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)

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)

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)

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)

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.

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)

Latest version of my SEP entry on ontological arguments, which first appeared in 1996. General discussion of ontological arguments. Includes a brief historical overview, a taxonomy of different kinds of ontological arguments, a brief survey of objections to the different kinds of ontological arguments identified in the taxonomy, and more extended discussions of Anselm's ontological argument (Proslogion 2), Godel's ontological argument, and Plantinga's ontological argument.

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).

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)

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 discusses: (1) Anselm’s ontological argument and its criticism by Gaunilo; (2) Plantinga’s ontological argument and its criticism by Mackie and Sobel; and (3) a simplified version of Gödel’s ontological argument. It also looks carefully at (4) Kant’s attempt to show that it is impossible for there to be a successful ontological argument.

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.

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 chapter contains sections titled: The History of the Argument The Modal Ontological Argument The Possibility of a Perfect Being: Leibniz The Possibility of a Perfect Being: Gödel The Rationality of Belief in a Perfect Being: Plantinga Summary Works cited.

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 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.

This was an attempt to show what is wrong with Anselm’s ‘Ontological Argument’ for the existence of God. My present view is that Peter Millican has given us a similar, but much better line of attack in his “The One Fatal Flaw….” Paper.

The first aim of this paper is to try and determine what St. Anselm meant in his original argument in the Proslogion. This needs to be done because not only are the writers who expound his demonstration divided in their interpretations of it, and these interpretations quite different, but, very strangely, one does not find that they mention that there is any ambiguity and that other writers construe Anselm's words differently from themselves. Since there are in fact two arguments (...) in the classical formulation of his proof, I want to try and disentangle them, decide which of them Anselm intendedand what is its connection with the demonstration in the third chapter of the Proslogion, and thus to offer an opinion on what exactly the ontological argument of St. Anselm is. Then on the basis of the clarification, I hope to show what it was that Gaunilo and St. Thomas attacked, to touch briefly on the relation of the 17th century ontological argument to Anselm's, and in general to indicate the curious round-aboutness of the historical criticism of Anselm. Anselm argued and Gaunilo and Thomas refuted something, though not, I think, precisely what Anselm meant, or, at any rate, not all that he meant. Actually it was Descartes that Thomas refuted by refuting half of Anselm. Kant refuted the other half of Anselm in an irrelevant criticism of Descartes. So that it could properly be said that Anselm's proof has been broken by philosophers who were either only partly aware of what he was maintaining, or who never had him in mind at all. Finally, because the important thing about the ontological argument, in my opinion, is not whether it is a demonstration of God's existence, but whether its refutation is a demonstration of the impossibility of God's existence, and because this question seems to arise most sharply in connexion with Anselm's position, I want to say something about that. (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)

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)

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.

Much of the difficulty surrounding Anselm's ontological argument has been generated by ignoring its metaphysical framework. Examination of the Proslogion and Monologion shows it to be a platonic argument, thus disposing of the 'Lost Island' objection, among others. Contemporary modal interpretations are neither correct versions of, nor advances upon, the original: Hartshorne confuses modal status with truth-value. The Proslogion II argument is valid; it is the sense of Anselm's definition and conclusion which is at issue. Since (...) the argument explicates why God, defined as that which is maximally real, must be real, questions of the logic of 'existence' are irrelevant; however, what 'x is real' means depends on what sort of entity x is, so that Anselm's conclusion remains uncertain until and unless it can be shown in what the non-fictionality of God might consist. Indeed, can God be anything but a fiction? If he can, then he must be real, for, I contend, 'God is eternal and self-sufficient' (which Anselm, as a platonist, claims for him) entails that 'God is real' is necessary. Whether it is necessarily true or necessarily false, then, depends on the coherence of otherwise of 'God', a coherence Anselm assumes. The assumption is disguised by his - and his commentators' - failure to distinguish between the bearer of the name 'God', and the supreme reality described as 'god'. We may thus ask, Can God be god? To discover what sort of entity God is, we must see what may be predicated of him: but neither analogia entis, the language-game theory, nor analogia fidei solve the problem of how attributes may be predicated of a being who is god. The question of God's reality remains unresolved; but Anselm's argument at least shows how and why it is a conceptual question. (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)

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 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)

This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience (...) with the rigor of mathematical proof. Moreover, since the incompleteness theorems concern formal deductive systems, it would help for a potential reader of this book to have some familiarity with at least elementary logic.There are, of course, a number of informal expositions of the incompleteness theorems. I do not see the need to engage in a comparative study. The second, and more important, goal of the present book is to discuss some alleged consequences of the incompleteness theorems and, in particular, to debunk thoroughly claims in philosophy, religion, and literary criticism that are supposed to be established, or at least bolstered, by incompleteness. The execution of this part of the book is masterly. The arguments are clear and compelling.The book has eight chapters, each between eight and twenty-eight pages. The opening chapter gives a very brief account of the incompleteness theorems and a sketch of Gödel's life and work. Chapter 2 gives the main overview of the incompleteness theorems, plus …. (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)

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)

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)

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.).

The most eminent exponents of the ontological argument for the existence of God have been characterized as well by a common emotional ingredient — a concern with individual guilt. Anselm, Josiah Royce, Karl Barth, and Norman Malcolm in their respective ways have made the experience of guilt a central one in their metaphysical standpoints. The hypothesis is therefore advanced that the validity which such thinkers have found in the ontological argument is the expression of a frame (...) of mind which we can call 'logical masochism'; under the influence of such emotions, the ontologian bows his logical powers submissively before an entity the existence of which he cannot question because of the guilt that such questioning would arouse. Under such social circumstances, on the other hand, as during the eighteenth century, when the psychology of individual guilt subsided, the appeal of the ontological argument likewise declines. (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)

First, I suggest that it is possible to make some further improvements upon the Gödelian ontological arguments that Pruss develops. Then, I argue that it is possible to parody Pruss's Gödelian ontological arguments in a way that shows that they make no contribution towards 'lowering the probability of atheism and raising the probability of theism'. I conclude with some remarks about ways in which the arguments of this paper can be extended to apply to the whole family of (...) Gödelian ontological arguments. (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)

“Doing Gender,” Candance West and Don Zimmerman’s famous 1987 article, has become a folk concept—a trope or commonsense resource within the sociology of gender. Yet at the same time, most gender scholars overlook its ethnomethodological premise, visible in both poststructuralist misunderstandings of its argument outside the discipline of sociology and what I term a realist misunderstanding of it in the study of structures and identities within the discipline. Reading West and Zimmerman queerly while clarifying ethnomethodology’s ontology, I refocus attention (...) for critical scholarship on ethnomethodology’s analytic sensibilities for research on gender, race, and sexuality, among other embodiments. Specifically, ethnomethodology reframes a vision of actors as relational, practical actors; repositions gender as accountable, jointly produced social relations, not individual identity; and foregrounds resistance in addition to conformity. Hence, my gender is not mine; it is ours. Ethnomethodology’s ontological shift in temporality to reality-in-production enables interpretive-materialism: a queer, anti-racist, intersectional sociology that is future-facing and in motion. (shrink)