Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that (...) led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results. (shrink)

Some recently-proposed counterexamples to the traditional definition of essential property do not require a separate logic of essence. Instead, the examples can be analysed in terms of the logic and theory of abstract objects. This theory distinguishes between abstract and ordinary objects, and provides a general analysis of the essential properties of both kinds of object. The claim ‘x has F necessarily’ becomes ambiguous in the case of abstract objects, and in the case of ordinary objects there are various ways (...) to make the definition of ‘F is essential to x’ more fine-grained. Consequently, the traditional definition of essential property for abstract objects in terms of modal notions is not correct, and for ordinary objects the relationship between essential properties and modality, once properly understood, addresses the counterexample. (shrink)

The author revises the formulation of propositional modal logic by interposing a domain of structured propositions between the modal language and the models. Interpretations of the language (i.e., ways of mapping the language into the domain of propositions) are distinguished from models of the domain of propositions (i.e., ways of assigning truth values to propositions at each world), and this contrasts with the traditional formulation. Truth and logical consequence are defined, in the first instance, as properties of, and relations among, (...) propositions. (shrink)

§0. A familiar if obscure idea: an indicative conditional presents its consequent as holding in the actual world on the supposition that its antecedent so holds, whereas a subjunctive conditional merely presents its consequent as holding in a world, typically counterfactual, in which its antecedent holds. Consider this pair.

Identity, we're told, is the binary relation that every object bears to itself, and to itself only. But how can a relation be binary if it never relates two objects? This puzzled Russell and led Wittgenstein to declare that identity is not a relation between objects. The now standard view is that Wittgenstein's position is untenable, and that worries regarding the relational status of identity are the result of confusion. I argue that the rejection of identity as a binary relation (...) is perfectly tenable. To this end, I outline and defend a logical framework that is not committed to an objectual identity relation but is nevertheless expressively equivalent to first-order logic with identity. After it has thus been shown that there is no indispensability argument for objectual identity, I argue that we have good reasons for doubting the existence of such a relation, and rebut a number of attempts at discrediting these reasons. (shrink)

What sort of logic do we get if we adopt a supervaluational semantics for vagueness? As it turns out, the answer depends crucially on how the standard notion of validity as truth preservation is recasted. There are several ways of doing that within a supervaluational framework, the main alternative being between “global” construals (e.g., an argument is valid iff it preserves truth-under-all-precisifications) and “local” construals (an argument is valid iff, under all precisifications, it preserves truth). The former alternative is by (...) far more popular, but I argue in favor of the latter, for (i) it does not suffer from a number of serious objections, and (ii) it makes it possible to restore global validity as a defined notion. (shrink)

Todd (2016) proposes an analysis of future-directed sentences, in particular sentences of the form 'will(φ)', that is based on the classic Russellian analysis of definite descriptions. Todd's analysis is supposed to vindicate the claim that the future is metaphysically open while retaining a simple Ockhamist semantics of future contingents and the principles of classical logic, i.e. bivalence and the law of excluded middle. Consequently, an open futurist can straightforwardly retain classical logic without appeal to supervaluations, determinacy operators, or any further (...) controversial semantical or metaphysical complication. In this paper, we will show that this quasi-Russellian analysis of 'will' both lacks linguistic motivation and faces a variety of significant problems. In particular, we show that the standard arguments for Russell's treatment of definite descriptions fail to apply to statements of the form 'will(φ)'. (shrink)

Pierre Saint-Germier | : Nous discutons la thèse, acceptée par de nombreux théoriciens des intuitions rationnelles, selon laquelle ces dernières s’accompagnent d’une apparence de nécessité. L’existence d’intuitions rationnelles ayant pour objet des propositions contingentes jette un doute sur l’adéquation de cette thèse. Le problème peut trouver une solution dans le cadre d’une théorie faillibiliste des intuitions rationnelles, pourvu que l’on admette des illusions modales inéliminables. En nous appuyant sur une explication bidimensionnelle de l’a priori contingent, nous défendons une solution différente (...) consistant à reconsidérer le contenu des apparences associées aux intuitions rationnelles : ce que l’on pourrait prendre pour des apparences de nécessité n’est autre qu’une manifestation de leur caractère a priori. | : We discuss the view, accepted by many rational intuition theorists, that rational intuitions involve appearances of necessity. The fact that we can have rational intuitions of contingent propositions casts a doubt on the adequacy of this view. This fact can be accommodated by a fallibilist theory of rational intuitions, provided one accepts that these intuitions generate ineliminable modal illusions. Drawing on two-dimensional explanations of the contingent a priori, we defend an alternative view according to which the appearances involved in rational intuitions are not appearances of necessity but rather an expression of their a priori status. (shrink)

The deontic status of tautologies and contradictions is one of the major puzzles for authors of early works on deontic logic. It is well-known that von Wright addresses this problem by adopting a Principle of Deontic Contingency, which says that tautologies are not necessarily obligatory and contradictions are not necessarily forbidden. A more radical solution is proposed by Anderson within a reductionist approach to deontic logic and consists in restricting the range of application of deontic modalities to contingent propositions. Anderson’s (...) solution has not received much attention in the literature, despite reflecting a typical feature of ordinary deontic reasoning, where non-contingent propositions are rarely, if ever, taken into account. In the present article we explore some of its formal consequences, providing a taxonomy of the properties of the Andersonian operators of obligation and permission for contingent propositions, O′ and P′, in the class of normal alethic systems. (shrink)

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that (...) have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts. (shrink)

I argue that propositions are contingent existents. Some propositions that in fact exist might not have existed and there might have been propositions that are distinct from every actually existing proposition. This is because some propositions are singular propositions, which are propositions containing ordinary objects as constituents, and so are ontologically dependent on the existence of those objects; had those objects not existed, then the singular propositions would not have existed. I provide both a philosophical and technical understanding of the (...) contingent status of propositions. (shrink)

Karen Bennett has recently argued that the views articulated by Linsky and Zalta (Philos Perspect 8:431–458, 1994) and (Philos Stud 84:283–294, 1996) and Plantinga (The nature of necessity, 1974) are not consistent with the thesis of actualism, according to which everything is actual. We present and critique her arguments. We first investigate the conceptual framework she develops to interpret the target theories. As part of this effort, we question her definition of ‘proxy actualism’. We then discuss her main arguments that (...) the theories carry a commitment to actual entities that do not exist. We end by considering and addressing a worry that might have been the driving force behind Bennett’s claim that Linsky and Zalta’s view is not fully actualistic. (shrink)

A formula is a contingent logical truth when it is true in every model M but, for some model M , false at some world of M . We argue that there are such truths, given the logic of actuality. Our argument turns on defending Tarski’s definition of truth and logical truth, extended so as to apply to modal languages with an actuality operator. We argue that this extension is the philosophically proper account of validity. We counter recent arguments to (...) the contrary presented in Hanson’s ‘Actuality, Necessity, and Logical Truth’ (Philos Stud 130:437–459, 2006 ). (shrink)

In his paper, Leuenberger discerns two salient conceptions of logical validity. Strikingly, neither of these conceptions involves modality. He goes on to use these conceptions as a framework to explore certain recent investigations in the logic of modality, where he ingeniously articulates and proves interesting theses about the logic of contingentism. While I think there’s much of interest in Leuenberger’s results, and that his conception of de facto validity gives a unified account of philosophers’ talk of the logic of time (...) and modality, in this note I suggest that perhaps he is too hasty to dismiss the modal conception of validity and that, moreover, his concept of de facto validity may be too inclusive. (shrink)

The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which actualists object. Though these philosophers have introduced variations (...) on Kripke-models to eliminate their objectionable features, the most well-known variations all have difficulties of their own. The present authors reexamine simple QML and discover that, in addition to having a possibilist interpretation, it has an actualist interpretation as well. By introducing a new sort of existing abstract entity, the contingently nonconcrete, they show that the seeming drawbacks of the simplest QML are not drawbacks at all. Thus, simple QML is independent of certain metaphysical questions. (shrink)

This paper investigates and develops generalizations of two-dimensional modal logics to any finite dimension. These logics are natural extensions of multidimensional systems known from the literature on logics for a priori knowledge. We prove a completeness theorem for propositional n-dimensional modal logics and show them to be decidable by means of a systematic tableau construction.

In a recent paper, Pruss proves the validity of the rule beta-2 relative to Lewis’s semantics for counterfactuals, which is a significant step forward in the debate about the consequence argument. Yet, we believe there remain intuitive counter-examples to beta-2 formulated with the actuality operator and rigidified descriptions. We offer a novel and two-dimensional formulation of the Lewisian semantics for counterfactuals and prove the validity of a new transfer rule according to which a new version of the consequence argument can (...) be formulated. This new transfer rule is immune to the counter-examples involving the actuality operator and rigidified descriptions. However, we show that counter-examples to this new rule can also be generated, demanding that the Lewisian semantics be generalized for higher dimensions where counter-examples can always be generated. (shrink)

In this paper we present tableau methods for two-dimensional modal logics. Although models for such logics are well known, proof systems remain rather unexplored as most of their developments have been purely axiomatic. The logics herein considered contain first-order quantifiers with identity, and all the formulas in the language are doubly-indexed in the proof systems, with the upper indices intuitively representing the actual or reference worlds, and the lower indices representing worlds of evaluation—first and second dimensions, respectively. The tableaux modulate (...) over different notions of validity such as local, general, and diagonal, besides being general enough for several two-dimensional logics proposed in the literature. We also motivate the introduction of a new operator into two-dimensional languages and explore some of the philosophical questions raised by it concerning the relations there are between actuality, necessity, and the a priori, that seem to undermine traditional intuitive interpretations of two-dimensional operators. (shrink)

We present a new puzzle about logical truth, necessity, and moral responsibility. We defend one solution to the puzzle. A corollary of our preferred solution is that prominent arguments for the incompatibility of determinism and moral responsibility are invalid.

The most common first- and second-order modal logics either have as theorems every instance of the Barcan and Converse Barcan formulae and of their second-order analogues, or else fail to capture the actual truth of every theorem of classical first- and second-order logic. In this paper we characterise and motivate sound and complete first- and second-order modal logics that successfully capture the actual truth of every theorem of classical first- and second-order logic and yet do not possess controversial instances of (...) the Barcan and Converse Barcan formulae as theorems, nor of their second-order analogues. What makes possible these results is an understanding of the individual constants and predicates of the target languages as strongly Millian expressions, where a strongly Millian expression is one that has an actually existing entity as its semantic value. For this reason these logics are called ‘strongly Millian’. It is shown that the strength of the strongly Millian second-order modal logics here characterised afford the means to resist an argument by Timothy Williamson for the truth of the claim that necessarily, every property necessarily exists. (shrink)

This paper recalls some applications of two-dimensional modal logic from the 1980s, including work on the logic of Actually and on a somewhat idealized version of the indicative/subjunctive distinction, as well as on absolute and relative necessity. There is some discussion of reactions this material has aroused in commentators since. We also survey related work by Leslie Tharp from roughly the same period.

The central question of this article is how to combine counterfactual theories of knowledge with the notion of actuality. It is argued that the straightforward combination of these two elements leads to problems, viz. the problem of easy knowledge and the problem of missing knowledge. In other words, there is overgeneration of knowledge and there is undergeneration of knowledge. The combination of these problems cannot be solved by appealing to methods by which beliefs are formed. An alternative solution is put (...) forward. The key is to rethink the closeness relation that is at the heart of counterfactual theories of knowledge. (shrink)

Does general validity or real world validity better represent the intuitive notion of logical truth for sentential modal languages with an actuality connective? In (Philosophical Studies 130:436–459, 2006) I argued in favor of general validity, and I criticized the arguments of Zalta (Journal of Philosophy 85:57–74, 1988) for real world validity. But in Nelson and Zalta (Philosophical Studies 157:153–162, 2012) Michael Nelson and Edward Zalta criticize my arguments and claim to have established the superiority of real world validity. Section 1 (...) of the present paper introduces the problem and sets out the basic issues. In Sect. 2 I consider three of Nelson and Zalta’s arguments and find all of them deficient. In Sect. 3 I note that Nelson and Zalta direct much of their criticism at a phrase (‘true at a world from the point of view of some distinct world as actual’) I used only inessentially in Hanson (Philosophical Studies 130:436–459, 2006), and that their account of the philosophical foundations of modal semantics leaves them ill equipped to account for the plausibility of modal logics weaker than S5. Along the way I make several general suggestions for ways in which philosophical discussions of logical matters–especially, but not limited to, discussions of truth and logical truth for languages containing modal and indexical terms–might be facilitated and made more productive. (shrink)

The traditional view that all logical truths are metaphysically necessary has come under attack in recent years. The contrary claim is prominent in David Kaplan’s work on demonstratives, and Edward Zalta has argued that logical truths that are not necessary appear in modal languages supplemented only with some device for making reference to the actual world (and thus independently of whether demonstratives like ‘I’, ‘here’, and ‘now’ are present). If this latter claim can be sustained, it strikes close to the (...) heart of the traditional view. I begin this paper by discussing and refuting Zalta’s argument in the context of a language for propositional modal logic with an actuality connective (section 1). This involves showing that his argument in favor of real world validity his preferred explication of logical truth, is fallacious. Next (section 2) I argue for an alternative explication of logical truth called general validity. Since the rule of necessitation preserves general validity, the argument of section 2 provides a reason for affirming the traditional view. Finally (section 3) I show that the intuitive idea behind the discredited notion of real world validity finds legitimate expression in an object language connective for deep necessity. (shrink)

This paper argues that a prominent—and oft-thought to be persuasive—argument against general validity as the best account of validity for languages containing the actuality operator is flawed, the flaw arising out of inadequate attention to the formalisation of mood distinctions.

In this paper I argue that deontic modals are hyperintensional, i.e. logically equivalent contents cannot be substituted in their scope. I give two arguments, one deductive and the other abductive. First, I show that the contrary thesis leads to falsity; second, I argue that a hyperintensional theory of deontic modals fares better than its rivals in terms of elegance, theoretical simplicity and explanatory power. I then propose a philosophical analysis of this thesis and outline some consequences. In Section 1 I (...) introduce and define deontic modality and hyperintensionality. In Section 2 I give a reductio for the hyperintensionality of deontic modals. If the argument is sound, a useful corollary is that deontic modals are also non-intensional, and therefore possible-world semantics accounts are illfitted for them. I then show how the main result can be strengthened or weakened by varying the definition of logical validity. In Section 3 I give an abductive argument for the hyperintensionality of deontic modals, arguing that with a single move we are able to solve many paradoxes and puzzles traditionally troubling deontic logic. I present a version of a hyperintensional deontic logic in an appendix, which I prove is sound and complete with respect to a version of truthmaker semantics. (shrink)

We consider a natural-language sentence that cannot be formally represented in a first-order language for epistemic two-dimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the first-order language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal.

According to the well-known Kripke-Putnam view developed in Naming and Necessity and ‘The Meaning of Meaning’, proper names and ‘natural kind terms’ - words for natural substances, species, and phenomena - are non-descriptional and rigid. A singular term is rigid if it has the same referent in every possible world, and is non-descriptional if, roughly speaking, its referent is not secured by purely descriptive conditions analytically tied to the term. Thus, ‘the inventor of bifocals’ is nonrigid and descriptional, while ‘the (...) unique even and prime integer’ is rigid and descriptional, and ‘Noam Chomsky’ is rigid and non-descriptional. (shrink)

In this paper I explored three well-known cases that seem to cast doubt on the notion that a speaker is always at the place of the utterance when the utterance occurs. I gave a few examples produced in Second Life environment, which cannot be handled correctly by evaluating the expression at issue with respect to the traditional view, i.e., the kaplanian framework—where the agent and the utterer will always be identical, and the referent of “I” will always be the utterer. (...) The same happens to “here” and “now”; in each case the character of the indexical is a function from a contextual parameter to the referent of the expression. An example of logical truth peculiar to indexicals is the sentence “I am here now”, since, according to that view, no utterance can take place in a context whose agent is not in its location in its time. Finally I concluded by claiming that particular kinds of utterances produced in virtual environments (VE) break with the logical truth status of “I am here now”. (shrink)

In this paper, I motivate the addition of an actuality operator to relevant logics. Straightforward ways of doing this are in tension with standard motivations for relevant logics, but I show how to add the operator in a way that permits one to maintain the intuitions behind relevant logics. I close by exploring some of the philosophical consequences of the addition.

Modal logic is one of philosophy’s many children. As a mature adult it has moved out of the parental home and is nowadays straying far from its parent. But the ties are still there: philosophy is important to modal logic, modal logic is important for philosophy. Or, at least, this is a thesis we try to defend in this chapter. Limitations of space have ruled out any attempt at writing a survey of all the work going on in our field—a (...) book would be needed for that. Instead, we have tried to select material that is of interest in its own right or exemplifies noteworthy features in interesting ways. Here are some themes that have guided us throughout the writing: • The back-and-forth between philosophy and modal logic. There has been a good deal of give-and-take in the past. Carnap tried to use his modal logic to throw light on old philosophical questions, thereby inspiring others to continue his work and still others to criticise it. He certainly provoked Quine, who in his turn provided—and continues to provide—a healthy challenge to modal logicians. And Kripke’s and David Lewis’s philosophies are connected, in interesting ways, with their modal logic. Analytic philosophy would have been a lot different without modal logic! • The interpretation problem. The problem of providing a certain modal logic with an intuitive interpretation should not be conflated with the problem of providing a formal system with a model-theoretic semantics. An intuitively appealing model-theoretic semantics may be an important step towards solving the interpretation problem, but only a step. One may compare this situation with that in probability theory, where definitions of concepts like ‘outcome space’ and ‘random variable’ are orthogonal to questions about “interpretations” of the concept of probability. • The value of formalisation. Modal logic sets standards of precision, which are a challenge to—and sometimes a model for—philosophy. Classical philosophical questions can be sharpened and seen from a new perspective when formulated in a framework of modal logic. On the other hand, representing old questions in a formal garb has its dangers, such as simplification and distortion. • Why modal logic rather than classical (first or higher order) logic? The idioms of modal logic—today there are many!—seem better to correspond to human ways of thinking than ordinary extensional logic. (Cf. Chomsky’s conjecture that the NP + VP pattern is wired into the human brain.) In his An Essay in Modal Logic (1951) von Wright distinguished between four kinds of modalities: alethic (modes of truth: necessity, possibility and impossibility), epistemic (modes of being known: known to be true, known to be false, undecided), deontic (modes of obligation: obligatory, permitted, forbidden) and existential (modes of existence: universality, existence, emptiness). The existential modalities are not usually counted as modalities, but the other three categories are exemplified in three sections into which this chapter is divided. Section 1 is devoted to alethic modal logic and reviews some main themes at the heart of philosophical modal logic. Sections 2 and 3 deal with topics in epistemic logic and deontic logic, respectively, and are meant to illustrate two different uses that modal logic or indeed any logic can have: it may be applied to already existing (non-logical) theory, or it can be used to develop new theory. (shrink)

The notion of validity for modal languages could be defined in two slightly different ways. The first is the original definition given by S. Kripke, for which a formula φ of a modal language L is valid if and only if it is true in every actual world of every interpretation of L. The second is the definition that has become standard in most textbook presentations of modal logic, for which a formula φ of L is valid if and only (...) if it is true in every world in every interpretation of L. For simple modal languages, “Kripkean validity” and “Textbook validity” are extensionally equivalent. According to E. Zalta, however, Textbook validity is an “incorrect” definition of validity, because: (i) it is not in full compliance with Tarski’s notion of truth; (ii) in expressively richer languages, enriched by the actuality operator, some obviously true formulas count as valid only if the Kripkean notion is used. The purpose of this paper is to show that (i) and (ii) are not good reasons to favor Kripkean valid- ity over Textbook validity. On the one hand, I will claim that the difference between the two should rather be seen as the result of two different conceptions on how a modal logic should be built from a non-modal basis; on the other, I will show the advantages, for the question at issue, of seeing the actuality operator as belonging to the family of two-dimensional operators. (shrink)

Anselm's argument for the existence of God in Proslogion Chap.II starts from the contention that `lq when a Fool hears `something-than-which-nothing-greater-can-be-thought', he understands what he hears, and what he understands is in his mind. This is a special feature of the Pros.II argument which distinguishes the argument from other ontological arguments set up by, for example, Descartes and Leibniz. This is also the context which makes semantics necessary for evaluation of the argument. It is quite natural to ask `lq What (...) is understood by the Fool, and what is in his mind? It is essential for a proper consideration of the argument to identify the object which is understood by the Fool, and so, is in his mind. A semantics gives answers to the questions of `lq What the Fool understands? and `lq What is in the Fool's mind? If we choose a semantics as a meta-theory to interpret the Pros.II argument, it makes an effective guide to identify the object. It is a necessary condition for a proper evaluation of the Pros.II argument to fix our universe of discourse, especially since, in the argument, we are involved in such talk about existing objects as Anselm's contention that `when a Fool hears `something-than-which-nothing-greater-can-be-thought', he understands what he hears, and what he understands is in his mind. The ontology to which a semantic theory commits us will be accepted as our scope of objects when we introduce our semantic theory to interpret the Pros.II argument, and this ontological boundary constrains us to identify the object in a certain way. Consistent application of an ontology, most of all, is needed for the evaluation of the logical validity of an argument. If we take Frege's three-level semantics, we are ontologically committed to intensional entities, like meaning, as well as extensional entities. Sluga contends that Frege's anti-psychologism for meanings should not be interpreted as vindicating reification of intensional entities in relation to Frege's contextualism, that Frege's anti-psychologism with his contextualism is nothing but a linguistic version of Kantian philosophy for the transcendental unity of a judgement. There is, however, another possible interpretation of Frege's contextualism. According to Dummett, the significance of Frege's contextualism must be understood as a way of explanation for a word's having meaning. If Dummett's view is cogent, we could say that Frege's contextualism does not prevent our interpreting his semantics as being committed to intensional entities. We need not worry that Frege's over all semantics, especially with his contextualism, would internally deny the ontological interpretation of his theory. We see Anselm's argument for the existence of God in Pros.II is an invalid argument if we introduce Frege's three-level semantics, i.e. if we acknowledge meanings of words as entities in our universe of discourse. We can also employ extensional semantics for the interpretation of the Pros.II argument. According to extensionalists, like Quine and Kripke, we need not assume intensional entities, like meaning, to be part of our ontological domain. They argue that we can employ our language well enough without assuming intensional entities. If we choose extensional semantics as a meta-theory to interpret the Pros.II argument, it commits us only to extensional entities as objects in the Universe of our interpretation. In Sections 1.4 and 1.5, I show that extensional semantics makes the Pros.II argument a valid argument for the existence of God. `lq Necessary existence is the central concept of Anselm's argument for the existence of God in Proslogion Chap.III. It has been said that, even if the argument is formally valid, it cannot stand as a valid argument for the existence of God, since `lq necessary existence is an absurd concept like `lq round square. And further that even if there is a meaningful combination of concepts for `lq necessary existence, it cannot quality as a subject of an a priori argument. As objections to the interpretations which make the Pros.III argument valid, it has been argued that even if there is a concept of `lq necessary existence which is meaningful and there is another concept of `lq necessary existence which is suitable as a subject of an a priori argument, there is no concept of `lq necessary existence which is meaningful and at the same time suitable as a subject of an a priori argument. In Chap.2 and Chap.3, I try to show that there can be concepts of `lq necessary existence which are proof against these objections. Anselm's arguments for the existence of God in Proslogian Chap.II and Chap.III are logically valid arguments on some logical principles. Some fideists, K. Barth, for example, argue that Anselm's arguments for the existence of God in Proslogion are not proofs for the existence of God even if they are logically valid arguments. I raise the question how this attitude could be possible, in Chap.4 and Chap.5. Barth's fideistic interpretation of Anselm's Proslogion arguments does not find any flaw in the validity of the arguments, and it accepts the meaningfulness and truth of the premises even to the fool in Proslogion. If this is the case, i.e. if Barth's interpretation accepts the validity of the arguments and the truth of the premises, I raise the question, how can the arguments not be interpreted as proofs for the existence of God? How is it possible that the function of the arguments is not that of proving the existence of God? According to Wittgensteinian fideism, premises in the arguments should not be intelligible to those who do not believe in God's existence already, and so the real function of the arguments is the elucidation, the understanding of believer's belief, rather than proving articles of belief to unbelievers. Barth's fideistic interpretation of the arguments, however, fully recognizes the meaningfulness and truth of the premises in the arguments as well as the validity of the arguments. I argue that there could be a justification for the Barthian fideism. As Malcolm notices, there are still atheists who understand Anselm's arguments as valid, but the only possibility for the people who recognize the validity of Anselm's arguments still to remain atheists has been thought to be to challenge the truth of premises employed in the arguments. Now, of the atheistic possibility, we can change the direction of our attention, that is, to the question about the function of a logically valid argument itself. What has not been thought of in relation to Anselm's arguments is the significance of logical truth or the logical validity of an argument. We have not asked such questions as `lq What does a logical truth say? and `lq What does a logically valid argument guarantee with true premises? Let us assume that even the premises are accepted by atheists. Do they all convert to theism? If that were so, the disagreement between atheist and believer over the ontological arguments should turn only on the truth of premises. If that is not so, there is some point in raising this other question. If there are people who, recognizing the premises and validity of an argument, are still reluctant to accept the conclusion, we have reason to question the function of a valid argument. I argue that there is a way of being consistently reasonable while accepting the premises and the validity of the ontological arguments and yet remaining an atheist or an agnostic. (shrink)