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)

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

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)

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)

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)

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.

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)

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)

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 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.

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.

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 \forall\exists\forall, while the axiom system based on congruence and order can beformulated using only \forall\exists-axioms.

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)

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.

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)

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)

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)

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)

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)

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.

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)

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)

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)

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)

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)

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 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)

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?

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)

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)