A simplified variant of Gödel’s ontological argument is presented. The simplified argument is valid already in basic modal logics K or KT, it does not suffer from modal collapse, and it avoids the rather complex predicates of essence (Ess.) and necessary existence (NE) as used by Gödel. The variant presented has been obtained as a side result of a series of theory simplification experiments conducted in interaction with a modern proof assistant system. The starting point for these (...) experiments was the computer encoding of Gödel’s argument, and then automated reasoning techniques were systematically applied to arrive at the simplified variant presented. The presented work thus exemplifies a fruitful human-computer interaction in computational metaphysics. Whether the presented result increases or decreases the attractiveness and persuasiveness of the ontological argument is a question I would like to pass on to philosophy and theology. (shrink)

Over the past decade, variants of Gödel's ontological arguments have been critically examined using modern symbolic AI technology. Computers have unearthed new insights about them and even contributed to the exploration of new, simplified variants of the argument, which now need to be further investigated by theologians and philosophers. In this article, I provide a brief, informal overview of these contributions and engage in a discussion of the possible future role of AI technology for the rigorous assessment of (...) arguments in theology and philosophy. (shrink)

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 ultrafilters, and a careful distinction between extensions and intensions of positive properties. (shrink)

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)

Gödel's ontological argument is related to Gödel's view that causality is the fundamental concept in philosophy. This explicit philosophical intention is developed in the form of an onto-theological Gödelian system based on justification logic. An essentially richer language, so extended, offers the possibility to express new philosophical content. In particular, theorems on the existence of a universal cause on a causal "slingshot" are formulated.

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 Godel’s axioms.

In his "Ontological proof", Kurt Gödel introduces the notion of a second-order value property, the positive property P. The second axiom of the proof states that for any property φ: If φ is positive, its negation is not positive, and vice versa. I put forward that this concept of positiveness leads into a paradox when we apply it to the following self-reflexive sentences: (A) The truth value of A is not positive; (B) The truth value of B is positive. (...) Given axiom 2, sentences A and B paradoxically cannot be both true or both false, and it is also impossible that one of the sentences is true whereas the other is false. (shrink)

The paper shows that it is possible to obtain a "slingshot" result in Gödel's theory of positiveness in the presence of the theorem of the necessary existence of God. In the context of the reconstruction of Gödel's original "slingshot" argument on the suppositions of non-Fregean logic, this is a natural result. The "slingshot" result occurs in sufficiently strong non-Fregean theories accepting the necessary existence of some entities. However, this feature of a Gödelian theory may be considered not as a (...) trivialisation, but as an intended consequence of Gödel's ontotheological views. (shrink)

This paper presents detailed formalizations of ontological arguments in a simple modal natural deduction calculus. The first formal proof closely follows the hints in Scott’s manuscript about Gödel’s argument and fills in the gaps, thus verifying its correctness. The second formal proof improves the first one, by relying on the weaker modal logic KB instead of S5 and by avoiding the equality relation. The second proof is also technically shorter than the first one, because it eliminates unnecessary detours (...) and uses Axiom 1 for the positivity of properties only once. The third and fourth proofs formalize, respectively, Anderson’s and Bjørdal’s variants of the ontological argument, which are known to be immune to modal collapse. (shrink)

This chapter contains sections titled: The Validity of Anselm's Ontological Argument The Truth of the Anselmian Premises On Whether Anselm's Ontological Argument Begs the Question On Parodies The Validity of the Ontological Argument of Descartes and Leibniz On the Truth of the Descartes–Leibniz Premises Critiques of the Descartes–Leibniz Ontological Argument Ontological Arguments of the Twentieth Century Gödel's Ontological Argument On Whether Gödel's Argument is Sound The Modal Perfection (...) class='Hi'>Argument The Temporal‐Contingency Argument Conclusion References Appendix 1. Logic Matters Appendix 2. Formal Proofs of Some Modal Arguments. (shrink)

This paper aims to show that Godel's ontological argument can be parodied in much the same kind of way in which Gaunilo parodied Anselm's Proslogion argument. The parody in this paper fails; there is a patch provided in "Reply to Gettings" (Analysis 60, 4, 2000, 363-7).

Gödel's ontological argument is a formal argument for a being defined in terms of the concept of a positive property. I shall defend several versions of Gödel's argument, using weaker premises than Anderson's (1990) version, and avoiding Oppy's (1996 and 2000) parody refutations.

This paper discusses recent work on higher-order ontological arguments, including work on arguments due to Gödel, Maydole and Pruss. After setting out a range of these arguments, the paper seeks to highlight the principal difficulties that these kinds of arguments confront. One important aim of the paper is to cast light on Gödel's ontological argument by way of an examination of a range of related higher-order arguments.

This book is a unique contribution to the philosophy of religion. It offers a comprehensive discussion of one of the most famous arguments for the existence of God: the ontological argument. The author provides and analyses a critical taxonomy of those versions of the argument that have been advanced in recent philosophical literature, as well as of those historically important versions found in the work of St Anselm, Descartes, Leibniz, Hegel and others. A central thesis of the (...) book is that ontological arguments have no value in the debate between theists and atheists. There is a detailed review of the literature on the topic (separated from the main body of the text) and a very substantial bibliography, making this volume an indispensable resource for philosophers of religion and others interested in religious studies. (shrink)

Gödel's modal ontological argument is the centerpiece of an extensive examination of intensional logic. First, classical type theory is presented semantically, tableau rules for it are introduced, and the Prawitz/Takahashi completeness proof is given. Then modal machinery is added to produce a modified version of Montague/Gallin intensional logic. Finally, various ontological proofs for the existence of God are discussed informally, and the Gödel argument is fully formalized. Parts of the book are mathematical, parts philosophical.

Nous publions sous ce titre la traduction française de l'essai philosophique de Kurt Gödel intitulé: «Is Mathematics Syntax of Language?» Inédit jusqu'à présent, l'original paraîtra dans le 3e volume des Collected Works de Gödel, dont la publication est imminente. Nous savons par Hao Wang que, le 15 mai 1953, Paul Arthur Schilpp avait invité Gödel à apporter sa contribution au volume consacré à Carnap dans The Library of Living Philosophers. Le manuscrit de Gödel «Carnap and the Ontology of Mathematics», devait (...) être remis au plus tard le 2 avril 1954. Dés le 2 juillet 1953, Gödel donnait son accord à Schilpp pour la rédaction d'un article intitulé «Some Observations on the Nominalistic View of the Nature of Mathematics.» Cependant, le volume consacré à Carnap paraîtra en 1963 sans qu'y figure la contribution de Gödel. En effet — toujours selon Wang — Gödel avait informé Schilpp, le 2 février 1959, qu'il renonçait à l'envoi de l'article, dont il avait composé, surtout en 1954–1955, plusieur s versions sous le titre «Is Mathematics Syntax of Language?» Pour expliquer cet abandon, Wang avance les raisons suivantes: En premier lieu, Gödel était bien convaincu d'avoir réfuté la thése syntaxique, mais il n'était pas satisfait de son argumentation en faveur du platonisme. Deuxiémement, les critiques de Gödel s'adressent aux théses soutenues par Carnap en 1934 plutôt qu'à celles que ce dernier défendra vingt ans aprés, et qui paraissent moins hostiles au platonisme de son interlocuteur. II pouvait paraître douteux, en troisieme lieu, que Carnap ait eu le temps de rédiger, en 1959, la réponse qu'il aurait du faire, conformément aux régies de la collection, à Gödel. Or, en 1943 déjà, Gödel avait voulu retirer sa contribution au volume consacré à Russell dans la même serie, parce que ce dernier n'avait pu répondre à l'article de Gödel dont la rédaction lui avait été communiquée trop tardivement. (shrink)

This paper introduces the special issue on Formal Approaches to the Ontological Argument of the Journal of Applied Logics (College Publications). The issue contains the following articles: Formal Approaches to the Ontological Argument, by Ricardo Sousa Silvestre and Jean-Yves Béziau; A Brief Critical Introduction to the Ontological Argument and its Formalization: Anselm, Gaunilo, Descartes, Leibniz and Kant, by Ricardo Sousa Silvestre; A Mechanically Assisted Examination of Begging the Question in Anselm’s Ontological Argument, (...) by John Rushby; A Tractarian Resolution to the Ontological Argument, by Erik Thomsen; On Kant’s Hidden Ambivalence Toward Existential Generalization in his Critique of the Ontological Argument, by Giovanni Mion; The Totality of Predicates and the Possibility of the Most Real Being, by Srećko Kovač; An Even More Leibnizian Version of Gödel’s Ontological Argument, by Kordula Świętorzecka and Marcin Łyczak; A Case Study On Computational Hermeneutics: E. J. Lowe’s Modal Ontological Argument, by David Fuenmayor. (shrink)

Gödel’s Ontological argument is distinctive because it is the most sophisticated and formal of ontological arguments and relies heavily on the notion of positive property. Gödel uses a third-order modal logic with a property abstraction operator and property quantification into modal contexts. Gödel describes positive property as "independent of the accidental structure of the world"; "pure attribution," as opposed to privation; "positive in the 'moral aesthetic sense.'" Pure attribution seems likely to be related to the Leibnizian concept (...) of perfection.By a careful examination of the formal semantics of third-order modal logic with property abstraction together with a Completeness result for third-order modal logic with property abstraction for faithful models that I previously developed in 2000 in my work, Gödel’s Ontological Argument, I argue that it is not possible to develop a sufficient applied third-order modal semantics for Gödel’s ontological argument. As I explore possible approaches for an applied semantics including anti-Realist accounts of the semantics of modal logic compatible with Actualism, I argue that Gödel makes implicit philosophical assumptions which commit him to both possibilism (the belief in merely possible objects) and modal realism (the belief in possible worlds). (shrink)

This paper examines the so-called modal ontological argument. It pays special attention to the role that the symmetry and transitivity of the accessibility relation play in the argument, and examines various approaches to a defense of the “possibility premise,” the premise of the argument that states that the existence of a perfect being is metaphysically possible. It contains an analysis of Gödel’s attempt to show that this premise is true, and of a recent formulation by David (...) Johnson of Gödel’s argument. (shrink)

In this paper I will suggest an interpretation of Spinoza's ontological argument on which the argument, properly construed, is valid, and Spinoza, if granted the claim that it is possible that God exists, is successful in obtaining the conclusion of the argument. The interpretations given by H.A. Wolfson, G.H.R. Parkinson, and William A. Earle will then be argued to be deficient on textual and logical grounds. Leibniz’ assessment of the argument, namely that it “permits us (...) only to conclude that God's existence follows if his possibility is already proven,” is thus held to be correct. (shrink)

Hargrove’s ontological argument for Aesthetic Foundation of Wilderness Preservation was the extension of the thinking paradigm from human beings to moral consideration and aesthetic appreciation of wilderness. The thinking paradigm was based on the framework of traditional Western philosophy, which includes metaphysics, epistemology, and ethics. In this article, I will examine Hargrove’s ontological argument by making a little comparison with the perspective of a Chinese philosophical framework. This paper will be divided into three parts. In the (...) part I, I will summarize Hargrove’s ontological argument for natural beauty and environmental preservation. In part II, I will examine the metaphysical framework in Hargrove’s ontological argument. I argue it is the extension of Plato’s metaphysics of the ultimate reality to wilderness. In Part III, I will examine the ethical and aesthetic framework in his ontological argument. I point out Hargrove’s ethical framework comes from ethical rationalism and his aesthetic framework is the legacy of Plato’s aesthetics. By comparison, Chinese metaphysics stresses the process and Chinese ethics stresses care. The two different directions promote the aesthetic appreciation of agricultural land rather than wilderness. (shrink)

Anselm’s ontological argument is an argument for the existence of God. This paper presents Iris Murdoch’s ontological argument for the existence of the Good. It discusses her interpretation of Anselm’s argument, her distinctive appropriation of it, as well as some of the merits of her version of the argument. In doing so, it also shows how the argument integrates some key Murdochian ideas: morality’s wide scope, the basicness of vision to morality, moral (...) realism, and Platonism. (shrink)

I argue that spinoza's ontological argument is successful when it is understood to have two premises: (i) it is possible for god to exist, (ii) it is necessary that, if god exists, he necessarily does. the argument is valid in s5. spinoza is in a position to establish the second premise of the argument on the basis of his definitions and axioms. the first premise was assumed to be true, but, as leibniz noted, it must be (...) established for the conclusion of the argument to be forthcoming. this is one of the main problems with the argument, and it requires a solution to the problem of the attributes. certain alternate interpretations of spinoza are then argued to be deficient on textual and logical grounds. (shrink)

This article is a reply to Michael Gettings' criticisms of a previous paper of mine on Godel's ontological argument. (All relevant bibliographical details may be found in the article.) I provide a patch to my previous -- faulty -- attempt to provide a parody of Godel's ontological argument on the model of Gaunilo's parody of Anselm's Proslogion 2 argument.

This paper is a reply to Stephen Makin's response to my previous criticism of his defense of a conceptual ontological argument. (All relevant bibliographical details are provided in this paper.).

Saint Anselm’s ontological argument is usually interpreted either (1) as an attempt to deductively prove God’s existence or (2) as a form of prayer, which is not intended to “prove” God’s existence, but rather to deepen the devotion of those who already believe. In this paper I attempt to find a mean between these two interpretations, showing that while Anselm’s argument is not a deductive proof, it is nevertheless a proof of God’s existence. I argue that Anselm’s (...) ontological argument is analogous to Aristotle’s to elegktikōs apodeixai (retorsive argument) for the truth of the principle of non-contradiction in Metaphysics IV: an argument that does not move from premises to conclusion, but rather demonstrates the truth of its conclusion by showing that its conclusion is always presupposed. I argue that interpreting Anselm’s ontological argument in this way exempts it from the most common objections against it. (shrink)

We must agree, I think, with Frank Ebersole that there is something preposterous in supposing that the God of religious belief, the God who handed down tablets to Moses on Mt. Sinai, etc., should be proven to exist by the ontological argument. Indeed, when we place the one, the ontological argument, by the side of the other, the God of religious belief, there seems hardly to be any connection between them. But if we agree to this (...) perception of things, what sense can we make of the ontological argument, for example, the “pull” that it sometimes exerts on us? (shrink)

This essay takes up a challenge recently posed by Graham Oppy: to clearly express, in premise-conclusion form, Hegel's version of the ontological argument. In addition to employing this format, it seeks to supplement existing treatments by locating a core component of Hegel's argument in a slightly different place than is common. Whereas some prominent recent treatments (Williams, Bubbio, Melechar) focus on Hegel's definition of the Absolute as the Concept, from the third part of his Science of Logic (...) (the Doctrine of the Concept), mine focuses on earlier definitions from the first (the Doctrine of Being). As I hope to show, there are even more resources in Hegel's Logic for an ontological argument than those emphasized in recent treatments: the concept, the Idea, etc. Already in the first third of the Logic, we find a compelling response to a famous Kantian counter-argument to the ontological proof. The counter-argument is summed up in the phrase ‘existence [Sein] is not a real predicate’. Hence, Hegel's response as I interpret it will take the form of a competing analysis of Being, a Lehre vom Sein (Doctrine of Being). What do we learn when we put the ontology back into Hegel's ontological argument? That Being is neither predicate, nor subject, nor copula, but a monist (or ‘infinite’) category. The larger importance of this exercise to our understanding of Hegel's thought lies in the way it clarifies his profound debt to even non-idealist conceptions of God, such as the one espoused by Spinoza. (shrink)

This paper is my second contribution to the Szatkowski volume. In the first paper, I provide a critical discussion of Bob Maydole's ontological arguments. In this second paper, I respond to Maydole's critical response to my first paper. My overall verdict is that Maydole does not successfully defend his arguments against my critical attack.

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)

This text is expository. We explain Gödel’s ‘Master Argument’ for incompleteness as distinguished from the 'official' proof of his 1931 paper, highlight its attractions and limitations, and explain how some of the limitations may be transcended by putting it in a more abstract form that makes no reference to truth.

Anselm's ontological argument is one of the most fascinating, most controversial, and most misunderstood arguments in the entire history of Western thought. By centring the argument firmly in the Neoplatonic tradition within which Anselm was writing, Understanding Anselm's Ontological Argument sheds fresh light and clarity on this enigmatic piece of philosophy. It argues that, far from resting upon a fallacy or illegitimately attempting to define God into existence, Anselm's argument is a powerful and plausible (...) philosophical proof, and deserves to be taken seriously as such. Written to be understandable for specialists and non-specialists alike, Understanding Anselm's Ontological Argument is ideal for scholars and researchers in philosophy of religion and philosophy in the Middle Ages (especially Neoplatonism) as well as for medievalists in general. (shrink)

Gödel’s slingshot-argument proceeds from a referential theory of definite descriptions and from the principle of compositionality for reference. It outlines a metasemantic proof of Frege’s thesis that all true sentences refer to the same object—as well as all false ones. Whereas Frege drew from this the conclusion that sentences refer to truth-values, Gödel rejected a referential theory of definite descriptions. By formalising Gödel’s argument, it is possible to reconstruct all premises that are needed for the derivation of Frege’s (...) thesis. For this purpose, a reference-theoretical semantics for a language of first-order predicate logic with identity and referentially treated definite descriptions will be defined. Some of the premises of Gödel’s argument will be proven by such a reference-theoretical semantics, whereas others can only be postulated. For example, the principle that logically equivalent sentences refer to the same object cannot be proven but must be assumed in order to derive Frege’s thesis. However, different true (or false) sentences can refer to different states of affairs if the latter principle is rejected and the other two premises are maintained. This is shown using an identity criterion for states of affairs according to which two states of affairs are identical if and only if they involve the same objects and have the same necessary and sufficient condition for obtaining. (shrink)

After introductory reminder of and comments on Gödel’s ontological proof, we discuss the collapse of modalities, which is provable in Gödel’s ontological system GO. We argue that Gödel’s texts confirm modal collapse as intended consequence of his ontological system. Further, we aim to show that modal collapse properly fits into Gödel’s philosophical views, especially into his ontology of separation and union of force and fact, as well as into his cosmological theory of the nonobjectivity of the lapse (...) of time. As a result, modal collapse should not be conceived in Gödel so much as a deficit, but rather as a kind of the rise of modalities to the “perfect” being. We further show that, in accordance with Gödel’s ontology, the concepts of modality and time should be derived in terms of the “fundamental philosophical concept” of cause. To give an example of how a formalization of such causative Gödelian ontology and ontological proof might look, we propose the transformation of GO into a kind of causally re-interpreted justification logic. (shrink)

Al-Muḥaqqiq al-Iṣfahānī (1878-1942) and Spinoza (1632-1677), two prominent intellectuals of the Islamic and Western worlds respectively, have proposed different versions of the ontological argument for the existence of God. I present five versions of al-Muḥaqqiq al-Iṣfahānī’s argument in three general dimensions: first, the concept of the necessary being (wājib al-wujūd) as a mental concept; second, the concept of the necessary being as a representation of something external; and finally, the reality of the necessary being or what externally (...) exists by its essence. Only one of these versions is compatible with al-Muḥaqqiq al-Iṣfahānī’s words. On the other hand, Spinoza has presented six arguments in which he has deployed the concept of God in three ways: the concept of God as a concept, the concept of God as a representation of something external, and the mental existence of this concept. In this paper, I compare the accurate construal of al-Muḥaqqiq al-Iṣfahānī’s argument with Spinoza’s six arguments, whereby I make a case for a strong similarity between the grounds and forms of the ontological argument as formulated by these two intellectuals. (shrink)

Gödel argued that the incompleteness theorems entail that the mind is not a machine, or that certain arithmetical propositions are absolutely undecidable. His view was that the mind is not a machine, and that no arithmetical propositions are absolutely undecidable. I argue that his position presupposes that the idealized mathematician has an ability which I call the recursive-ordinal recognition ability. I show that we have this ability if, and only if, there are no absolutely undecidable arithmetical propositions. I argue that (...) there are such propositions, but that no recognizable example of one can be identified, even in principle. (shrink)

One of the perennial defenses of God’s existence is the ontological argument, associated particularly with St. Anselm. Ontological arguments are generally discounted in today’s philosophical circles, but a remarkably sophisticated version was developed by the preeminent mathematical logician Kurt Gödel. This short paper evaluates the major objection to ontological proofs and finds Gödel’s formulation convincing.

Iris Murdoch develops a version of the Ontological Argument as a moral argument for the existence of a transcendent and perfect Platonic Good. I argue that her version of the argument over-emphasises moral goodness as a distant and intangible ideal to which we are inevitably attracted, and towards which we may progress, but which, apart from occasional revelations in saintly lives and great art, is normally only available in glimpses and intimations, and which remains mysterious. The (...) argument is better construed as concerning the moral reality of human beings as sacred or inviolable, and the moral demands this makes upon us—a reality that is proximate, palpable, inescapable and highly familiar. (shrink)

We offer a reading of Anselm's Ontological Argument inspired by Wittgenstein which focuses on the fact that the “argument” occurs in a prayer addressed to God, making it a strange argument since as a prayer it seems to presuppose its conclusion. We reconstruct the argument as expressive. Within the religious perspective, the issues are to be focused on the right object not to present an argument for the existence of God. While this sort of (...) reading lets us understand much about the argument, it also opens new avenues of criticism, one of which is the problem of worship. (shrink)

Anselm described god as “something than which nothing greater can be thought” [1, p. 93], and Descartes viewed him as “a supreme being” [7, p. 122]. I first capture those characterizations formally in a simple language for monadic predicate logic. Next, I construct a model class inspired by Stoic and medieval doctrines of grades of being [8, 20]. Third, I prove the models sufficient for recovering, as internal mathematics, the famous ontological argument of Anselm, and show that (...) class='Hi'>argument to be, on this formalization, valid. Fourth, I extend the models to incorporate a modality fit for proving that any item than which necessarily no greater can be thought is also necessarily real. Lastly, with the present approach, I blunt the sharp edges of notable objections to ontological arguments by Gaunilo and by Grant. A trigger warning: every page of this writing flouts the old saw “Existence is not a predicate” and flagrantly. (shrink)