Ontological Proofs." Studia Logica no. 79:317-355.
"The aim of this paper is to prove strong completeness theorems for several Anderson-like variants of Gödel's theory wrt. classes of modal structures, in which: (i). 1st order terms
receive only rigid extensions in the constant objectual 1st order domain; (ii). 2nd order terms receive nonrigid extensions in preselected world-relative objectual domains of 2nd order and rigid
intensions in the constant conceptual 2nd order domain."
Wang, Hao. 1996. A Logical Journey. From Gödel to Philosophy. Cambridge: MIT Press.
See Chapter 3.2 Religion and Gödel's ontological proof pp. 111-121.