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Formal reconstructions of St. Anselm’s ontological argument
Abstract
In this paper, we discuss formal reconstructions of Anselm’s ontological argument. We first present a number of requirements that any successful reconstruction should meet. We then offer a detailed preparatory study of the basic concepts involved in Anselm’s argument. Next, we present our own reconstructions—one in modal logic and one in classical logic—and compare them with each other and with existing reconstructions from the reviewed literature. Finally, we try to show why and how one can gain a better understanding of Anselm’s argument by using modern formal logic. In particular, we try to explain why formal reconstructions of the argument, despite its apparent simplicity, tend to become quite involved.
... One of them is the modal ontological argument (hereinafter MOA), an argument formalizable in a simple zero-order language of (applied) modal logic or an (appropriately enriched) standard first-order language of the theory of possible worlds. 1 1 These versions basically have their origins in Hartshorne (1944), Malcolm (1960) and Plantinga (1974) (although their intuitive formulation already appears in St. Anselm's Proslogion III; see Eder andRamharter 2015, p. 2815;Oppy 2019). They are clearly distinguished here from arguments of the Gödelian kind formulated within second-or higher-order modal theories of positive properties (see Sobel 1987;Benzmüller 2020). ...
... Regardless of whether or not these formulae are used in a particular MOA-version, it is the basic system for explicating the meaning of the modal operators L and M. Therefore, we will take it to be a logical basis for MOA. 14 Now, the MOA-structure can be presented as an arrangement <T, (1), , X, g>, where is a sentence of the language of the (applied) modal logic including the sentence g, and X is a set of sentences resulting from the substitution of propositional variables by the sentence g 10 The operators M and L can also be understood as formal counterparts of Anselm's "it is conceivable that" and "it is not conceivable that not", respectively (see Eder andRamharter 2015, p. 2814). In turn, for the constant g, the following interpretation is sometimes proposed: "There is a necessarily existent being that has all perfections essentially" (van Inwagen, 2012, p. 158;2018, p. 242). ...
... Cf., for example, van Inwagen's remark that the formula 'Lp p' "must be valid in every modal system in which the sentential operators represent possibility and necessity in any intuitive sense" (van Inwagen 2018, p. 242). Cf. also Eder and Ramharter (2015), where the authors state that the T-system would "seem to be mandatory on any modal conception of conceivability which can claim to be faithful to Anselm's reasoning" (p. 2814). ...
This paper deals with some metaphilosophical aspects of the modal ontological argument originating from Charles Hartshorne. One of the specific premises of the argument expresses the idea that the existence of God is not contingent. Several well-known versions of the argument have been formulated that appeal to different ways of clarifying the latter. A question arises: which of the formally correct and relevant versions is proper or basic? The paper points to some criteria of formal correctness, and distinguishes two types of relevance for these versions: strong and weak. Its aim is to furnish a strictly worked out answer to the question, taking into account each of these types. As a result, a very simple, formally correct and (weakly) relevant version of the modal ontological argument is formulated. The results obtained are also used to criticize a popular belief about the relations in which the main versions of the modal ontological argument stand to one another. Key words: structure of argument, formal correctness of argument, simplicity of argument, relevance of argument, modal logic, mini-theories of God.
... Consider, for example, Meyer's [6] attempt to rehabilitate the cosmological argument with a system that extends ZFC-this may serve as an example to which one can apply GST. Another issue that might yield another system is given by the applicability of mathematics to natural sciences-a phenomenon to which Wigner [13] referred as "The Unreasonable Effectiveness of Mathematics in the Natural Sciences," and one that may play a role in a possible argument for G. 10 It is reasonable, therefore, to require that F contain the part of arithmetic that is necessary to GST. 11 To prove F G in a weak system F is pointless; systems F that meet the postulate above usually satisfy the conditions for GST and allow us to infer that for consistent F, F ¬ G. We now have a tentative answer to our questions. ...
... An analysis of reconstructions of Anselm's argument shows that higher order logic is required in order to capture his assumptions. See [11]. ...
... I am not assuming here that the arguments that follow these routes are valid! But we want F to be rich enough to allow for the formulations of such arguments.11 Noteworthy, Wigner[13] ties the unreasonable effectiveness of mathematics in physics with the consistency of mathematics. ...
According to a common view, belief in God cannot be proved and is an issue that must be left to faith. Kant went even further and argued that he can prove this unprovability. But any argument implying that a certain sentence is not provable is challenged by Gödel’s second theorem (GST). Indeed, one trivial consequence of GST is that for any formal system F that satisfies certain conditions and for every sentence K that is formulated in F it is impossible to prove, from F, that K is not provable. In the article, I explore the general issue of proving the unprovability of the existence of God. Of special interest is the question of the relation between the existence of God and the conditions that F must satisfy in order to allow for its subjection to GST.
... 1 For more on the concept of omnipotence and the paradox of the stone see [6]. 2 We are here using the terms "contradictory" and "consistent" as applied also to concepts. A concept C is consistent or coherent if and only if the set composed by "There is an object x which is ...
... Theists must of course try refute claims like this; ideally, they must provide arguments showing that the concept of God is coherent or consistent. 2 One of the most influential theist arguments (which is in fact a family of related arguments) in the history of philosophy is the ontological argument. First proposed by Anselm of Canterbury in the Eleventh Century, the ontological argument has been either analyzed or reformulated in the modern period by philosophers such as Descartes, Spinoza, Leibniz, Hume and Kant. 3 There has been a revival in the interest in the ontological argument in the Twentieth century; besides a growing literature on the topic, contemporary thinkers such Norman Malcolm [10], Charles Hartshorne [5], David Lewis [9], Alvin Plantinga [16] and Kurt Gödel [4] have either offered fresh views on the ontological argument or proposed new versions of it. ...
... 49-57], [1,15,7], [20, pp. 60-65], [12,2]. Second, there have been many new formulations of the ontological argument directly embedded in formal frameworks. ...
This paper presents the special issue on Formal Approaches to the Ontological Argument and briefly introduces the ontological argument from the standpoint of logic and philosophy of religion (more specifically the debate on the rationality of theistic belief).
... There can be interactions between quantifiers and the modal qualifiers so this needs to be done with care. For example, a standard step in modal formulations of of Anselm's Proslogion II argument for the existence of God [4] (the "Ontological Argument," not to be confused with his Proslogion III argument which we used for illustration in the previous section) is to consider "some thing x than which there is nothing greater." This might be formulated as ¬∃y : 3(y > x), which can be read as "there is no y that is greater than x in any (accessible) possible world." ...
... Eder and Ramharter propose the following definition [4,Section 4.2] for the predicate G that recognizes maximally great things under this new interpretation. Here g(x) is the "greatness" of x and is an ordering (actually an uninterpreted predicate) on greatness. ...
Modal logics allow reasoning about various modes of truth: for example, what it means for something to be possibly true, or to know that something is true as opposed to merely believing it. This report describes embeddings of propositional and quantified modal logic in the PVS verification system. The resources of PVS allow this to be done in an attractive way that supports much of the standard syntax of modal logic, while providing effective automation. The report introduces and formally specifies and verifies several standard topics in modal logic such as relationships between the standard modal axioms and properties of the accessibility relation, and attributes of the Barcan Formula and its converse in both constant and varying domains.
... In the spirit of the principle of charity, we may justify adding further complexity to the argument's formalization if we later find out that it is required for its validity. 19 Here, the concepts of necessariness and contingency are meant as properties of beings, in contrast to the concepts of necessity and possibility which are modals. We will see later how both pairs of concepts can be related in order to validate this argument. ...
... More specifically, Eder and Ramharter[19] propose several criteria aimed at judging the adequacy of formal reconstructions of St. Anselm's ontological argument. They also show how such reconstructions help us gain a better understanding of this argument. ...
Computers may help us to better understand (not just verify) arguments. In this article we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language argument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding light on the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem reasoning technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argument’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In computational hermeneutics, the utilization of automated reasoning tools effectively boosts our capacity to expose the assumptions we indirectly commit ourselves to every time we engage in rational argumentation and it fosters the explicitation and revision of our concepts and commitments.
... Verification systems are tools from computer science that are generally used for exploration and verification of software or hardware designs and algorithms; they comprise a specification language, which is essentially a rich (usually higher-order) logic, and a collection of powerful deductive engines (e.g., satisfiability solvers for combinations of theories, model checkers, and automated and interactive theorem provers). I have previously explored renditions of the Argument due to Oppenheimer and Zalta (1991) and Eder and Ramharter (2015) using the PVS verification system (Rushby, 2013(Rushby, , 2016, and those provide the basis for the work reported here. Benzmüller and Woltzenlogel-Paleo (2014) have likewise explored modal arguments due to Gödel and Scott using the Isabelle and Coq verification systems Mechanized analysis confirms the conclusions of most earlier commentators: the Argument is valid. ...
I use mechanized verification to examine several first- and higher-order formalizations of Anselm's Ontological Argument against the charge of begging the question. I propose three different but related criteria for a premise to beg the question in fully formal proofs and find that one or another applies to all the formalizations examined. I also show that all these formalizations entail variants that are vacuous, in the sense that they apply no interpretation to "than which there is no greater" and are therefore vulnerable to Gaunilo's refutation. My purpose is to demonstrate that mechanized verification provides an effective and reliable technique to perform these analyses; readers may decide whether the forms of question begging and vacuity so identified affect their interest in the Argument or its various formalizations. This version updates the paper that originally appeared as Chapter 13 in "Beyond Faith and Rationality: Essays on Logic, Religion and Philosophy" published by Springer to respond to criticisms by Oppenheimer and Zalta.
... • It follows Eder and Ramharter's criteria (Eder & Ramharter, 2015) for the formal reconstruction of Anselm's ontological argument very closely. ...
In 1962 Charles Hartshorne published a modal logic proof formalizing Anselm of Canterbury's ontological argument for the necessary existence of God. This article presents Kurt G\"odel's notes on this proof which have now been discovered in his Nachlass among other theological material, and discusses possible influences on the development of G\"odel's own ontological proof. To complete the picture, strong connections between Anselm of Canterbury's and G\"odel's conceptions of God and his positive properties are pointed out.
... O. Boulnois, L. Honnefelder, S. Knuuttila, J. Marenbon, A. Vos, and many others, with their historical reading of medieval arguments coupled with the intention to explicate those with references to later philosophy and contemporary parlance and by means of some newer tools, stand on the historical-reconstructionist side of the pole, somewhere between the middle and the extreme [Boulnois 1999[Boulnois , 2007[Boulnois , 2014Honnefelder 1990Honnefelder , 1998Honnefelder , 2008Knuuttila 1993Knuuttila , 2017Marenbon 1997Marenbon , 2003Marenbon , 2013Vos 2006Vos , 2018. R. Campbell, K. Rogers, D. Smith, E. Stump, and others, with their interpretation of the medievals' arguments being significantly influenced by the agenda of contemporary philosophy, would take their position between the center and the rational-reconstructionist extreme [Campbell 2018;Eder & Ramharter 2015;Rogers 2007Rogers , 2008Rogers , 2015Smith 2014;Stump 2001Stump , 2003Stump , 2014. Thus, one's research can be located anywhere on this axis, depending on the emphasis either on the historical form and meaning or on the contemporary meaning and significance of the medieval thinker's rationes that the researcher makes. ...
The last thirty years of scholarship in western medieval philosophical historiography have seen a number of reflections on the methodological paradigms, schools, trends, and dominant approaches in the field. As a contribution to this ongoing assessment of the existing methods of studies in medieval philosophy and theology and a supplement to classifications offered by M. Colish, J. Inglis, C. König-Pralong, J. Marenbon, A. de Libera, and others, the article offers another explanatory tool. Here is a description of an imaginary system of methodological coordinates that systematizes the current tendencies by placing them in a three-dimensional system of axes. Every axis corresponds to a certain aspect of the historical and systematic research in medieval thought and symbolizes a possible movement between two extremes representing opposite methodological values and directions. The methods and approaches practiced in recent studies in medieval philosophy and theology might be schematically located inside this general system of argumentational, focal (or objectival), and (con)textual axes with their intersection identified with what some scholars call the “integral” model of study. This explanatory tool allows one to see how current approaches and methods form a panoply of axes that belong together in one complex grid and helps to visualize the tapestry of existing approaches in medieval philosophical historiography.
... A pertinent observation to be made about Adams's analysis concerns his choice of taking (8) as premise. As I have said, sentences from (4) to (8) can be taken very reasonably as a preliminary argument: while (4) is an anticipation of the conclusion and (8) is the conclusion, sentences (5) to (7) seem to be meant to support (8). That Adams skips this and takes (8) instead as premise is significant for a couple of reasons. ...
The purpose of this paper is twofold. First, it aims at introducing the ontological argument through the analysis of five historical developments: Anselm's argument found in the second chapter of his Proslogion, Gaunilo's criticism of it, Descartes' version of the ontological argument found in his Meditations on First Philosophy, Leibniz's contribution to the debate on the ontological argument and his demonstration of the possibility of God, and Kant's famous criticisms against the (cartesian) ontological argument. Second, it intends to critically examine the enterprise of formally analyzing philosophical arguments and, as such, contribute in a small degree to the debate on the role of formalization in philosophy. My focus will be mainly on the drawbacks and limitations of such enterprise; as a guideline, I shall refer to a Carnapian, or Carnapian-like theory of argument analysis.
The philosophical proof of the existence of God has been a central topic of debate throughout history, with various arguments put forth to support the existence of a higher being. This argument, such as the ontological, cosmological, and teleological arguments, aims to demonstrate that the existence of God can be logically inferred from the nature of the universe and human experience. The implications of these philosophical proofs for Christian faith and practice are profound. For Christians, the philosophical proof of God’s existence can strengthen their faith by providing rational justification for their beliefs. It can also deepen their understanding of God’s nature and attributes, leading to a more profound relationship with the divine. Additionally, these proofs can guide moral and ethical decision-making, as they provide a foundation for understanding the source of objective moral and ethical values. The philosophical proof of God’s existence can inspire believers to live out their faith more authentically and passionately. It can also foster a sense of awe and wonder at the complexity and beauty of the universe, leading to a deeper appreciation of God’s creation. Overall, the philosophical proof of the existence of God has the potential to enrich and enliven Christian faith and practice, offering believers a firm intellectual foundation for their spiritual journey.
My purpose in this paper is to shed some light on two questions: In what sense is logic philosophical? And what is philosophical logic? I take these two questions as coextensive: An answer to one of them is also (or can easily be converted into) an answer to the other. I approach the problem from three perspectives: a conceptual, a descriptive, and a prescriptive perspective. In other words, I try to answer the following questions: (i) In what sense can logic be taken as philosophical? (ii) In what sense has logic been taken as philosophical? (iii) In what sense should logic be taken as philosophical? To this end, I analyze excerpts from five works in which meanings are attributed to the expression “philosophical logic.” The meanings are then object of critical analysis: I try to assess which of these semantical alternatives do not hold up as satisfactory answers to (iii). The result of this analysis is then used as a starting point for my own answer to question (iii).
In 1962 Charles Hartshorne published a modal logic proof formalizing Anselm of Canterbury’s ontolgical argument for the necessary existence of God. This article presents Kurt Gödel’s notes on this proof which have now been discovered in his Nachlass among other theological material, and discusses possible influences on the development of Gödel’s own ontological proof. To complete the picture, strong connections between Anselm of Canterbury’s and Gödel’s conceptions of God and his positive properties are pointed out.KeywordsKurt GödelOntological proofCharles HartshorneModal logicAnselm of Canterbury
Formulations of Anselm’s ontological argument have been the subject of a number of recent studies. We examine these studies in light of Anselm’s text and (a) respond to criticisms that have surfaced in reaction to our earlier representations of the argument, (b) identify and defend a more refined representation of Anselm’s argument on the basis of new research, and (c) compare our representation of the argument, which analyzes that than which none greater can be conceived as a definite description, to a representation that analyzes it as an arbitrary name.
Computers may help us to better understand (not just verify) arguments. In this chapter we defend this claim by showcasing the application of a new, computer-assisted interpretive method to an exemplary natural-language argument with strong ties to metaphysics and religion: E. J. Lowe’s modern variant of St. Anselm’s ontological argument for the existence of God. Our new method, which we call computational hermeneutics, has been particularly conceived for use in interactive-automated proof assistants. It aims at shedding light on the meanings of words and sentences by framing their inferential role in a given argument. By employing automated theorem proving technology within interactive proof assistants, we are able to drastically reduce (by several orders of magnitude) the time needed to test the logical validity of an argument’s formalization. As a result, a new approach to logical analysis, inspired by Donald Davidson’s account of radical interpretation, has been enabled. In computational hermeneutics, the utilization of automated reasoning tools effectively boosts our capacity to expose the assumptions we indirectly commit ourselves to every time we engage in rational argumentation and it fosters the explicitation and revision of our concepts and commitments.
I use mechanized verification to examine several first- and higher-order formalizations of Anselm’s Ontological Argument against the charge of begging the question. I propose three different but related criteria for a premise to beg the question in fully formal proofs and find that one or another applies to all the formalizations examined. I also show that all these formalizations entail variants that are vacuous, in the sense that they apply no interpretation to “than which there is no greater” and are therefore vulnerable to Gaunilo’s refutation. My purpose is to demonstrate that mechanized verification provides an effective and reliable technique to perform these analyses; readers may decide whether the forms of question begging and vacuity so identified affect their interest in the Argument or its various formalizations.
We use a mechanized verification system, PVS, to examine the argument from Anselm’s Proslogion Chapter III, the so-called “Modal Ontological Argument.” We consider several published formalizations for the argument and show they are all essentially similar. Furthermore, we show that the argument is trivial once the modal axioms are taken into account. This work is an illustration of Computational Philiosophy and, in addition, shows how these methods can help detect and rectify errors in modal reasoning.
Scholars were greatly indebted to Max Charlesworth for publishing in 1965 the Latin text of Anselm’s Proslogion, together with his own translation and commentary. The intense discussion this argument has received since then has, however, clarified a number of points about the logic of this argument. Its first premise is not a definition of God, and that identification is one of the conclusions of a three-stage argument. Also, the much-discussed issue of the relation of Chap. 3 to Chap. 2 has now been clarified: that the premise with which Anselm begins Chap. 3 is entailed by the conclusion of Chap. 2. For that reason, substituting a description of anything other than God for Anselm’s formula, such as Gaunilo’s Lost Island, entails that that thing both can and could not be thought not to exist. So, no such substitution is legitimate.
We claim that Kant's doctrine of the "transcendental ideal of pure reason" contains, in an anticipatory sense, a second-order theory of reality (as a second-order property) and of the highest being. Such a theory, as reconstructed in this paper, is a transformation of Kant's metatheoretical regulative and heuristic presuppositions of empirical theories into a hypothetical ontotheology. We show that this metaphysical theory, in distinction to Descartes' and Leibniz's ontotheology, in many aspects resembles Gödel's theoretical conception of the possibility of a supreme being. The proposed second-order modal formalization of the Kantian doctrine of the "transcendental ideal" of the supreme being includes some specific features of Kantian general logic like "categorical", "hypothetical", and exclusive disjunctive propositions as basic forms.
Fitting and Mendelsohn present a thorough treatment of first-order modal logic, together with some propositional background. They adopt throughout a threefold approach. Semantically, they use possible world models; the formal proof machinery is tableaus; and full philosophical discussions are provided of the way that technical developments bear on well-known philosophical problems.
The book covers quantification itself, including the difference between actualist and possibilist quantifiers; equality, leading to a treatment of Frege's morning star/evening star puzzle; the notion of existence and the logical problems surrounding it; non-rigid constants and function symbols; predicate abstraction, which abstracts a predicate from a formula, in effect providing a scoping function for constants and function symbols, leading to a clarification of ambiguous readings at the heart of several philosophical problems; the distinction between nonexistence and nondesignation; and definite descriptions, borrowing from both Fregean and Russellian paradigms.
Introduction When Anselm completed his Monologion, he submitted it to his teacher, Lanfranc, for his approval. Although we do not have the text of Lanfranc's reply, it seems to have called for Anselm to give appropriate sources for his assertions. In response to Lanfranc's criticism, Anselm sought to justify himself this way: It was my intention throughout this disputation to assert nothing which could not be immediately defended either from canonical Dicta or from the words of St. Augustine. And however often I look over what I have written, I cannot see that I have asserted anything that is not to be found there. Indeed, no reasoning of my own, however conclusive, would have persuaded me to have been the first to presume to say those things which you have copied from my work, nor several other things besides, if St. Augustine had not already proved them in the great discussions in his De trinitate.
Philosophy and theology both ask what sort of being God is. One way toward an answer begins from the idea that God is in all respects perfect, and fills out the concept of God by reasoning about what a perfect being would be like. Anselm's is perhaps the name most associated with this program. The perfect-being project had a long history before Anselm. Plato, Aristotle, and such Stoics as Zeno and Cicero all offered perfect-being arguments. But Anselm probably had no access to these. It is likely that passages in Augustine and Boethius suggested the perfect-being project to him. Anselm took up the perfect-being project in Monologion 15. Prior to this, the Monologion has argued that there is something “through which all good things are good” (Mon. 1) – something that plays the role of a property of goodness all good things share. But, Anselm suggests, whatever it is that makes all good things good must be a great good itself. (This suggestion is not backed up. Perhaps Anselm had some such thought as this in mind: delete this thing from reality, and all goodness goes with it. Perhaps an item’s goodness is in some proportion to how much less good things would be without it.) If this thing is good, it must be good through itself, as it is that through which all good things are good. So there is, Anselm thinks, a good thing whose goodness is entirely due to its own intrinsic character – not a function of its relations to anything else. Anselm asserts that this is the best of all goods, just because it is not good through anything other than itself (Mon. 1). The highest good turns out to be the efficient cause of all things other than itself (Mon. 7). So while it plays the role of a property of goodness, it is not after all a property. Properties are not causes.
This book contains 15 papers by the influential American philosopher, David Lewis. All previously published (between 1966 and 80), these papers are divided into three groups: ontology, the philosophy of mind, and the philosophy of language. Lewis supplements eight of the fifteen papers with postscripts in which he amends claims, answers objections, and introduces later reflections. Topics discussed include possible worlds, counterpart theory, modality, personal identity, radical interpretation, language, propositional attitudes, the mind, and intensional semantics. Among the positions Lewis defends are modal realism, materialism, socially contextualized formal semantics, and functionalism of the mind. The volume begins with an introduction in which Lewis discusses his philosophical method.
In his paper “St. Anselm’s ontological argument succumbs to Russell’s paradox” [Int. J. Philos. Relig. 52, 123–128 (2002)], C. Viger presents a critique of Anselm’s argument from the second chapter of Proslogion. Viger claims there that he manages to show that the greater than relation that Anselm used in his proof leads to inconsistency. I argue firstly, that Viger does not show what he maintains to show, secondly, that the flaw is not in the nature of Anselm’s reasoning but in Viger’s (mis)understanding of Anselm as well as in Viger’s (mis)application of some set-theoretical notions. I also describe some features of Anselmian greater than relation, which indeed plays a crucial role in his ontological argument. Last but not least, I present the argument itself.
In this book, Graham Oppy examines arguments for and against the existence of God. He shows that none of these arguments is powerful enough to change the minds of reasonable participants in debates on the question of the existence of God. His conclusion is supported by detailed analyses of the arguments as well as by the development of a theory about the purpose of arguments and the criteria that should be used in judging whether or not arguments are successful. Oppy discusses the work of a wide array of philosophers, including Anselm, Aquinas, Descartes, Locke, Leibniz, Kant, Hume and, more recently, Plantinga, Dembski, White, Dawkins, Bergman, Gale and Pruss.
This book includes arguments for and against belief in God. The arguments for the belief are analyzed in the first six chapters and include ontological arguments from Anselm through Godel; the cosmological arguments of Aquinas and Leibniz; and arguments from evidence for design and miracles. The next two chapters consider arguments against belief. The last chapter examines Pascalian arguments for and against belief in God. This book is a valuable resource for philosophers of religion and theologians and interests logicians and mathematicians as well.
Yujin Nagasawa accuses me of attributing to Anselm a principle (the 'principle of the superiority of existence', or PSE) which
is not present in his text and which weakens, rather than strengthens, his Ontological Argument. I am undogmatic about the
interpretative issue, but insist on a philosophical point: that Nagasawa's rejection of PSE does not help the argument, and
appears to do so only because he overlooks the same ambiguity that vitiates the original. My conclusion therefore remains:
that the fatal flaw in Anselm's argument—as in many other variants—is a relatively shallow ambiguity rather than a deep metaphysical
mistake.
Peter Millican (2004) provides a novel and elaborate objection to Anselm's ontological argument. Millican thinks that his objection is more powerful than any other because it does not dispute contentious 'deep philosophical theories' that underlie the argument. Instead, it tries to reveal the 'fatal flaw' of the argument by considering its 'shallow logical details'. Millican's objection is based on his interpretation of the argument, according to which Anselm relies on what I call the 'principle of the superiority of existence' (PSE). I argue that (i) the textual evidence Millican cites does not provide a convincing case that Anselm relies on PSE and that, moreover, (ii) Anselm does not even need PSE for the ontological argument. I introduce a plausible interpretation of the ontological argument that is not vulnerable to Millican's objection and conclude that even if the ontological argument fails, it does not fail in the way Millican thinks it does.
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 Argument The Temporal-Contingency Argument Conclusion References Appendix 1. Logic Matters Appendix 2. Formal Proofs of Some Modal Arguments
1. The modal argument Anselm’s ontological proof for the existence of God has a precise modal structure. By formalizing the argument, it is possible to identify the intensional modal fallacy it contains. The deductive invalidity of Anselm’s inference embodied in the fallacy defeats his argument, even if, contrary to Kant’s famous objection, existence is included as a ‘predicate’ or identitydetermining constitutive property of particulars. Norman Malcolm in ‘Anselm’s ontological arguments’ distinguishes two forms of Anselm’s inference. 1 Malcolm acknowledges that ‘There is no evidence that [Anselm] thought of himself as offering two different proofs’. 2 Some commentators have indeed understood what Malcolm refers to as the second ontological proof as Anselm’s official or final formulation, interpreting the first version as a preliminary attempt to express or preparatory remarks for the demonstration’s later restatement. 3 Anselm presents the so-called second ontological proof in the Proslogion III: For there can be thought to exist something whose non-existence is inconceivable; and this thing is greater than anything whose non-existence is conceivable. Therefore, if that than which a greater cannot be thought could be thought not to exist, then that than which a greater cannot be thought would not be that than which a greater cannot be thought ‐ a contradiction. Hence, something than which a greater cannot be thought exists so truly that it cannot even be thought not to exist. And You are this being, O Lord our God. Therefore, Lord my God, You exist so truly that You cannot even be thought not to exist. 4
L'a. propose une refutation de la preuve ontologique de Saint Anselme. Pour ce faire il montre que l'argumentation d' Anselme repose a la base sur une notion incoherente. Utilisant le raisonnement par l'absurde, l'a. utilise le paradoxe de Russell pour mener a bien sa demonstration.
Anselm's Ontological Argument fails, but not for any of the various reasons commonly adduced. In particular, its failure has nothing to do with violating deep Kantian principles by treating ‘exists’ as a predicate or making reference to ‘Meinongian’ entities. Its one fatal flaw, so far from being metaphysically deep, is in fact logically shallow, deriving from a subtle scope ambiguity in Anselm's key phrase. If we avoid this ambiguity, and the indeterminacy of reference to which it gives rise, then his argument is blocked even if his supposed Meinongian extravagances are permitted . Moreover it is blocked in a way which is straightforward and compelling (by contrast with the Kantian objections), and which generalizes easily to other versions of the Ontological Argument. A significant moral follows. Fear of Anselm's argument has been hugely influential in motivating ontological fastidiousness and widespread reluctance to countenance talk of potentially non-existing entities. But if this paper is correct, then the Ontological Argument cannot properly provide any such motivation. Some of the most influential contributions to ontology, from Kant to Russell and beyond, rest on a mistake.
This paper was published in Philosophical Perspectives 8: Logic and Language, J. Tomberlin (ed.), Atascadero: Ridgeview Press, 1994, pp. 431--458. The authors would like to acknowledge support from the Social Sciences and Humanities Research Council of Canada (SSHRC Grant #410-91-1817), and generous support from the Center for the Study of Language and Information (CSLI) at Stanford University. We would also like to thank Harry Deutsch, Greg Fitch, Chris Menzel, and Nathan Tawil for giving us insightful comments on an earlier draft of this paper
this paper, we develop a reading of Anselm's Proslogium that contains no modal inferences. Rather, the argument turns on the di#erence between saying that there is such a thing as x and saying that x has the property of existence. We formally represent the claim that there is such a thing as x # Originally published in Philosophical Perspectives 5: The Philosophy of Religion, James Tomberlin (ed.), Atascadero: Ridgeview, 1991: 509--529; selected for republication in The Philosopher's Annual: 1991 , Volume XIV (1993): 255--275. The authors would like to thank Chris Menzel for encouraging us to write this paper and William Uzgalis, Edgar Morscher, and Marleen Rozemond for some excellent suggestions on how to improve it. We would also like to acknowledge generous support from the Center for the Study of Language and Information at Stanford University
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