This is Francesco Orilia's book review of Edward N. Zalta's 1983 book, Abstract Objects: An Introduction to Axiomatic Metaphysics.

In this book, Zalta attempts to lay the axiomatic foundations of metaphysics by developing and applying a (formal) theory of abstract objects. The cornerstones include a principle which presents precise conditions under which there are abstract objects and a principle which says when apparently distinct such objects are in fact identical. The principles are constructed out of a basic set of primitive notions, which are identified at the end of the Introduction, just before the theorizing begins. The main (...) reason for producing a theory which defines a logical space of abstract objects is that it may have a great deal of explanatory power. It is hoped that the data explained by means of the theory will be of interest to pure and applied metaphysicians, logicians and linguists, and pure and applied epistemologists. (shrink)

알베르투스는 형이상학을 모든 학문들의 원리를 제공하는 기초 학문으로, 그 연구 대상은 “가장 먼저 유출된 존재”로 이해한다. 그리고 공리적 방법으로 존재와 선의 관계를 규명한 보에티우스의 『데헵도마디부스』를 자신의 형이상학적 체계 내에서 핵심적인 요소로 수용한다. 제일원인이 만물의 단일한 근거이고 세계가 여기서 창조되었다는 사상은 공리들과 그것들로 조합된 다수의 학적 지식들의 논증 질서에 대응한다. 알베르투스는 『원인론』의 제일원인을 그 자신인 것을 산출하는 원인으로 이해하는데, 어떠한 원인도 가지지 않고 피조물과도 결합되지 않는다는 점이 제일원인의 본질적인 특성이다. 이를 알베르투스는 『원인론 주해』와 『형이상학 주해』에서 보에티우스의 네 가지 공리를 가지고 해명한다.본 (...) 논문은 알베르투스의 형이상학에서 공리적 방법(axiomatic method)이 어떻게 적용되는지를 분석하는 연구다. 특히, 보에티우스의 『데헵도마디부스』와 『원인론』이 알베르투스 형이상학의 공리적 틀을 형성하는 중요한 원천임을 규명한다. 기존연구들이 알베르투스주의자들의 사유에서 공리적 방법의 흔적을 찾는 데 집중했다면, 본 연구는 알베르투스 자신의 텍스트에서 이 방법론이 구체적으로 어떻게 나타나며, 그의 형이상학 체계를 형성하는 데 어떤 역할을 했는지에 주목했다. 알베르투스는 아리스토텔레스의 형이상학을 수용하면서도 프로클로스적인 요소, 즉 기하학적 방법론을 활용하여 존재의 근거를 공리적 구조로 정립한다. 이를 통해 알베르투스의 형이상학적 사유가 보에티우스와 『원인론』을 통해 어떻게 연역 체계로서 구성되었는지를 밝히고, 나아가 그의 사상이 근대 형이상학 발전에 미친 영향을 가늠하고자 한다. (shrink)

What is the aim of philosophy? There may be too many philosophical branches, traditions, practices, and programs to admit of a single overarching aim. Here I focus on a fairly traditional philosophical project that has recently received increasingly sophisticated articulation, especially by Frank Jackson (1998) and David Chalmers (2012). In §1, I present the project and suggest that it is usefully thought of as ‘total axiomatics’: the project of attempting to axiomatize the total theory of the world. In §2, I (...) raise a problem for the project that I call the ‘problem of multiple axiomatizations.’ I consider some initially alluring but ultimately unpromising approaches to this problem in §3. In §4, I defend a surprising approach to the problem, according to which competing axiomatizations of the total theory of the world are effectively evaluated for their aesthetic virtues. (shrink)

In this paper, the authors describe their initial investigations in computational metaphysics. Our method is to implement axiomatic metaphysics in an automated reasoning system. In this paper, we describe what we have discovered when the theory of abstract objects is implemented in PROVER9 (a first-order automated reasoning system which is the successor to OTTER). After reviewing the second-order, axiomatic theory of abstract objects, we show (1) how to represent a fragment of that theory in PROVER9's first-order (...) syntax, and (2) how PROVER9 then finds proofs of interesting theorems of metaphysics, such as that every possible world is maximal. We conclude the paper by discussing some issues for further research. (shrink)

Axiomatic Formal Ontology is a fairly comprehensive systematic treatise on general metaphysics. The axiomatic method is applied throughout the book. Its main theme is the construction of a general non-set-theoretical theory of intensional entities. Other important matters discussed are the metaphysics of modality, the nature of actual existence, mereology and the taxonomy of entities.

Ernst Mayr argued that the emergence of biology as a special science in the early nineteenth century was possible due to the demise of the mathematical model of science and its insistence on demonstrative knowledge. More recently, John Zammito has claimed that the rise of biology as a special science was due to a distinctive experimental, anti-metaphysical, anti-mathematical, and anti-rationalist strand of thought coming from outside of Germany. In this paper we argue that this narrative neglects the important role played (...) by the mathematical and axiomatic model of science in the emergence of biology as a special science. We show that several major actors involved in the emergence of biology as a science in Germany were working with an axiomatic conception of science that goes back at least to Aristotle and was popular in mid-eighteenth-century German academic circles due to its endorsement by Christian Wolff. More specifically, we show that at least two major contributors to the emergence of biology in Germany—Caspar Friedrich Wolff and Gottfried Reinhold Treviranus—sought to provide a conception of the new science of life that satisfies the criteria of a traditional axiomatic ideal of science. Both C.F. Wolff and Treviranus took over strong commitments to the axiomatic model of science from major philosophers of their time, Christian Wolff and Friedrich Wilhelm Joseph Schelling, respectively. The ideal of biology as an axiomatic science with specific biological fundamental concepts and principles thus played a role in the emergence of biology as a special science. (shrink)

It is argued that existing axiomatic theories of utility do not provide the utility principle or the principle of maximising expected utility with a formal justification. It is also argued that these theories only put mild constraints on a decision-maker in a decision-context. Finally, it is argued that the prospects are not particularly bright for finding formal non-circular arguments for the utility principle that do not rely on the law of large numbers.

In normative reasoning one typically refers to intervals of time across which norms are intended to hold, as well as to alternative possibilities representing hypothetical developments of a given scenario. Thus, deontic modalities are naturally intertwined with temporal and metaphysical ones. Furthermore, contemporary debates in philosophy suggest that a proper understanding of fundamental ethical principles, such as the Ought-Implies-Can thesis, requires a simultaneous analysis of these three families of concepts. In the present article we propose a general formal framework which (...) allows for fine-grained multimodal reasoning in the normative domain. We provide an axiomatization for a novel system of propositional logic encoding the way in which possibilities and norms arising from different sources change over intervals of time. The usefulness of our framework is illustrated by analysing an ancient and particularly challenging `cold case', the Paradox of the Court. (shrink)

Lately, an increased interest in formal devices has led to an attempt on the part of some mathematicians to do without those aspects of mathematics which require intuition. One consequence of this movement has been a new conception of pure mathematics as a science of axiomatic systems. According to this conception, there is no reality beyond an axiomatic system which the statements of mathematics are about; the fact that a statement is a theorem in the system is all (...) that is of interest. This is a sort of nominalist position, since the contrary position seems to be committed to a belief in the existence of a domain of mathematical entities, which are just as suspect as the universale commonly discussed in metaphysical treatises. More often it is called formalism. (shrink)

Following recent developments in the literature on axiomatic theories of truth, we investigate an alternative to the widespread habit of formalizing the syntax of the object-language into the object-language itself. We first argue for the proposed revision, elaborating philosophical evidences in favor of it. Secondly, we present a general framework for axiomatic theories of truth with theories of syntax. Different choices of the object theory O will be considered. Moreover, some strengthenings of these theories will be introduced: we (...) will consider extending the theories by the addition of coding axioms or by extending the schemas of O, if present, to the entire vocabulary of our theory of truth. Finally, we touch on the philosophical consequences that the theories described can have on the debate about the metaphysical status of the truth predicate and on the formalization of our informal metatheoretic reasoning. (shrink)

The series offers a publication forum for innovative works on all topics of analytic philosophy. The focus is on the disciplines of theoretical philosophy: metaphysics, ontology, epistemology, philosophy of language, logic. Furthermore, works that additionally include contributions to the history of philosophy are also welcome.

I propose a way to accommodate plausible examples of cyclic grounding chains that have been proposed in the literature while preserving a good measure of the metaphysical foundationalist’s grounding hierarchy with a foundational level. This precludes the need to adopt a strong form of coherentism, such as the view that everything grounds everything, or even the view that everything grounds everything other than itself. I do this by developing axiomatizations of grounding which allow for localized cycles of ground at the (...) non-foundational and foundational levels, but which still place certain foundationalism-like constraints on the grounds that grounded facts must have. In addition, I point to considerations, previously put forth in favor of foundationalism and a strong form of coherentism, that also help support the views I develop. (shrink)

This essay compares the fundamental metaphysical principles, the Categoreal Scheme of A. N. Whitehead's Process and Reality with the axiomatic-deductive scheme of Whitehead and Russell's Principia Mathematica to reveal the influence of mathematical logic on Whitehead's metaphysics.

Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type forallexistsforall, while the axiom system based on congruence and order can beformulated using only forallexists-axioms.

The Transcendental Psychology Approach to the study of perception has been developed by A.I. Mirakyan at the Psychological Institute (Moscow, Russia) about 30 years ago. This article considers the results of theoretical and experimental investigations and provides a historical overview of the approach’s development. Started with the investigations of constancy in perception, it went beyond the traditional Product Basis Paradigm (relying on perceptual features for finding perceptual mechanisms) into Philosophical Metaphysics of “nothing” and “something” concepts for revealing the form-generating (...) principles as fundamental axiomatics of the Transcendental Psychology Approach. Several principles were developed and justified: structure-process anisotropy, spatial-temporal discreteness, the formation of anisotropic (particularly, symmetric) relations, the coexistence of alternatives, and some others. Principles are explanatory for the regulations of sensory-perceptual processes and are the direct object of further specification and experimental verification using hypothetical transcendental models of perceptual structures. The theory suggests that the internal mechanisms of these models would not naturally manifest themselves in experiments within the functional range of perception, and to see the phenomena, it is necessary to bring the perceptual system out of its natural functional range. The form-generating processes are named adiaphorous in the sense that they specifically generate new structures and forms, regardless of the characteristics of products used and produced in the processes. In general, it is possible to speak about the class of so-called structurally-generative processes that are specific to the process of transition between the system-generating structures studied by different hierarchically interrelated sciences. The proposed two-staged qualitative model of the perceptual process consists of two substantially different parts: the direct sensory perception and a process of form designation or sensory name assignment. Further investigations of structurally-generative processes seem likely to shed light on the mechanisms of brain function and to contribute strategically to new directions in philosophical psychology and neuroscience. (shrink)

Although this excellent introductory and intermediate level text is intended for students of mathematics, it could serve well in any course for philosophers on that level. The first two chapters present the propositional and predicate calculi, along with an informal discussion of some of the set-theoretic concepts needed to study logic. The third chapter discusses what exactly an axiomatic system is, and examples of various mathematical systems cast in axiomatic form are provided; the discussion here, as elsewhere in (...) the book, is generally self-contained and presupposes only an ability to deal with abstract ideas. The study of the propositional calculus is resumed in chapter four, but its formulation is now more rigorous and thoroughly axiomatic; in a similar manner the predicate calculus is treated in the next chapter, and the usual metatheorems are proved about each, but the specific techniques differ from the usual. The last chapter concentrates on the completeness proof of quantification theory, relating it to some of the axiomatic algebraic systems considered earlier. An appendix sketches some sophisticated results of Vaught and Robinson on model theory and completeness. In sum, this is one of the very best of a plethora of logic texts on the market, and it should serve well for years to come.—P. J. M. (shrink)

Reichenbach wrote this book just after taking the first course Einstein ever taught on the theory of relativity. His important and influential work The Philosophy of Space and Time was written several years later and relied in part on the axiomatization of the special and general theories of relativity already worked out in this book. For special relativity Reichenbach divides his axioms into two sets, the light axioms which relate light signals to the topology and metric of time and space, (...) and the matter axioms which do the same for rigid rods and clocks. Thus the axioms focus on the conventions uniting theory and observation and give more insight into the physical foundations of the theory than do most axiomatizations. Reichenbach's approach has been criticized by Weyl and others, but at least some of the criticism seems to rest on a misunderstanding of what Reichenbach was attempting to do. Now that this early work has been translated into English, there is hope that it will be more widely read. There is no doubt that students of the theory or of recent philosophical discussion of space and time will profit from a careful reading of this book.--R. H. K. (shrink)

The book presents results from research conducted by Zygmunt Zawirski on the theory of knowledge, quantum mechanics, logic, ontology and metaphysics. 0The works undertaken in the field of logic, methodology and philosophy of science, and in particular the philosophy of nature and natural science testify to a solid preparation for the fundamental task of developing contemporary scientific philosophy. The emerging mathematical natural science did not have those possibilities which emerged in the 20th Century and which Zygmunt Zawirski (1882-1948) used. (...) In the development of scientific metaphysics, he took into account both the achievements of modern logic, mathematics and physics. Zawirski builds scientific metaphysics by referring to empiricism, broadly understood experience. Modern metaphysics should meet high standards of precision and uniqueness, which is why Zawirski attempts to apply the axiomatic method to both the analysis of the theory of physics and the scientific metaphysics. (shrink)

In the TACITUS project for using commonsense knowledge in the understanding of texts about mechanical devices and their failures, we have been developing various commonsense theories that are needed to mediate between the way we talk about the behavior of such devices and causal models of their operation. Of central importance in this effort is the axiomatization of what might be called commonsense metaphysics. This includes a number of areas that figure in virtually every domain of discourse, such as (...) granularity, scales, time, space, material, physical objects, shape, causality, functionality, and force. Our effort has been to construct core theories of each of these areas, and then to define, or at least characterize, a large number of lexical items in terms provided by the core theories. In this paper we discuss our methodological principles and describe the key ideas in the various domains we are investigating. (shrink)

The principle _determinatio est negatio_—that determination is achieved through negation—has philosophical roots extending back to Plato and Aristotle, and it later influenced early modern thinkers such as Francisco Suárez and Spinoza. This paper has two aims. The first demonstrates how the principle of negation functions as a tool for conceptual determination across various philosophical frameworks, and the second demonstrates that the principle plays a key role in the analysis and resolution of the Burali-Forti paradox within the context of the naïve (...) and axiomatic set theories. In the first section, the analysis focuses on the evolution of the principle from ancient philosophy to early modern metaphysics, examining Plato’s dialectics, Aristotle’s metaphysical distinctions, Suárez’s scholastic theories, and Spinoza’s monist metaphysics. The second section shifts to mathematics, where _determinatio est negatio_ plays a key role in resolving set-theoretical paradoxes, particularly the Burali-Forti paradox. By exploring Cantor’s solution and its reliance on the distinction between consistent and inconsistent sets, this study demonstrates how this principle is essential for avoiding self-referential inconsistencies. The contributions of Zermelo and von Neumann, who developed axiomatic frameworks to address these paradoxes, further elaborate the philosophical foundations of set theory. Ultimately, the study reveals a deep connection between metaphysical principles and the mathematical treatment of paradoxes, emphasizing the ongoing relevance of _determinatio est negatio_ in both domains. (shrink)

What and how should individuals resist in political situations? While this question, or versions of it, recurs regularly within Western political philosophy, answers to it have often relied on dyads founded upon dogmatically held ideals. In particular, there is a strain of idealist political philosophy, inaugurated by Plato and finding contemporary expression in the work of Alain Badiou, that employs dyads (such as the distinction between truth and doxa or the privilege of thought over sense) that tend to reduce the (...) complexities of practices of resistance to concepts of commitment. Although these dyads have been challenged by, amongst others, poststructuralist theorists, this has often been at the cost of losing their structuralist heritage. This thesis develops an ontology proper to structuralism that engenders non-idealist and non-dogmatic, yet ethical, practices of resistance against commitment orientated accounts of resistance and the return of classical ontological dyads. The thesis begins with an examination of the extent to which a dogmatic use of idealism grounds the work of a prominent contemporary theorist, Alain Badiou. In developing his neo-Maoist metapolitics, Badiou follows both Platonic ontology and the Marxist tradition of dialectics by claiming that political practice can only be carried out in truth by paying fidelity to an event that ruptures the presented order of things. Chapter one opens with an exploration of Badiou's mathematic meta-ontology to draw out its three foundational dyads (truth/doxa; sense/intelligibility; is/is not). It is argued that although Badiou makes important criticisms of the preponderant trends of political philosophy, he is unable to support his own account of politics due to his dogmatic reliance on idealist principles. Chapter two begins by developing two accounts: first, of the relations between Badiou's work and that of his former teacher Louis Althusser and, secondly, the relations between Althusser's thought and that of Gilles Deleuze, in particular his reading of David Hume. Discussion centres around the importance of the role that time plays within the works of all three authors, particularly in regard to the idea of the void. The chapter concludes with the argument that Hume's temporal idea of human nature is the key to a symptomatic reading of Althusser that accounts for the persistence of ideas in the latter's social theory. In chapter three, Deleuze's reading of Hume's idea of relations is developed to take into account Bergson's theory of time. Read in contrast to Quentin Meillassoux's speculative realism, the chapter argues that Deleuze's account of temporal relations informs Althusser's social theory to create the ontological grounds for non-dogmatic and non-idealist practices of resistance. These practices are developed in chapter four with an unlikely turn to John Stuart Mill's idea of genius, the metaphysical property of the individual that signifies the discovery of new truth. The chapter begins with an argument that there is an under-developed account of ethics in Deleuze's work. Distinguishing the idea of genius from both Mill's moral philosophy, as well as from utilitarian thought more generally, the idea of genius is sutured onto Deleuze's ontological account of individuation. Read alongside Althusser's social theory, which accounts for the non-idealist conceptualisation of situations, this suture creates an ethically oriented structuralist ontology. The thesis concludes with the argument that the idea of genius is the ethical imperative that motivates practices of resistance. When individuals are understood as embodied within situations, practices of resistance are conceptualised not against other components of a situation, but contra them, taking them into account in order to amplify, multiply and transform the individual's potential within a situation. (shrink)

Many contemporary philosophers accept Hume's Dictum, according to which there are no metaphysically necessary connections between distinct, intrinsically typed entities. Tacit in Lewis 's work is a potential motivation for HD, according to which one should accept HD as presupposed by the best account of the range of metaphysical possibilities---namely, a combinatorial account, applied to spatiotemporal fundamentalia. Here I elucidate and assess this Ludovician motivation for HD. After refining HD and surveying its key, recurrent role in Lewis ’s work, I (...) present Lewis ’s appeal to HD as providing a broadly axiomatic generating basis for the space of metaphysical modality, and canvas the prima facie advantages of the resulting combinatorial principle---HD ---as being principled, extensionally adequate and modally reductive. Most criticisms of Lewis 's combinatorialism have targeted seeming ways in which the theory overgenerates the desired space; I rather argue that HD seriously undergenerates the desired space in three different ways. For each way I argue that available means of overcoming the undergeneration either fail to close the gap, undermine the claim that HD is a principled generator of metaphysical modal space, undermine the reductive status of Lewis 's combinatorialism, or call into question the truth of HD. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms (...) and predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

The present paper investigates the philosophical relationship between John von Neumann’s Nohidden-variable theorem and Bell’s inequalities. Bell erroneously takes the axiomatic method as implying a finality claim and thus ignores von Neumann’s strongly pragmatist stance towards mathematical physics. If one considers, however, Hilbert’s axiomatic method as a critical enterprise, Bell’s theorem improves von Neumann’s by defining a more appropriate notion of ‘ hidden variable’ that permits one to include Bohm’s interpretation which recovers the predictive content of quantum mechanics. (...) Contrary to Bell’s belief, accepting this model does not require adopting the metaphysically realist Böhm picture. If one takes the latter as a physical research programme one sees that it only partly disputes a common domain of facts with the mathematically oriented research programme of von Neumann. (shrink)

Another exceptionally fine text by Suppes. Designed for those who can follow a mathematical argument, but presupposes no special knowledge of mathematics or symbolic logic. The system developed is that of Zermelo-Fraenkel. Especially noteworthy is the discussion of the exact role played by the various axioms.--N. D. B., Jr.

This book tackles the issues that arise in connection with intensional logic -- a formal system for representing and explaining the apparent failures of certain important principles of inference such as the substitution of identicals and existential generalization -- and intentional states --mental states such as beliefs, hopes, and desires that are directed towards the world. The theory offers a unified explanation of the various kinds of inferential failures associated with intensional logic but also unifies the study of intensional contexts (...) and intentional states by grounding the explanation of both phenomena in a single theory. When an axiomatized realm of abstract entities is added to the metaphysical structure of the world, we can use them to identify and individuate the contents of directed mental states. The special abstract entities can be viewed as the objectified contents of mental files, and they play a crucial role in the analysis of truth conditions of the sentences involved in inferential failures. (shrink)

There is, of course, one clear sense in which Spinoza’s axiomatic method is a method of presentation: this is the sense which contrasts a method of presentation with a method of discovery. In the Ethics, Spinoza is stating and explaining his views, not describing how he arrived at them or telling us how to make discoveries for ourselves. Nor does he elsewhere present the axiomatic method as a method of discovery. In the seventeenth century, the distinction between a (...) method of discovery and a method of presentation was often described as a distinction between "analytic" and "synthetic" method; Arnauld and Nicole write. (shrink)

In this article, I discuss parthood status in mereologi- cally interpreted Daoist metaphysics, based on the Daodejing. I depart from the dao and you interrela- tion, which mereologically overlap by sharing parts. I consider the case of a complete overlap, which (a) challenges proper parthood, according to which a part cannot be identical with the whole that it com- poses, and (b) entails the question of identity that, while complying with classical mereology, cannot be consis- tent with Daoist (...) class='Hi'>metaphysics. The discussion leads to abandoning proper parthood and antisymmetry axiom from classical axiomatics. It also shows a plausible fur- ther direction for mereological reconstruction. (shrink)

Finean essentialists take metaphysical necessity to be metaphysically explained by essence. But whence the explanatory power of essence? A recent wave of criticism against the Finean account has put pressure on essentialists to answer this question. Wallner and Vaidya (2020) have responded by offering an axiomatic account of the explanatory power of essence. This paper discusses their account in light of some recent criticism by Bovey (2022). Building on work by Glazier (2017), Bovey succeeds in showing that Wallner and (...) Vaidya’s account is in need of modification and clarification. In this paper, I take up Bovey’s challenge and argue that Wallner and Vaidya have the resources available to incorporate the necessary modifications. I discuss the details of this modification, specifically, the question of what kind of metaphysical explanation is at work in the explanation of the necessity of essence itself. To get clear on this, I will introduce the distinction between generative and non-generative metaphysical explanation. In this way the paper contributes to both, the debate about whether essences can explain necessity and to the ongoing debate about metaphysical explanation. (shrink)

The reason for this spell--which was already felt in Anselm's life time-cannot be solely Anselm's subject matter, for this has been treated by many before and after with less than intriguing effects. It must be, to a large degree, his method. But what can there be so exciting about a logical demonstration?

The Guggenheim’s spring retrospective of the seminal Swedish painter, Hilma Af Klint, has, naturally, evoked a multitude of art critics and visual culture scholars who laud her radical abstraction which, at the beginning of the 20th century, preceded Kandinsky, Malevich, Mondrian. Yet, where much attention has been given to the symbology and motifs riddling Klint’s work – bold, private, untethered and nonrepresentational as they are – there has been a modicum of nuanced thought on how, exactly, esotericism and theology fomented (...) Klint’s pedagogical projects. Jillian Steinhauer, for instance, has underscored Klint’s naturalistic watercolors; featuring flowers and lifelike drawings of women; Summer Landscape (1888) suggests Klint’s technical aptitude for rendering light. Jadranka Ryle, in a clever article titled “Reinventing the Yggdrasil,” invokes Klint’s motif of the Yggdrasil, or the holy tree of Norse mythology, as a political axle for Nordic romanticism. Ryle makes the claim that by redirecting the Yggdrasil towards androgynous abstraction, Klint’s engagement with Norse mythology subverts the nationalist and patriarchal romanticism that previously characterized the nineteenth century’s use of the Yggdrasil. Others have adroitly uncovered biographical fragments from Klint’s life, detailing how she and four friends formed De Fem (“the Five”), an exclusively female coven that met weekly, praying, meditating, and holding seances to commune with spiritual guides. Rather than poise the Swedish maverick within the cannon of abstraction qua a political praxis, feminist subject-being, or historiography – projects that have been exhaustively overwrought, to the point of rendering Klint’s work both mute and reductively technical– I would like to take this time to closely examine simply one of Klint’s paintings and foreground it as an axiomatic modernist injunction between metaphysics and epistemology. (shrink)

In this article I intend to show that certain aspects of A.N. Whitehead's philosophy of organism and especially his epochal theory of time, as mainly exposed in his well-known work Process and Reality, can serve in clarify the underlying assumptions that shape nonstandard mathematical theories as such and also as metatheories of quantum mechanics. Concerning the latter issue, I point to an already significant research on nonstandard versions of quantum mechanics; two of these approaches are chosen to be critically presented (...) in relation to the scope of this work. The main point of the paper is that, insofar as we can refer a nonstandard mathematical entity to a kind of axiomatical formalization essentially 'codifying' an underlying mental process indescribable as such by analytic means, we can possibly apply certain principles of Whitehead's metaphysical scheme focused on the key notion of process which is generally conceived as the becoming of actual entities. This is done in the sense of a unifying approach to provide an interpretation of nonstandard mathematical theories as such and also, in their metatheoretical status, as a formalization of the empirical-experimental context of quantum mechanics. (shrink)

In the epicenter of his attempt to justify the “objective validity” of morality and freedom in the Critique of Practical Reason, Kant introduces a so-called fact of reason, which is rendered as the fact that human beings are consciou s of the moral ought’s categorical authority. However, few parts of Kant’s thinking have bemused commentators as much as this. In this article, the author explores a set of intersecting problems related to the fact of reason: (1) the problem of its (...) general argumentative role in Kant’s practical philosophy, (2) the problem of the fact as a brute fact. He argues that both problems can be understood and resolved only if we regard Kant’s introduction of the fact of reason as an implicit attempt to articulate intrinsic problems with reductive explanations of morality. His main claim, which has been surprisingly absent in most interpretations of Kant’s fact of reason, will be that the fact of reason functions as an undeniable yet improvable fact (similar to mathematical principles) from which other practical truths (such as the objective reality of freedom) can be derived. As Kant says unambiguously in the Jäsche-Logic : the “reality” of the moral law “is an axiom.”. (shrink)

Metaphysics as the foundation of science, philosophy as an exact method to discover this foundation: These are central themes in the Kant-Friesian philosophy which, at the beginning of the 20th century, Leonard Nelson, using the methods of mathematical axiomatics, further developed into interdisciplinary research programmes. Nelson carried out this research programme in his ethics but his untimely death prevented a systematic presentation of his natural philosophy and his doctrine of method. The present volume contains four transcripts of his seminars (...) on natural philosophy which he held over a period of 15 years. The transcripts, which may be taken as a comparatively complete presentation of Nelson's natural philosophy, supplement the writings published during his lifetime and convey a representative picture of his natural philosophy. // Metaphysik als Fundament der Wissenschaften, Philosophie, eine exakte Disziplin, die dieses Fundament offenlegt: Leitmotive des Kant-Fries’schen Denkgebäudes, die Leonard Nelson zu Beginn des 20. Jahrhunderts in interdisziplinäre Forschungsprogramme umsetzt. Die dabei angewandten – an der mathematischen Axiomatik ausgerichteten – Methoden haben nichts an Aktualität eingebüßt. Kann Nelson dieses Vorhaben bezüglich seiner Ethik verwirklichen, so verhindert sein früher Tod eine systematische Darstellung seiner Naturphilosophie und Methodenlehre. Der vorliegende Band enthält vier Mitschriften von Lehrveranstaltungen Nelsons. Diese Schriften, die die zu seinen Lebzeiten veröffentlichten Arbeiten zur Naturphilosophie ergänzen und erweitern sollen, decken einen Zeitraum von ca. 15 Jahren ab und geben somit einen repräsentativen Einblick in Nelsons naturphilosophisches Schaffen. Die vorliegende Zusammenstellung naturphilosophischer Nachlassschriften darf als verhältnismäßig vollständig gelten. (shrink)

How does vagueness interact with metaphysical modality and with restrictions of it, such as nomological modality? In particular, how do definiteness, necessity (understood as restricted in some way or not), and actuality interact? This paper proposes a model-theoretic framework for investigating the logic and semantics of that interaction. The framework is put forward in an ecumenical spirit: it is intended to be applicable to all theories of vagueness that express vagueness using a definiteness (or: determinacy) operator. We will show how (...) epistemicists, supervaluationists, and theorists of metaphysical vagueness like Barnes and Williams (2010) can interpret the framework.  We will also present a complete axiomatization of the logic we recommend to both epistemicists and local supervaluationists. . (shrink)

The metaphysics of dust -- Realist atomism -- Problems of the composition of phenomena -- Positivist atomism -- Critical atomism -- Axiomatic atomism.

Kalinowski conceives metaphysics as a set of propositions which can be axiomatized. This paper exposes the typical rules of an axiomatic system comparing them with the metaphysical system developed by Kalinowski. These rules are linguistic rules, but also admission-rules and transformation-rules. They always refer to the metaphysical propositions (wich are first or second propositions, as it is usual in logical systems). This paper tries to evaluate Kalinowski's attempt.

The received view says that possibility is the dual of necessity: a proposition is (metaphysically, logically, epistemically etc.) possible iff it is not the case that its negation is (metaphysically, logically, epistemically etc., respectively) necessary. This reading is usually taken for granted by modal logicians and indeed seems plausible when dealing with logical or metaphysical possibility. But what about epistemic possibility? We argue that the dual definition of epistemic possibility in terms of epistemic necessity generates tension when reasoning about non-idealized (...) agents and is a problem of concern for most hyperintensional epistemic logics that alleviate the problem of logical omniscience. The tension is particularly evident when knowledge is taken as a primitive to define other epistemic concepts, such as justification and belief, as done in the knowledge-first tradition. We propose a non-dual interpretation of epistemic possibility, employing a hyperintensionality filter similar to the one that makes the corresponding epistemic necessity operator hyperintensional. We employ the proposed semantics to model Stalnaker's belief as epistemic possibility of knowledge and provide a sound and complete axiomatization for a hyperintensional version of his bimodal logic of knowledge and belief. (shrink)

Kalinowski conceives metaphysics as a set of propositions which can be axiomatized. This paper exposes the typical rules of an axiomatic system comparing them with the metaphysical system developed by Kalinowski. These rules are linguistic rules, but also admission-rules and transformation-rules. They always refer to the metaphysical propositions (wich are first or second propositions, as it is usual in logical systems). This paper tries to evaluate Kalinowski's attempt.

This book offers a clear, analytic, and innovative interpretation of Heidegger's late work. This period of Heidegger's philosophy remains largely unexplored by analytic philosophers, who consider it filled with inconsistencies and paradoxical ideas, particularly concerning the notions of Being and nothingness. This book takes seriously the claim that the late Heidegger endorses dialetheism--namely the position according to which some contradictions are true--and shows that the idea that Being is both an entity and not an entity is neither incoherent nor logically (...) trivial. The author achieves this by presenting and defending the idea that reality has an inconsistent structure. In doing so, he takes one of the most discussed topics in current analytic metaphysics, grounding theory, into a completely unexplored area. Additionally, in order to make sense of Heidegger's concept of nothingness, the author introduces an original axiomatic mereological system that, having a paraconsistent logic as a base logic, can tolerate inconsistencies without falling into logical triviality. This is the first book to set forth a complete and detailed discussion of the late Heidegger in the framework of analytic metaphysics. It will be of interest to Heidegger scholars and analytic philosophers working on theories of grounding, mereology, dialetheism, and paraconsistent logic. (shrink)

In my dissertation, I address a variety of issues in the metaphysics of space and related areas. I begin by discussing the popular thesis that regions of space are identical to sets of points in space. I present three arguments against this thesis and conclude that we should be skeptical of it. In its place, I propose an axiomatic theory of regions of space that is consistent with both reductive accounts of their nature and with accounts that treat (...) them as sui generis entities. -/- I next explore the consequences of the aforementioned considerations. In particular, I describe five different sorts of structure each of which is such that the claim that space could have that structure is consistent with the axiomatic theory previously proposed. I claim that this fact, together with the skepticism concerning reductive accounts argued for earlier, shows that we should take seriously the claim that space could have any of these structures. -/- Having argued that we should be skeptical of the thesis that regions of space are identical to sets of points in space and suggesting that space could have different sorts of structure, I discuss how best to analyze continuity. I present an analysis of continuity inspired by remarks of Richard Cartwright in his 1975 paper ‘Scattered Objects’. I argue that this Cartwrightian analysis should be rejected because it identifies regions of space with sets of points in space, and I present a modified version of the analysis that does not do so. I note, however, that there is an intuitive notion of continuity that is not captured by this modified Cartwrightian analysis. I present and defend an analysis of continuity that better captures this intuitive notion. -/- I then turn to the issue of how to analyze what it is for a region of space to be open and what it is for a region of space to be closed. Here I argue that Cartwright’s analyses of these notions are incorrect. I then present a series of alternative analyses, revising each in response to objections. This process culminates with my proposed analyses of what it is for a region of space to be open and what it is for a region of space to be closed. -/- Finally, I discuss the Maximally Continuous Account of Simples (MaxCon), originally formulated and defended by Ned Markosian in his 1998 paper ‘Simples’. I argue that Markosian’s version of MaxCon, which identifies regions of space with sets of points in space and relies on the Cartwrightian analysis of continuity, should be rejected. I then formulate a new version of MaxCon that builds on the views defended earlier in my dissertation and defend this new version of MaxCon from objections. (shrink)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

In “Models and Reality”, H. Putnam formulated his model-theoretic argument against “metaphysical realism”. The article proposes a meta-reconstruction of Putnam’s model-theoretic argument in the light of two mutually compatible interpretations of it–elaborated by Manuel Garcia-Carpintero and Igor van Douven. A critical reflection on these interpretations and their adequacy for Putnam’s argument allows us to expose new theses coherent with Putnam’s reasoning and indicate new paths to improve this argument for our reconstruction task. In particular, we show that Putnam’s position may (...) be coherent with van Douven’s versions of Global Descriptivism under some conditions, but Putnam cannot reject realism as quickly as Carpintero suggests. We show that Suszko’s canonic axiomatic system and Sneed’s concept of theory may provide valuable support for Putnam’s argument. Finally, we critically evaluate Carpintero’s theses about the genesis of unintended interpretations of our languages, adopting the machinery of the Upward Skolem–Loewenheim Theorem and Knight’s Theorem. (shrink)

Bird’s Ultimate Argument sought to show that Armstrong’s N relationships involving categorical universals can’t entail nomic regularities. In N’s place Bird offered the non-categorical SR relation. Two kinds of objection have been raised: either Bird’s own alternative metaphysics fails in just the same way as Armstrong’s or the target of Bird’s argument may anyway have a way out of the problem. My aim is to reclaim the victory for Bird. I argue that the responses in defence of Armstong’s N (...) relationships fail to acknowledge that Bird was explicitly concerned with Armstrong’s commitment to a categoricalist view of universals. Moreover, Bird’s alternative account does not suffer the same problem since his metaphysics of properties is essentialist. Nevertheless, Bird’s account does need elaborating on to explainwhySR relationships entail their regularities. I offer Schaffer’s Axiomatic Solution as a candidate for this purpose. (shrink)