We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a (...) new characteristic of relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains. (shrink)

As Friedman has argued, Kant's argument for the ideality of space turns on the nondeductive character of geometrical reasoning in Euclid's system. Since geometry can be axiomatized, this argument fails. But ("pace" Russell) Leibniz's argument based on the unreality of constitutive relations is not thereby answered as well. I argue that what is needed in response to Leibniz is a properly post-Kantian conception of concepts as inferentially articulated. This conception, I suggest, is based on the same fundamental insight that (...) underlies the axiomatization of geometry. (shrink)

Stephan Körner; X*—On The Logic Of Relations, Proceedings of the Aristotelian Society, Volume 77, Issue 1, 1 June 1977, Pages 149–164, https://doi.org/10.1093/a.

The present contribution shows that a Hilbert-style axiomatization for dynamic logic of relation changers is complete for the standard Kripke semantics not by a well-known rewriting technique but by the idea of an auxiliary semantics studied by van Benthem and Wang et al. A key insight of our auxiliary semantics for dynamic logic of relation changers can be described as: “relation changers are bounded morphisms.” Moreover, we demonstrate that this semantic insight can be used to provide a modular (...) cut-free labelled sequent calculus for the logic in the sense that our calculus can be regarded as a natural expansion of a labelled sequent calculus of iteration-free propositional dynamic logic. (shrink)

The middle years of the nineteenth century saw two crucial develop ments in the history of modern logic: George Boole's algebraic treat ment of logic and Augustus De Morgan's formulation of the logic of relations. The former episode has been studied extensively; the latter, hardly at all. This is a pity, for the most central feature of modern logic may well be its ability to handle relational inferences. De Morgan was the first person to work (...) out an extensive logic of relations, and the purpose of this book is to study this attempt in detail. Augustus De Morgan (1806-1871) was a British mathematician and logician who was Professor of Mathematics at the University of London (now, University College) from 1828 to 1866. A prolific but not highly original mathematician, De Morgan devoted much of his energies to the rather different field of logic. In his Formal Logic (1847) and a series of papers "On the Syllogism" (1846-1862), he attempted with great ingenuity to reformulate and extend the tradi tional syllogism and to systematize modes of reasoning that lie outside its boundaries. Chief among these is the logic of relations. De Mor gan's interest in relations culminated in his important memoir, "On the Syllogism: IV and on the Logic of Relations," read in 1860. (shrink)

The work of joachim jungius on the logic of relations was not as original as some authors have thought, But he did make it clear that relational inferences should be distinguished from categorical inferences; and he was the first to recognize the argument 'a rectis ad obliqua', An example of which is 'all circles are figures, Therefore whoever draws a circle draws a figure'.

In this essay, I discuss some observations by Peirce which suggest he had some idea of the substantive metalogical differences between logics which permit both quantifiers and relations, and those which do not. Peirce thus seems to have had arguments?which even De Morgan and Frege lacked?that show the superior expressiveness of relational logics.

SummaryThe following paper is an attempt to deal with relations as elements of certain closed structures which I call relation schemata. A relation schema is essentially a lattice whose alements are m‐ary relations over a domain of individuals. The approach is based upon methods well known from linear algebra and throws light upon some analogies between classic syllogistic and relational logic; it also suggests a natural way of extending classic syllogistic to a system of 42 syllogisms.RésuméL'article suivant (...) est une tentative de traiter les relations comme des éléments de certaines structures fermées que j'appelle schémas relationnels. Un schéma relationnel est essentiellement un treillis dont les éléments sont des relations n‐aires sur un domaine d'individus. L'essai s'inspire de méthodes bien connues en algèbre linéaire et éclaire des analogies entre la syllogistique classique et la logique relationnelle; il suggère aussi une maniére naturelle d'étendre cette syllogistique classique à un système à 42 syllogismes.ZusammenfassungDie folgende Arbeit versucht, Relationen als Elemente gewisser geschlossener Strukturen, die ich Relationenschemata nenne, darzustellen. Ein Relationenschema ist im wesentlichen ein Verband, dessen Elemente m‐stellige Relationen ilber einern Individuenbereich sind. Die hier gegebene Darstellung beruht auf bekannten Methoden der linearen Algebra, weist ferner einige zwischen der klassischen Syllogistik und der Relationenlogik bestehende Analogien auf und legt eine natürliche Erweiterung der klassischen Syllogistik auf 42 Syllogismen nahe. (shrink)

Foundations of Relational Realism presents an intuitive interpretation of quantum mechanics, based on a revised decoherent histories interpretation, structured within a category theoretic topological formalism. -/- If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of intuitive (...) vision, if not intuitive comprehension. For many physicists and philosophers, however, the currently fashionable tendency toward exotic interpretation of the theoretical formalism is recognized not as a mark of ascent for the tower of physics, but rather an indicator of sway—one that must be dampened rather than encouraged if practical progress is to continue. -/- In this unique two-part volume, designed to be comprehensible to both specialists and non-specialists, the authors chart out a pathway forward by identifying the central deficiency in most interpretations of quantum mechanics: That in its conventional, metrical depiction of extension, inherited from the Enlightenment, objects are characterized as fundamental to relations—i.e., such that relations presuppose objects but objects do not presuppose relations. The authors, by contrast, argue that quantum mechanics exemplifies the fact that physical extensiveness is fundamentally topological rather than metrical, with its proper logico-mathematical framework being category theoretic rather than set theoretic. -/- By this thesis, extensiveness fundamentally entails not only relations of objects, but also relations of relations. Thus, the fundamental quanta of quantum physics are properly defined as units of logico-physical relation rather than merely units of physical relata as is the current convention. Objects are always understood as relata, and likewise relations are always understood objectively. In this way, objects and relations are coherently defined as mutually implicative. The conventional notion of a history as “a story about fundamental objects” is thereby reversed, such that the classical “objects” become the story by which we understand physical systems that are fundamentally histories of quantum events. -/- These are just a few of the novel critical claims explored in this volume—claims whose exemplification in quantum mechanics will, the authors argue, serve more broadly as foundational principles for the philosophy of nature as it evolves through the twenty-first century and beyond. (shrink)

In this paper, we present the history of and the research directions in relating logic. For this purpose we will describe Epstein's Programme, which postulates accounting for the content of sentences in logical research. We will focus on analysing the content relationship and Epstein's logics that are based on it, which are special cases of relating logic. Moreover, the set-assignment semantics will be discussed. Next, the Torunian Programme of Relating Semantics will be presented; this programme explores the various (...) non-logical relationships in logical research, including those which are content-related. We will present a general description of relating logic and semantics as well as the most prominent issues regarding the Torunian Programme, including some of its special cases and the results achieved to date. -/- . (shrink)

In this paper I present some of Robert N. McLaughlin's critique of a truth functional approach to conditionals as it appears in his book On the Logic of Ordinary Conditionals. Based on his criticism I argue that the basic principles of logic together amount to epistemological and metaphysical implications that can only be accepted from a logical atomist perspective. Attempts to account for conditional relations within this philosophical framework will necessarily fail. I thus argue that it is (...) not truth functionality as such that is the problem, but the philosophical foundation of modern logic. (shrink)

This is the first critical study of The Logic of Sense, Gilles Deleuze's most important work on language and ethics, as well as the main source of his vital philosophy of the event.James Williams explains the originality of Deleuze's work with careful definitions of all his innovative terms and a detailed description of the complex structure he constructs. This reading makes connections to his ground-breaking work on literature, to his critical but also progressive relation to the sciences, and to (...) his controversial denial of the priority of standard logics, human values and 'meaning' in thinking.This book will open new debates and develop current ones around Deleuze's work in philosophy, politics, literature, linguistics, cultural studies and sociology. (shrink)

The late nineteenth- and early twentieth-century debate over ‘internal’ and ‘external’ relations is well explored, as far as its course in Britain is concerned. F. H. Bradley’s idealistic internalism, on the one hand, and Bertrand Russell’s realistic externalism, on the other, were at the center of this debate. Less well known, however, is that there was also a discussion about relations in the United States at the time. The central figures in this discussion were Edward Gleason Spaulding and (...) John Dewey. Like Russell, Spaulding advocated a realist-inspired externalism, while Dewey criticized this viewpoint from a pragmatist perspective. The aim of the present paper is to reconstruct the exchange between Spaulding and Dewey and to elaborate the specifics of this exchange. In doing so it will emerge, among other things, that, in contrast to Bradley’s idealist rejection of externalism, Dewey’s pragmatist attempt at a refutation was more in line with common sense and science and that Spaulding’s version of externalism differed markedly from Russell’s in its strong empirical orientation. Overall, an undeservedly forgotten chapter in the history of early twentieth-century American philosophy is revisited and reassessed. (shrink)

The paper relates evidence and justification logics, both philosophically and technically. On the philosophical side, it is suggested that the difference between the approaches to evidence in the two families of logics can be explained as a result of their focusing on two different notions of support provided by evidence. On the technical side, a justification logic with operators pertaining to both kinds of support is shown to be sound and complete with respect to a special class of awareness (...) models. In addition, a realization theorem with respect to K is shown to hold for the logic. (shrink)

One of the most significant discoveries of early twentieth century mathematical logic was a workable definition of ‘ordered pair’ totally within set theory. Norbert Wiener, and independently Casimir Kuratowski, are usually credited with this discovery. A definition of ‘ordered pair’ held the key to the precise formulation of the notions of ‘relation’ and ‘function’ — both of which are probably indispensable for an understanding of the foundations of mathematics. The set-theoretic definition of ‘ordered pair’ thus turned out to be (...) a key victory for logicism, providing one admits set theory is logic. The definition also was instrumental in achieving the appearance of ontological economy — since it seemed only sets were needed — although this feature was emphasized only later. (shrink)

In The Logic of Life, François Jacob reconstructed the history of heredity from the seventeenth century to the present, emphasizing the role of physics in the development of biology. Quantum mechanics provided questions, methods, and techniques to molecular biologists. In the 1960s, physics also provided the organizational model. Jacob worked on the creation of the European Molecular Biology Laboratory, on the model of CERN (European Organization for Nuclear Research). I argue that reflection on the relation between molecular biology and (...) physics played a part in the making of The Logic of Life. Between the mid-1960s and the early 1970s, Jacob focused on relations between the “genealogy” of his molecular biology work and the development of physics at scientific, institutional, and historical levels. He thus brought molecular biology to the attention of a larger public, following the history of the Institut Pasteur, whose public mission had to be reaffirmed to grant its survival. With The Logic of Life, Jacob became an influential intellectual who engaged with scholars in human sciences. He promoted his view of the development of molecular biology as a model for a possible new discipline: bioanthropology. (shrink)

Here, we discuss historical, philosophical and technical problems associated with relating logic and relating semantics. To do so, we proceed in three steps. First, Section 1 is devoted to providing an introduction to both relating logic and relating semantics. Second, we address the history of relating semantics and some of the main research directions and their philosophical applications. Third, we discuss some technical problems related to relating semantics, particularly whether the direct incorporation of the relation into the language (...) of relating logic is needed. The starting point for our considerations presented here is the 1st Workshop On Relating Logic and the selected papers for this issue. (shrink)

In this paper, first, we determine the number of sublogics of variable inclusion of an arbitrary finitary logic ⊢ with a composition term. Then, we investigate their position into the lattice of co...

The aim of the paper is twofold. First, we want to recapture the genesis of the logics of order. The origin of this notion is traced back to the work of Jerzy Kotas, Roman Suszko, Richard Routley and Robert K. Meyer. A further development of the theory of logics of order is presented in the papers of Jacek K. Kabziński. Quite contemporarily, this notion gained in significance in the papers of Carles Noguera and Petr Cintula. Logics of order are named (...) there _logics of weak implications_. They play a crucial role in their monograph (Noguera and Cintula _Logic and Implication. An Introduction to the General Algebraic Study of Non-Classical Logics_, Trends in Logic 57, Springer, Berlin, 2021). But, more importantly, the other goal is to define some subclasses of the logics of order in reference to later results of Jacek K. Kabziński and Michael Dunn. The original conception of implication is due to Kabziński. Implication is a stronger notion than the notion of the connective of order aka weak implication. As a result, the three subclasses of logics of order are isolated: logics of implication, logics of symmetry, and tonoidal logics. These notions are uniformly defined and investigated from various viewpoints in terms of consequence operations. The emphasis is put on their semantics. (shrink)

Examines the link between nonmonotonic inference relations and theory revision operations, focusing on the correspondence between abstract properties which each may satisfy.

This paper investigates the reason why, in the tradition of Western philosophy, a logic of relations was developed only in the second half of the nineteenth century. To this end, it moves along two different but interconnected paths: on the one hand, it attempts to reconstruct the main views concerning the ontology of relations during the middle ages; on the other, it focuses on the treatment of so-called oblique terms in the logical works of some preeminent authors (...) belonging to the scholastic and late-scholastic tradition. From the ontological point of view, realists and nominalist both denied that polyadic expressions of the language signify polyadic properties in the world extra. Some authors, such as Peter Auriol, claimed that polyadic expressions signify something merely mental, thus recognizing, even though in a limited ontological domain, the existence of full-fledged relations, that is, of ‘things’ simultaneousl... (shrink)

We consider the modality “ $\varphi $ is true in every $\sigma $ -centered forcing extension,” denoted $\square \varphi $, and its dual “ $\varphi $ is true in some $\sigma $ -centered forcing extension,” denoted $\lozenge \varphi $, which give rise to the notion of a principle of $\sigma $ -centered forcing. We prove that if ZFC is consistent, then the modal logic of $\sigma $ -centered forcing, i.e., the ZFC-provable principles of $\sigma $ -centered forcing, is exactly (...) $\mathsf {S4.2}$. We also generalize this result to other related classes of forcing. (shrink)

This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it (...) to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. (shrink)

This paper explores relational syllogistic logics, a family of logical systems related to reasoning about relations in extensions of the classical syllogistic. These are all decidable logical systems. We prove completeness theorems and complexity results for a natural subfamily of relational syllogistic logics, parametrized by constructors for terms and for sentences.

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency. Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What (...) George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency. (shrink)

Description: Ancient Indian logic by itself is a very vast subject. The ancient Sanskrit term nyaya which was first used in a different or in a much more general sense, was later specifically applied to the Nyaya school. The physics and physiology and psychology of the Nyaya doctrine are not specifically its own, being from the first indistinguishable from those of its sister Sastra, the Vaisesika. What characterizes it specifically is the development of the nyaya or five-membered method of (...) demonstration and its insistence on the four sources of knowledge, corresponding to the first four members of its demonstrative formula, or 'Syllogism.' The importance of Nyaya is in its doctrine of the pramanas, or sources of proof, and in its formulation of the nyaya, or method of demonstration. Indian Logic in its Early Schools by H.N. Randle is an interpretation of Indian logical theory in its historical development based on the study of original texts wherever reference to Nyaya as a method of logic has been made, and thus, covers a very wide field of Indian philosophy. An indispensable work for students of Indian philosophy, it is especially important for those who are interested in Nyayadarsana in its relation to the early logic of other schools. (shrink)

This paper is concerned with the semantics for the logics of ground that derive from a slight variant GG of the logic of (Fine, 2012) that have already been developed in (deRosset and Fine, 2023). Our aim is to outline that semantics and to provide a comparison with two related semantics for ground, given in (Correia, 2017) and (Kraemer, 2018). This comparison highlights the strengths and difficulties of these different approaches.

In this chapter, we begin by sketching in the broadest possible strokes the ideas behind two formal systems that have been introduced with to goal of explicating the ways in which reasons interact to support the actions and conclusions they do. The first of these is the theory of defeasible reasoning developed in the seminal work of Pollock; the second is a more recent theory due to Horty, which adapts and develops the default logic introduced by Reiter to provide (...) an account of reasons. However, the implementations are complex enough, in both cases, to prevent anything more than this sketch. And we would not want to give the impression that we think that work on the logic of reasons must follow the path mapped out in either of these theories—indeed, we feel that the field is wide open. In the remainder of the chapter, therefore, will concentrate on a number of issues bearing on the logic of reasons that are either not treated in the work of Pollock and Horty, or whose treatment there is, we feel, either inadequate or incomplete. These are: first, the question of whether it is necessary to understand logical interactions among reasons themselves, rather than simply between reasons and the actions or conclusions they support, and if so, what principles might govern these interactions; second, priority relations among reasons and the notion of reason accrual; and third, some problems posed by undercutting defeat. (shrink)

The logic of gift and gratuitousness in business activity raised by the encyclical Caritas in Veritate stresses a deeper critical evaluation of the category of relation. The logic of gift in business includes two aspects. The first is considering the logic of gift as a new conceptual lens in order to view business relationship beyond contractual logic. In this view, it is crucial to see the circulation of goods as instrumental for the development of relationships. The (...) second aspect is to qualify the relationships established through the gift, and to think about the motivation in gift-giving, which has an ethical content. We give because we have received, and through gift-giving we develop relationships that have a high ‘bonding value’. Analysing the logic of gift in business management may permit us to gain an understanding of the ambiguity of gift-giving in organizations. Looking at the relationships between organizations and employees, and organizations and customers, we can discover why the logic of gift is often misunderstood or abused in its application, and how it should be applied to be more consistent with the message of Caritas in Veritate. (shrink)

It is well known that the non-orthogonality relation between the states of a quantum system is reflexive and symmetric, and the modal logic \ is sound and complete with respect to the class of sets each equipped with a reflexive and symmetric binary relation. In this paper, we consider two properties of the non-orthogonality relation: Separation and Superposition. We find sound and complete modal axiomatizations for the classes of sets each equipped with a reflexive and symmetric relation that satisfies (...) each one of these two properties and both, respectively. We also show that the modal logics involved are decidable. (shrink)