The purpose of this note is to examine a recent axiomatization of classical particle mechanics, and its relation to an alternative axiomatization I had earlier proposed. A comparison of the two proposals casts some interesting light on the problems of operationalism in classical celestial mechanics.1. Comparison of the Two Axiomatizations. The basic differences between the two proposals arise from the nature of the undefined terms. Both systems take the set of particles, time, and position as primitive notions. Both systems assume (...) that there exists a set of particles having continuous, twice-differentiable paths over some time interval. In addition, CPM takes mass and force as primitive notions, and assumes that with each particle there is associated a mass and a set of forces such that Newton's Second Law is satisfied. A system with these properties is called in CPM “a system of particle mechanics.” If, in addition, the set of forces in the system satisfies Newton's Third Law, the system is called in CPM “Newtonian.”. (shrink)

The task of axiomatizing physical theories has attracted, in recent years, some interest among both empirical scientists and logicians. However, the axiomatizations produced by either one of these two groups seldom appear satisfactory to the members of the other. It is the purpose of this paper to develop an approach that will satisfy the criteria of both, hence permit us to construct axiomatizations that will meet simultaneously the standards and needs of logicians and of empirical scientists.

In this paper, we axiomatize the negatable consequences in dependence and independence logic by extending the systems of natural deduction of the logics given in [22] and [11]. We prove a characterization theorem for negatable formulas in independence logic and negatable sentences in dependence logic, and identify an interesting class of formulas that are negatable in independence logic. Dependence and independence atoms, first-order formulas belong to this class. We also demonstrate our extended system of independence logic by giving explicit derivations (...) for Armstrong's Axioms and the Geiger-Paz-Pearl axioms of dependence and independence atoms. (shrink)

In Zielonka (1981a, 1989), I found an axiomatics for the product-free calculus L of Lambek whose only rule is the cut rule. Following Buszkowski (1987), we shall call such an axiomatics linear. It was proved that there is no finite axiomatics of that kind. In Lambek's original version of the calculus (cf. Lambek, 1958), sequent antecedents are non empty. By dropping this restriction, we obtain the variant L0 of L. This modification, introduced in the early 1980s (see, (...) e.g., Buszkowski, 1985; Zielonka, 1981b), did not gain much popularity initially; a more common use of L0 has only occurred within the last few years (cf. Roorda, 1991: 29). In Zielonka (1988), I established analogous results for the restriction of L0 to sequents without left (or, equivalently, right) division. Here, I present a similar (cut-rule) axiomatics for the whole of L0.This paper is an extended, corrected, and completed version of Zielonka (1997). Unlike in Zielonka (1997), the notion of rank of an axiom is introduced which, although inessential for the results given below, may be useful for the expected non-finite-axiomatizability proof.The paper follows the same way of subject exposition as Zielonka (2000) but it is technically much less complicated. I restrict myself to giving bare results; all the ideological background is exactly the same as in case of the non-associative calculusNL0 and those who are interested in it are requested to consult the introductory section of Zielonka (2000). (shrink)

Hilbert's axiomatization program of physical theories met an interesting challenge when it confronted the rise of quantum mechanics in the mid-twenties. The novelty of the mathematical apparatus of the then newly born theory was to be matched only by its substantial lack of any definite physical interpretation. The early attempts at axiomatization, which are described here, reflect all the difficulty of the task faced by Jordan, Hilbert, von Neumann and others. The role of von Neumann is examined in considerable detail (...) as he can be viewed here as the most outstanding of Hilbert's heirs. Von Neumann, especially in his work devoted to the proof of the impossibility of hidden variables, not only continued Hilbert's program but pushed it to its very limits, blending axiomatic rigor and interpretative commitment. (shrink)

Suppose that entities composed of two independent components are qualitatively ordered by a relation that satisfies the axioms of conjoint measurement. Suppose, in addition, that each component has a concatenation operation that, together either with the ordering induced on the component by the conjoint ordering or with its converse, satisfies the axioms of extensive measurement. Without further assumptions, nothing can be said about the relation between the numerical scales constructed from the two measurement theories except that they are strictly monotonic. (...) An axiom is stated that relates the two types of measurement theories, seems to cover all cases of interest in physics, and is sufficient to establish that the conjoint measurement scales are power functions of the extensive measurement scales. (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)

Reconstructions of quantum theory are a novel research program in theoretical physics which aims to uncover the unique physical features of quantum theory via axiomatization. I focus on Hardy’s “Quantum Theory from Five Reasonable Axioms” (2001), arguing that reconstructions represent a modern usage of axiomatization with significant points of continuity to von Neumann’s axiomatizations in quantum mechanics. In particular, I show that Hardy and von Neumann share similar methodological ordering, have a common operational framing, and insist on the empirical basis (...) of axioms. In the reconstruction programme, interesting points of discontinuity with historical axiomatizations include the stipulation of a generalized space of theories represented by a framework and the stipulation of analytic machinery at two levels of generality (first by establishing a generalized mathematical framework and then by positing specific formulations of axioms). In light of the reconstruction programme, I show that we should understand axiomatization attempts as being context–dependent, context which is contingent upon the goals of inquiry and the maturity of both mathematical formalism and theoretical underpinnings within the area of inquiry. Drawing on Mitsch (2022)’s account of axiomatization, I conclude that reconstructions should best be understood as provisional, practical, representations of quantum theory that are well suited for theory development and exploration. However, I propose my context–dependent re–framing of axiomatization as a means of enriching Mitsch’s account. (shrink)

This book is a wonderful resource for historians and philosophers of mathematics and physics alike, not just for Hilbert's own work in physics, but also because Corry sets Hilbert in context, bringing out the people with whom Hilbert had contact, describing their work and possible links with Hilbert's work, and describing the activities going on around Hilbert. The historical thesis of this book is that Hilbert worked on a wide range of issues in physics for a period lasting more than (...) two decades, employing and developing his axiomatic approach throughout. One conclusion that follows from this is that Hilbert's 1915–1917 work relating to Einstein's General Theory of Relativity was a natural continuation of Hilbert's pre-existing interests and activities, and not a one-off foray into foreign territory. 1Of especial interest to philosophers of mathematics are two further theses. Corry stresses that for Hilbert geometry is an empirical science, and related to this argues first, that Hilbert intends the axiomatic method to be used in enhancing our understanding of the content of a given theory via relating the results of the axiomatic investigation back to the intuitive content of the axioms; and, second, that to understand Hilbert's axiomatic approach in mathematics we must pay serious attention to his work in physics.Corry also hopes to show ‘the significant and unique contribution of Hilbert to certain important developments in twentieth-century physics’ . 2 In the end, this assessment of Hilbert's contribution to physics is far from clear cut: the two cases where Hilbert goes into the details of a physical theory show him lacking feel for what is important physically with respect to that theory. Nevertheless, philosophers and historians of physics will find a great deal to interest them in the story of Hilbert's involvement in physics, and in the details …. (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)

We outline the rather complicated history of attempts at axiomatizing Jaśkowski’s discussive logic $$\mathbf {D_2}$$ D2 and show that some clarity can be had by paying close attention to the language we work with. We then examine the problem of axiomatizing $$\mathbf {D_2}$$ D2 in languages involving discussive conjunctions. Specifically, we show that recent attempts by Ciuciura are mistaken. Finally, we present an axiomatization of $$\mathbf {D_2}$$ D2 in the language Jaśkowski suggested in his second paper on discussive logic, by (...) following a remark of da Costa and Dubikajtis. We also deal with an interesting variant of $$\mathbf {D_2}$$ D2, introduced by Ciuciura, in which negation is also taken to be discussive. (shrink)

In the chapter on quantum logic in Volume 6 of Handbook of Philosophical Logic, Dalla Chiara and Giuntini make an interesting observation that there is a unified relational semantics underlying both the {¬,∧}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ {\lnot }, {\wedge } \}$$\end{document}-fragment of intuitionistic logic and ortho-logic. In this paper, we contribute to a systematic investigation of this relational semantics by providing an axiomatization of its logic.

In this essay, Alain Caillé reconstructs the “singular history of the MAUSS (Anti-Utilitarian Movement in the Social Sciences)” from when in early 1980 a group of friends from different disciplines (sociologists, economists, philosophers, etc.) decided to found the “Bulletin du MAUSS” to counter the growing hegemony of utilitarianism and economism in the human sciences and in the philosophical disciplines themselves. The Bulletin would initially become the “Revue du MAUSS trimestrielle” from 1988 to 1992 and from 1993 to 2022 the “Revue (...) du MAUSS semestrielle” until its current anglophone extension “MAUSS International”. The author takes Marcel Mauss’ definition of the gift as a “total social fact” and explains that the gift paradigm is a “translation operator” in the sense that it is continuously enriched in interaction with existing discourses. The theoretical operation that Caillé proposes is to explicate Mauss’s discovery by moving from a simple gift paradigm, based on the perspective of simple reciprocity between giver and recipient, to an expanded gift paradigm. In this way, it can dialogue with other paradigms and other currents of thought (theory of recognition, theory of care, theory of human development, etc.). And from here to a conception of the gift as “adonnement”, i.e. the commitment of human subjects to bring something new and unprecedented into the world. (shrink)

Every artificial-intelligence research project needs a working definition of "intelligence", on which the deepest goals and assumptions of the research are based. In the project described in the following chapters, "intelligence" is defined as the capacity to adapt under insufficient knowledge and resources. Concretely, an intelligent system should be finite and open, and should work in real time. ;If these criteria are used in the design of a reasoning system, the result is NARS, a non-axiomatic reasoning system. ;NARS uses a (...) term-oriented formal language, characterized by the use of subject-predicate sentences. The language has an experience-grounded semantics, according to which the truth value of a judgment is determined by previous experience, and the meaning of a term is determined by its relations with other terms. Several different types of uncertainty, such as randomness, fuzziness, and ignorance, can be represented in the language in a single way. ;The inference rules of NARS are based on three inheritance relations between terms. With different combinations of premises, revision, deduction, induction, abduction, exemplification, comparison, and analogy can all be carried out in a uniform format, the major difference between these types of inference being that different functions are used to calculate the truth value of the conclusion from the truth values of the premises. ;Since it has insufficient space-time resources, the system needs to distribute them among its tasks very carefully, and to dynamically adjust the distribution as the situation changes. This leads to a "controlled concurrency" control mechanism, and a "bag-based" memory organization. ;A recent implementation of the NARS model, with examples, is discussed. The system has many interesting properties that are shared by human cognition, but are absent from conventional computational models of reasoning. ;This research sheds light on several notions in artificial intelligence and cognitive science, including symbol-grounding, induction, categorization, logic, and computation. These are discussed to show the implications of the new theory of intelligence. ;Finally, the major results of the research are summarized, a preliminary evaluation of the working definition of intelligence is given, and the limitations and future extensions of the research are discussed. (shrink)

We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gs axiom of constructibility. In list theory there are strong arguments favoring Gs axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is (...) a natural and attractive modification of ALT where every object is constructible and countable. In order to substantiate our foundational interest in lists, we also compare sets and lists from the perspective of finite objects, arguing that lists are, from a certain point of view, conceptually simpler than sets. (shrink)

We may wonder about the status of logical accounts of the meaning of language. When does a particular proposal count as a theory? How do we judge a theory to be correct? What criteria can we use to decide whether one theory is “better” than another? Implicitly, many accounts attribute a foundational status to set theory, and set-theoretic characterisations of possible worlds in particular. The goal of a semantic theory is then to find a translation of the phenomena of (...) class='Hi'>interest into a set-theoretic model. Such theories may be deemed to have “explanatory” or “predictive” power if a mapping can found into expressions of set-theory that have the appropriate behaviour by virtue of the rules of set-theory (for example Montague 1973; Montague1974). This can be contrasted with an approach in which we can help ourselves to “new” primitives and ontological categories, and devise logical rules and axioms that capture the appropriate inferential behaviour (as in Turner 1992). In general, this alternative approach can be criticised as being mere “descriptivism”, lacking predictive or explanatory power. Here we will seek to defend the axiomatic approach. Any formal account must assume some normative interpretation, but there is a sense in which such theories can provide a more honest characterisation (cf. Dummett 199). In contrast, the set-theoretic approach tends to conflate distinct ontological notions. Mapping a pattern of semantic behaviour into some pre-existing set-theoretic behaviour may lead to certain aspects of that behaviour being overlooked, or ignored (Chierchia & Turner 1988; Bealer 1982). Arguments about the explanatory and predictive power of set-theoretic interpretations can also be questioned (see Benacerraf 1965, for example). We aim to provide alternative notions for evaluating the quality of a formalisation, and the role of formal theory. Ultimately, claims about the methodological and conceptual inadequacies of axiomatic accounts compared to set-theoretic reductions must rely on criteria and assumptions that lie outside the domain of formal semantics as such. (shrink)

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)

On December 10th, 1947, John von Neumann wrote to the Spanish translator of his Mathematical Foundations of Quantum Mechanics: 1Your questions on the nature of mathematical physics and theoretical physics are interesting but a little difficult to answer with precision in my own mind. I have always drawn a somewhat vague line of demarcation between the two subjects, but it was really more a difference in distribution of emphases. I think that in theoretical physics the main emphasis is on the (...) connection with experimental physics and those methodological processes which lead to new theories and new formulations, whereas mathematical physics deals with the actual solution and mathematical execution of a theory which is assumed to be correct per se, or assumed to be correct for the sake of the discussion. In other words, I would say that theoretical physics deals rather with the formation and mathematical physics rather with the exploitation of physical theories. However, when a new theory has to be evaluated and compared with experience, both aspects mix. (shrink)

The lack of specific arithmetical axioms in Book VII has puzzled historians of mathematics. It is hardly possible in our view to ascribe to the Greeks a conscious undertaking to axiomatize arithmetic. The view that associates the beginnings of the axiomatization of arithmetic with the works of Grassman [1861], Dedekind [1888] and Peano [1889] seems to be more plausible. In this connection a number of interesting historical problems have been raised, for instance, why arithmetic was axiomatized so late. This question (...) was first posed and quite conclusively answered by Yanovskaja [1956]. Her major and quite conclusive argument was that “algorithms in arithmetic have absolute character, while in geometry we have to do with reducibility algorithms”. In this paper we are going to draw attention on certain peculiarities of the construction of the arithmetical Books of Euclid’s "Elements" and show that Euclidean arithmetic is constructed by effective procedures. In spite of the use of inference by reductio ad absurdum and methods equivalent to mathematical induction, Euclidean arithmetic retains its finitary character. It is not full arithmetic, but a finitary fragment of classical arithmetic, and thereby there was not any internal reason for its axiomatization. (shrink)

There are two prominent viewpoints regarding the nature of rationality and how it should be evaluated in situations of interest: the traditional axiomatic approach and the newer ecological rationality. An obstacle to comparing and evaluating these seemingly opposite approaches is that they employ different language and formalisms, ask different questions, and are at different stages of development. I adapt a formal framework known as SCOP to address this problem by providing a comprehensive common framework in which both approaches may (...) be defined and compared. The main result is that the axiomatic and ecological approaches are in far greater agreement on fundamental issues than has been appreciated; this is supported by a pair of theorems to the effect that they will make accordant rationality judgements when forced to evaluate the same situation. I conclude that ecological rationality has some subtle advantages, but that we should move past the issues currently dominating the discussion of rationality. (shrink)

Inspired by an interesting quotation from the literature, we propose four modalities, called ‘sane belief’, ‘insane belief’, ‘reliable belief’ and ‘unreliable belief’, and introduce logics with each operator as the modal primitive. We show that the four modalities constitute a square of opposition, which indicates some interesting relationships among them. We compare the relative expressivity of these logics and other related logics, including a logic of false beliefs from the literature. The four main logics are all less expressive than the (...) standard modal logic over various model classes, and the logics of sane and insane beliefs are, respectively, equally expressive as the logics of unreliable and reliable beliefs on any class of models. The logics of reliable and unreliable beliefs are then combined into a bimodal logic, which turns out to be equally expressive as the standard modal logic. Despite this, we cannot obtain a complete axiomatization of the minimal bimodal logic, by simply translating the axioms and rules of the minimal modal logic $\textbf{K}$ into the bimodal language. We then introduce a schematic modality which unifies reliable and unreliable beliefs and axiomatize it over the class of all frames and also the class of serial frames. This line of research is finally extended to unify sane and insane beliefs and some axiomatizations are given. (shrink)

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 an earlier paper, [5], I gave semantics and tableau rules for a simple firstorder intensional logic called FOIL, in which both objects and intensions are explicitly present and can be quantified over. Intensions, being non-rigid, are represented in FOIL as (partial) functions from states to objects. Scoping machinery, predicate abstraction, is present to disambiguate sentences like that asserting the necessary identity of the morning and the evening star, which is true in one sense and not true in another.In this (...) paper I address the problem of axiomatizing FOIL. I begin with an interesting sublogic with predicate abstraction and equality but no quantifiers. In [2] this sublogic was shown to be undecidable if the underlying modal logic was at least K4, though it is decidable in other cases. The axiomatization given is shown to be complete for standard logics without a symmetry condition. The general situation is not known. After this an axiomatization for the full FOIL is given, which is straightforward after one makes a change in the point of view. (shrink)

This monograph is a detailed study, and systematic defence, of the Growing Block Theory of time (GBT), first conceived by C.D. Broad. The book offers a coherent, logically perspicuous and ideologically lean formulation of GBT, defends it against the most notorious objections to be found in the extant philosophical literature, and shows how it can be derived from a more general theory, consistent with relativistic spacetime, on the pre-relativistic assumption of an absolute and total temporal order. -/- The authors devise (...) axiomatizations of GBT and its competitors which, against the backdrop of a shared quantified tense logic, significantly improves the prospects of their comparative assessment. Importantly, neither of these axiomatizations involves commitment to properties of presentness, pastness or futurity. The authors proceed to address, and defuse, a number of objections that have been marshaled against GBT, including the so-called epistemic objection according to which the theory invites skepticism about our temporal location. The challenge posed by relativistic physics is met head-on, by replacing claims about temporal variation by claims about variation across spacetime. -/- The book aims to achieve the greatest possible rigor. The background logic is set out in detail, as are the principles governing the notions of precedence and temporal location. The authors likewise devise a novel spacetime logic suited for the articulation, and comparative assessment, of relativistic theories of time. The book comes with three technical appendices which include soundness and completeness proofs for the systems corresponding to GBT and its competitors, in both their pre-relativistic and relativistic forms. -/- The book is primarily directed at researchers and graduate students working on the philosophy of time or temporal logic, but is of interest to metaphysicians and philosophical logicians more generally. (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)

Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic. But a natural conception of the settheoretic universe is as an indefinite (or “potential”) totality, to which intuitionistic logic is more appropriately applied, while each set is taken to be a definite (or “completed”) totality, for which classical logic is appropriate; so on that view, set theory should be axiomatized on some correspondingly mixed basis. Similarly, in the (...) case of predicative analysis, the natural numbers are considered to form a definite totality, while the universe of sets (or functions) of natural numbers are viewed as an indefinite totality, so that, again, a mixed semi-constructive logic should be the appropriate one to treat the two together. Various such semiconstructive systems of analysis and set theory are formulated here and their proof-theoretic strength is characterized. Interestingly, though the logic is weakened, one can in compensation strengthen certain principles in a way that could be advantageous for mathematical applications. (shrink)

People tend to rank values of all kinds linearly from good to bad, but there is little reason to think that this is reasonable or correct. This book argues, to the contrary, that values are often partially ordered and hence frequently incomparable. Proceeding logically from a small set of axioms, John Nolt examines the great variety of partially ordered value structures, exposing fallacies that arise from overlooking them. He reveals various ways in which incomparability is obscured: using linear indices to (...) summarize partially ordered data, relying on an inadequately defined concept of parity, or conflating incomparability with vagueness. Incomparability can enrich and clarify a range of topics including the paradoxes of Derek Parfit, rational decision theory, and the infinite values of theology. Finally, Nolt shows how to generalize many of the concepts introduced earlier, explores the intricate depths of certain noteworthy partially ordered value structures, and argues for the finitude of value. Incomparable Values will be of interest to scholars and advanced students working in ethics, value theory, rational decision theory, and logic. (shrink)

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains so far an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still (...) lacking. We provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one's implicit commitments, and sheds light on the reason why this is so. We argue that the theory has interesting epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent proposals for the analysis of implicit commitment based on truth or epistemic notions. (shrink)

The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry , namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that motivated Hilbert’s axiomatic (...) investigations from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis. (shrink)

Meaning and Purpose of Life are perhaps the most thought about, if not talked about, issues on the planet since human beings have walked on earth. This book is another attempt to understand the Meaning and Purpose of Life using ideas of Vedanta in Indian philosophy, and of mainstream economics. Starting from first principles, Dr Agarwal explores the core concept of Brahman in Vedanta, and builds an axiomatic foundation for understanding the meaning and purpose of life using the fundamental ideas (...) of the Prasthana Traya, and in particular of The Bhagavad Gita and The Principal Upanishads. Dr Agarwal adheres to the philosophy of Karma Yoga as his chosen area of interest for the exploration of meaning and purpose, which allows him to seek linkages between the concepts of social welfare in Vedanta and economics. Ideas of lokasamgraha in Indian Philosophy and of externalities in economics provide a clear common ground between the two apparently disparate disciplines. Throughout the book, he emphasizes simple practical approaches for making life more meaningful and happier, while relying fundamentally on the basic ideas of Vedanta. (shrink)

Written before the impressive collapse of the socialist system in Eastern Europe, this book offers a quite objective and serious systematic analysis of the Marxian labor theory of value, Marx's main scientific legacy. After reconstructing the 'prototype' of this theory - which is the theory as it was left by Marx himself in Capital - the author proceeds to a careful and detailed analysis of its foundational problems, taking into account Bohm-Bawerk's important criticisms. After introducing advanced contemporary formal tools, the (...) author proceeds to a thorough discussion of the dialectical method, just in order to tackle the foundational problems of the theory. He provides a formally precise and well motivated definition of abstract labor, and then proceeds to prove the existence of a measurement of abstract labor - i.e. the existence of numerical labor-values. Using this result, the author provides rigorous axiomatic foundations for the theory of value and then proves the existence of a Marxian competitive equilibrium, which is tantamount to the proof of the possibility of reproduction for a capitalist economy. The author finishes the book by showing in detail how the problems of the prototype are solved, by reconstructing the Leontief model of the labor theory of value on the new logical bases. Written in a very clear style, in the language of contemporary philosophy of science, the book is of interest to philosophers of science and economists, applied logicians and all those interested in the scientific legacy of Karl Marx. (shrink)

Within the scope of interest of deontic logic, systems in which names of actions are arguments of deontic operators (deontic action logic) have attracted less interest than purely propositional systems. However, in our opinion, they are even more interesting from both theoretical and practical point of view. The fundament for contemporary research was established by K. Segerberg, who introduced his systems of basic deontic logic of urn model actions in early 1980s. Nowadays such logics are considered mainly within (...) propositional dynamic logic (PDL). Two approaches can be distinguished: in one of them deontic operators are introduced using dynamic operators and the notion of violation, in the other at least some of them are taken as primitive. The second approach may be further divided into the systems based on Boolean algebra of actions and the systems built on the top of standard PDL. In the present paper we are interested in the systems of deontic action logic based on Boolean algebra. We present axiomatizations of six systems and set theoretical models for them. We also show the relations among them and the position of some existing theories on the resulting picture. Such a presentation allows the reader to see the spectrum of possibilities of formalization of the subject. (shrink)

The object of this paper is to examine half and full connexive extensions of the basic regular conditional logic CR. Extensions of this system are of interest because it is among the strongest well-known systems of conditional logic that can be augmented with connexive theses without inconsistency resulting. These connexive extensions are characterized axiomatically and their relations to one another are examined proof-theoretically. Subsequently, algebraic semantics are given and soundness, completeness, and decidability are proved for each system. The semantics (...) is also used to establish independence results. Finally, a deontic interpretation of one of the systems is examined and defended. (shrink)

This paper introduces a generalized version of inquisitive semantics, denoted as GIS, and concentrates especially on the role of disjunction in this general framework. Two alternative semantic conditions for disjunction are compared: the first one corresponds to the so-called tensor operator of dependence logic, and the second one is the standard condition for inquisitive disjunction. It is shown that GIS is intimately related to intuitionistic logic and its Kripke semantics. Using this framework, it is shown that the main results concerning (...) inquisitive semantics, especially the axiomatization of inquisitive logic, can be viewed as particular cases of more general phenomena. In this connection, a class of non-standard superintuitionistic logics is introduced and studied. These logics share many interesting features with inquisitive logic, which is the strongest logic of this class. (shrink)

Among the numerous influences and reciprocal interactions between France and Italy at the beginning of the 20th century, it is interesting to investigate the complex case of Louis Rougier’s reception of Italian mathematical logic (including in particular the contributions by some members of the Peano school: Giuseppe Peano, Giovanni Vailati, Alessandro Padoa, and Mario Pieri). This paper aims to investigate the role and the influence of the Peano school on the inversion of this French tendency of philosophers to ignore logic (...) and mathematics. My claim is that Rougier was not only personally interested in Italian mathematics and logic, and in particular in the innovations made by Peano and his school, but used it to support a project of renewal of French logic and scientific philosophy. Rougier recalls the contributions of the school to axiomatics and to mathematical logic, in order to criticize the standard teaching of logic in France, mainly based on syllogistic theory, as well as the view, still defended by his teacher Goblot in 1907, that mathematical proofs are largely based on intuition. For this purpose, any reference to Hilbert’s axiomatics would have sufficed. The reason why Rougier refers also to Peano and his entourage is related to a common interest in linguistics and in epistemology. (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)

I this paper I give minimal axioms for the ontology of time, especially A-theories and B-theories and I derive philosophically interesting lemmas. The exercise is set-theoretical, defining all notions and indicating assumptions and philosophical points of disagreement, while being easy to translate to other formal expressions . The issue of a logic for A-theories of time is treated towards the end, where I sketch ‘copresent’ operators for capturing the idea of temporal passage. The main conclusion will be that, while circularity (...) threatens both families of theories, the growing block theory (belonging to the A-theory) is the least objectionable philosophically. (shrink)

Quasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the (...) extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts. (shrink)

Total Utilitarianism is the view that an action is right if and only if it maximizes the sum total of people’s well-being. A common objection to Total Utilitarianism is that it is insensitive to matters of distributive justice. For example, for a given amount of well-being, Total Utilitarianism is indifferent between an equal distribution and any unequal distribution, and if there would be a tiny gain in well-being by moving from an equal distribution to an unequal, we have a duty (...) to do so. To meet the objection from justice, Fred Feldman has suggested a desert adjusted version of Total Utilitarianism ---‘Justicism’ --- which in addition to the value of wellbeing takes into account factors concerning people’s desert.1 Feldman’s suggestion is novel and interesting but his theory has been severely criticized as a theory of distributive justice.2 In the present paper, I shall try to salvage what I think might be a kernel of truth in Feldman’s suggestion, or at least a kernel that is worthy of further investigation. (shrink)

This reissue of D. A. Gillies highly influential work, first published in 1973, is a philosophical theory of probability which seeks to develop von Mises’ views on the subject. In agreement with von Mises, the author regards probability theory as a mathematical science like mechanics or electrodynamics, and probability as an objective, measurable concept like force, mass or charge. On the other hand, Dr Gillies rejects von Mises’ definition of probability in terms of limiting frequency and claims that probability should (...) be taken as a primitive or undefined term in accordance with modern axiomatic approaches. This of course raises the problem of how the abstract calculus of probability should be connected with the ‘actual world of experiments’. It is suggested that this link should be established, not by a definition of probability, but by an application of Popper’s concept of falsifiability. In addition to formulating his own interesting theory, Dr Gillies gives a detailed criticism of the generally accepted Neyman Pearson theory of testing, as well as of alternative philosophical approaches to probability theory. The reissue will be of interest both to philosophers with no previous knowledge of probability theory and to mathematicians interested in the foundations of probability theory and statistics. (shrink)

Peano’s axioms for arithmetic, published in 1889, are ubiquitously cited in writings on modern axiomatics, and his Formulario is often quoted as the precursor of Russell’s Principia Mathematica. Yet, a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics remains to be made. In line with increased interest in the philosophy of mathematics for the investigation of mathematical practices, this them...

Peano’s axioms for arithmetic, published in 1889, are ubiquitously cited in writings on modern axiomatics, and his Formulario is often quoted as the precursor of Russell’s Principia Mathematica. Yet, a comprehensive historical and philosophical evaluation of the contributions of the Peano School to mathematics, logic, and the foundation of mathematics remains to be made. In line with increased interest in the philosophy of mathematics for the investigation of mathematical practices, this them...

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.

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.1 (2001) 121-140 [Access article in PDF] The Rhetoric of Modal Equivocacy in Cartesian Transubstantiation Julian Bourg Everyday language, in which words are not defined, is a medium in which nobody can express himself unequivocally. Robert Musil 1René Descartes's attempt to explain Eucharistic transubstantiation has long been understood as a dramatically significant moment in his tightrope walk across the medieval-to-modern divide. 2 Modeled (...) on the Last Supper in the New Testament, the Eucharistic rite had been explained in the Catholic tradition after the twelfth century in the scholastic philosophical language of substance and accident. 3 As the Council of Trent reaffirmed in 1551, this transubstantiation was a conversion "of the whole substance of the bread into the substance of the Body and of the whole substance of the wine into the substance of His blood... the appearances [speciebus] only of the bread and wine remaining." 4 Christ was [End Page 121] held to be "truly, really, and substantially contained under the species of those sensible things," under what still appeared to be bread and wine. 5Descartes's dilemma was that on one hand he was dead set against the hylomorphic theory of substance and accident, and on the other hand he found himself obliged by some of his interlocutors to comment on transubstantiation. It was a balancing act that pleased no one for very long, since despite his sometimes eager discussions of the Eucharist, he was not, as he admitted, "a theologian by profession." 6 The putative heterodoxy of Cartesian transubstantiation was principally responsible for his works being placed on the Index of Prohibited Books in 1663 and for his thought being banned from schools under Louis XIV in 1671. 7 His treatment of the Eucharistic question also distressed others in the seventeenth and early eighteenth centuries, such as Thomas Hobbes and Isaac Newton, who were unhappy that Descartes had bothered with such a theological non-issue. 8 For the many post-seventeenth-century advocates of a proto-secular Descartes as the godfather of modern philosophy, Cartesian transubstantiation has proved similarly inconvenient.The indisputable connections between Descartes and his scholastic forerunners, exemplified by the Eucharistic question, service a basic question of contemporary Cartesian scholarship. As Émile Boutroux put it, how was new wine poured from old bottles? 9 Accepting as axiomatic that there was something both radical and original in the Cartesian project but without falling back on pieties about his turn toward the modern subject, we must ask how Descartes put forth novel claims in conceptual language regarded by his contemporaries as resonant, if not consonant, with the scholastic intellectual culture of Catholic Reform France. More precisely, setting aside the not-very-defensible position that such language was merely incidental to Cartesian thought (a position that dehistoricizes Descartes as a precocious Enlightener), the dilemma then becomes explaining how original positions were elaborated within scholastic language. One important task is accounting for what Descartes did with that vocabulary--how he pointed words and concepts in new and surprising directions. The Eucharist becomes crucial in this perspective. [End Page 122]Though the question of Cartesian transubstantiation has been discussed for some time, the topic is worth revisiting again in light of two themes, rhetoric and the Cartesian theory of modes. My argument is that Descartes used one particular scholastic term, "mode," in equivocal ways and that this equivocation was in multiple senses rhetorical. This rhetoric coalesced and came to the fore around the Eucharist because, given the intellectual-political climate of the Catholic Reformation, the stakes of transubstantiation were so high. That Descartes used certain scholastic terms in equivocal or ambiguous ways should come as no surprise. Of interest, though, is the fact that by definition "mode" meant many things. This ambiguous term facilitated ambiguous intellectual positions about the Eucharist that allowed Descartes more generally (no matter what he "really" intended) to skirt between an orthodoxy that represented the past and an originality, not without dangers in his time, that seem to gesture toward our day. Foregrounded with regard to... (shrink)

ABSTRACT: A comprehensive introduction to ancient (western) logic from earliest times to the 6th century CE, with a focus on issues that may be of interest to contemporary logicians and covering important topics in Post-Aristotelian logic that are frequently neglected (such as Peripatetic hypothetical syllogistic, the Stoic axiomatic system of propositional logic and various later ancient developments).

This work is in two parts. The main aim of part 1 is a systematic examination of deductive, probabilistic, inductive and purely inductive dependence relations within the framework of Kolmogorov probability semantics. The main aim of part 2 is a systematic comparison of (in all) 20 different relations of probabilistic (in)dependence within the framework of Popper probability semantics (for Kolmogorov probability semantics does not allow such a comparison). Added to this comparison is an examination of (in all) 15 purely inductive (...) dependence relations. ————Part 1 leads in an axiomatic step-by-step development from the elementary classical truth value semantics of a sentential-logical language, called ‘L’, (chapter 1) to the elementary Kolmogorov probability semantics of L (chapter 2), which is then extended to four axiomatic semantical theories of dependence relations between the formulae of L. First the elementary Kolmogorov probability semantics of L is extended to a theory, called ‘Kdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 3). Then Kdd is extended to a theory, called ‘Kpd1’, of the degree to which formulae of L probabilistically depend on each other in regard to a given probability distribution on the set of all formulae of L (chapter 4). Kpd1, in its turn, gets extended to a theory, called ‘Kpd2’, of the relations of probabilistic dependence and independence, relativized to unary Kolmogorov probability functions defined on L (chapter 5). Then Kpd2 is extended to a theory, called ‘Kid’, of the relations of inductive dependence and inductive independence, again relativized to unary Kolmogorov probability functions defined on L (chapter 6). Finally, Kid is extended to a theory, called ‘Kpid’, of the relations of purely inductive positive and negative dependence, relativized to unary Kolmogorov probability functions defined on L (chapter 7). ——Chapter 1, which deals with the familiar notions of truth value functions, tautologies, consequence relations and relations of logical opposition, is naturally the shortest chapter of part 1.——In chapter 2, the elementary classical semantics of L is extended to the elementary Kolmogorov probability semantics of L, i.e. to an axiomatic theory of unary and of binary Kolmogorov probability functions defined on the set of formulae of L. Because of the elementary character of this theory, chapter 2 is also rather short.——Chapter 3 introduces the first theory on dependence relations, to wit: Kdd, the theory of deductive (in)dependence between formulae of L. I follow here the well-known idea of Popper and Miller, who have used it in a famous discussion on the nature of probabilistic support for their arguments that probabilistic support is deductive, not inductive. I develop Kdd in the form of about 100 theorems, making ample use of the fact that deductive independence is nothing but subcontrary opposition, and close with a remark on the fundamental difference between deductive and logical dependence—two relations the ideas of which are all too easily mixed up.——Chapters 4 and 5 deal extensively with the traditional ideas of probabilistic (in)dependence, applied to formulae rather than to events. As always, I proceed axiomatically in a step-by-step process under systematic viewpoints and obtain about 300 theorems in this way. In the formulation of the theorems, I took special care to state clearly and expressly so-called tacit assumptions, especially those concerning the probability values of the formulae said to be dependent on each other. These assumptions are usually missing in the literature, due either to economy of writing or to sloppiness of thinking. Presumably, both chapters contain little that is new, their value lying more in the systematic grouping and organic development of the theorems than in the newness of these.——In chapter 6, I extend the axiomatic theory about probabilistic (in)dependence which has been elaborated in chapter 5, to an axiomatic theory of inductive (in)dependence by requiring of the relation of inductive (in)dependence that it be probabilistic (in)dependence, but not also logical implication or logical opposition. I point out the differences between probabilistic and inductive (in)dependence by means of some 60 theorems and close my examination of inductive (in)dependence by considering its relationship to the notion of support in the philosophy of science.——Finally, in chapter 7, the last of part 1, I take the step from inductive dependence to what I call ‘purely inductive dependence’ by combining the idea of inductive dependence with that of deductive independence in a way which is suggested by writings of Popper and Miller. I arrive at two noteworthy theorems. Firstly, there is indeed no purely inductive support. But secondly, and perhaps amazingly, countersupport is purely inductive.————Whereas the probabilistic framework of part 1 of the present work is Kolmogorov probability semantics, the framework of part 2 is Popper probability semantics, which is not only worth examining as a fascinating alternative to orthodox Kolmogorov probability semantics, but also allows us to examine dependence relations more deeply, than Kolmogorov probability semantics does. Part 2 leads—again in an axiomatic step-by-step development—from the basic Popper probability semantics of L, called ‘Pb’, (chapter 8) via a probabilistic theory of logical attributes, called ‘Ps’, (chapter 9) to four axiomatic semantical theories of dependence relations between the formulae of L. First, Ps is extended to a theory, called ‘Pdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 10). Then Pdd is extended to a theory, called ‘Ppd’, of (in all) 20 relations of probabilistic (in)dependence, relativized to binary Popper probability functions defined on L (chapter 11). Ppd, in its turn, is extended to a theory, called ‘Pid’, of (in all) 10 relations of inductive dependence, again relativized to binary Popper probability functions defined on L (first part of chapter 12). Finally, Pid is extended to a theory, called ‘Ppid’, of (in all) 15 relations of purely inductive positive and negative dependence, relativized to binary Popper probability functions defined on L (second part of chapter 12).——Chapter 8, the first chapter of part 2 of the present work, is entirely preparatory. It introduces the axioms and about 180 theorems (150 of them together with their proofs) of basic Popper probability semantics in order to set this kind of semantics under way.——Then, in chapter 9, basic Popper probability semantics is extended to a probabilistic theory of logical properties of and relations between the formulae of L. Although I think that the way I did this extension is of some interest in itself, the main task of chapter 9 is again a preparatory one: to yield the indispensable lemmata (about 90 in number) for the theorems concerning probabilistic dependence relations in chapter 11 and concerning inductive dependence relations in chapter 12.——Chapter 10 brings the extension of Ps to the theory Pdd of deductive (in)dependence. Only half a dozen theorems are noted here for later use in the Pdd-extensions Ppd and Ppid. In view of the over 100 theorems already gained on this topic in the Kolmogorovian framework (cf. chapter 3), a similar extensive elaboration of Pdd would have been superfluous.——Chapter 11 is the most important one of part 2. It consists of a systematic comparison of 20 probabilistic (in)dependence concepts by means of about 230 theorems, obtained within the axiomatic theory Ppd, which is built up as an extension of Pdd. The main points of comparison were: differences in logical strength; reflexivity and symmetry; behaviour under the condition that the probability values of the formulae in question are extreme. It turned out that each of the examined concepts violates a strong and straightforward version of the intuitive requirement that probabilistic dependence should go with logical dependence. Whereas the corresponding chapter 5 in part 1 of the present work may not have led to new theorems, chapter 11 yields dozens of them in the process of comparison of concepts of dependence and independence which had—as far as I know—never before been treated in a single theoretical framework. With Popper probability semantics, this framework has become available, and here I have simply made full use of it.——In chapter 12, I extend the theory Ppd of probabilistic (in)dependence to the theories Pid and Ppid of inductive and purely inductive dependence, in a way very similar to that in which I have extended the theory Kpd2 to the theories Kid and Kpid in chapters 6 and 7. The first main result of Kpid (roughly: there is no purely inductive support) could be repeated for four of the five purely inductive positive dependence relations considered in chapter 12, whereas the second main result of Kpid (roughly: there is purely inductive countersupport ) could be repeated for each of the five examined purely inductive negative dependence relations. Chapter 12 closes with a brief recapitulation and critical discussion of the main results. (shrink)