The hexagon of opposition is an improvement of the square of opposition due to Robert Blanché. After a short presentation of the square and its various interpretations, we discuss two important problems related with the square: the problem of the I-corner and the problem of the O-corner. The meaning of the notion described by the I-corner does not correspond to the name used for it. In the case of the O-corner, the problem is not a wrong-name problem but a no-name (...) problem and it is not clear what is the intuitive notion corresponding to it. We explain then that the triangle of contrariety proposed by different people such as Vasiliev and Jespersen solves these problems, but that we don’t need to reject the square. It can be reconstructed from this triangle of contrariety, by considering a dual triangle of subcontrariety. This is the main idea of Blanché’s hexagon. We then give different examples of hexagons to show how this framework can be useful to conceptual analysis in many different fields such as economy, music, semiotics, identity theory, philosophy, metalogic and the metatheory of the hexagon itself. We finish by discussing the abstract structure of the hexagon and by showing how we can swing from sense to non-sense thinking with the hexagon. (shrink)
This paper introduces the special issue on Logic and Religion of the journal Logica Universalis (Springer). The issue contains the following articles: Logic and Religion, by Jean-Yves Beziau and Ricardo Silvestre; Thinking Negation in Early Hinduism and Classical Indian Philosophy, by Purushottama Bilimoria; Karma Theory, Determinism, Fatalism and Freedom of Will, by Ricardo Sousa Silvestre; From Logic in Islam to Islamic Logic, by Musa Akrami; Leibniz’s Ontological Proof of the Existence of God and the Problem of Impossible Objects, by Wolfgang (...) Lenzen; A Logical Analysis of the Anselm’s Unum Argumentum (from Proslogion), by Jean-Pierre Desclés; Monotonic and Non-monotonic Embeddings of Anselm’s Proof, by Jacob Archambault; Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy, by C. Benzmüller, L. Weber and B. Woltzenlogel Paleo. (shrink)
According to Boole it is possible to deduce the principle of contradiction from what he calls the fundamental law of thought and expresses as \. We examine in which framework this makes sense and up to which point it depends on notation. This leads us to make various comments on the history and philosophy of modern logic.
We discuss the origin and development of the universal logic project. We describe in particular the structure of UNILOG, a series of events created for promoting the universal logic project, with a school, a congress, a secret speaker and a contest. We explain how the contest has evolved into a session of logic prizes.
This is a collection of new investigations and discoveries on the theory of opposition (square, hexagon, octagon, polyhedra of opposition) by the best specialists from all over the world. The papers range from historical considerations to new mathematical developments of the theory of opposition including applications to theology, theory of argumentation and metalogic.
After recalling the distinction between logic as reasoning and logic as theory of reasoning, we first examine the question of relativity of logic arguing that the theory of reasoning as any other science is relative. In a second part we discuss the emergence of universal logic as a general theory of logical systems, making comparison with universal algebra and the project of mathesis universalis. In a third part we critically present three lines of research connected to universal logic: logical pluralism, (...) non-classical logics and cognitive science. (shrink)
We present a paraconsistent logic, called Z, based on an intuitive possible worlds semantics, in which the replacement theorem holds. We show how to axiomatize this logic and prove the completeness theorem.
The present book discusses all aspects of paraconsistent logic, including the latest findings, and its various systems. It includes papers by leading international researchers, which address the subject in many different ways: development of abstract paraconsistent systems and new theorems about them; studies of the connections between these systems and other non-classical logics, such as non-monotonic, many-valued, relevant, paracomplete and fuzzy logics; philosophical interpretations of these constructions; and applications to other sciences, in particular quantum physics and mathematics. Reasoning with contradictions (...) is the challenge of paraconsistent logic. The book will be of interest to graduate students and researchers working in mathematical logic, computer science, philosophical logic, linguistics and physics. (shrink)
Contradiction is often confused with contrariety. We propose to disentangle contrariety from contradiction using the hexagon of opposition, providing a clear and distinct characterization of three notions: contrariety, contradiction, incompatibility. At the same time, this hexagonal structure describes and explains the relations between them.
This paper introduces the special issue on Formal Approaches to the Ontological Argument of the Journal of Applied Logics (College Publications). The issue contains the following articles: Formal Approaches to the Ontological Argument, by Ricardo Sousa Silvestre and Jean-Yves Béziau; A Brief Critical Introduction to the Ontological Argument and its Formalization: Anselm, Gaunilo, Descartes, Leibniz and Kant, by Ricardo Sousa Silvestre; A Mechanically Assisted Examination of Begging the Question in Anselm’s Ontological Argument, by John Rushby; A Tractarian Resolution to the (...) Ontological Argument, by Erik Thomsen; On Kant’s Hidden Ambivalence Toward Existential Generalization in his Critique of the Ontological Argument, by Giovanni Mion; The Totality of Predicates and the Possibility of the Most Real Being, by Srećko Kovač; An Even More Leibnizian Version of Gödel’s Ontological Argument, by Kordula Świętorzecka and Marcin Łyczak; A Case Study On Computational Hermeneutics: E. J. Lowe’s Modal Ontological Argument, by David Fuenmayor. (shrink)
We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of the four basic logical notions. This is what we are doing here, as well as introducing a challenging fifth logical notion. We first recall (...) the context and origin of what are here called Tarski-Lindenbaum logical notions. In the second part, we present these notions in the simple case of a binary relation. In the third part, we examine in which sense these are considered as logical notions contrasting them with an example of a nonlogical relation. In the fourth part, we discuss the formulations of the four logical notions in natural language and in first-order logic without equality, emphasizing the fact that two of the four logical notions cannot be expressed in this formal language. In the fifth part, we discuss the relations between these notions using the theory of the square of opposition. In the sixth part, we introduce the notion of variety corresponding to all non-logical notions and we argue that it can be considered as a logical notion because it is invariant, always referring to the same class of structures. In the seventh part, we present an enigma: is variety formalizable in first-order logic without equality? There follow recollections concerning Jan Woleński. This paper is dedicated to his 80th birthday. We end with the bibliography, giving some precise references for those wanting to know more about the topic. (shrink)
We present the logic K/2 which is a logic with classical implication and only the left part of classical negation.We show that it is possible to define a classical negation into K/2 and that the classical proposition logic K can be translated into this apparently weaker logic.We use concepts from model-theory in order to characterized rigorously this translation and to understand this paradox. Finally we point out that K/2 appears, following Haack's distinction, both as a deviation and an extension of (...) K. (shrink)
In this paper we discuss the notion of “possible worlds” contrasting a philosophical idea due to Malebranche with a mathematical concept of modern logic due to Kripke from which many pseudo-philosophical discussions have arisen.
I discuss the origin and development of logic prizes around the world. In a first section I describe how I started this project by creating the Newton da Costa Logic Prize in Brazil in 2014. In a second section I explain how this idea was extended into the world through the manifesto A Logic Prize in Every Country! and how was organized the Logic Prizes Contest at the 6th UNILOG in Vichy in June 2018 with the participation of 9 logic (...) prizes winners from 9 countries. In a third section I discuss how this project will develop in the future with the creation of more logic prizes, an Encyclopædia of Logic, the book series Logic PhDs, as well as the creation of a World Logic Day, January 14, day of birth of Alfred Tarski and of death of Kurt Gödel. (shrink)
Dans un texte désormais célèbre, Ferdinand de Saussure insiste sur l’arbitraire du signe dont il vante les qualités. Toutefois il s’avère que le symbole, signe non arbitraire, dans la mesure où il existe un rapport entre ce qui représente et ce qui est représenté, joue un rôle fondamental dans la plupart des activités humaines, qu’elles soient scientifiques, artistiques ou religieuses. C’est cette dimension symbolique, sa portée, son fonctionnement et sa signification dans des domaines aussi variés que la chimie, la théologie, (...) les mathématiques, le code de la route et bien d’autres qui est l’objet du livre La Pointure du symbole. -/- Jean-Yves Béziau, franco-suisse, est docteur en logique mathématique et docteur en philosophie. Il a poursuivi des recherches en France, au Brésil, en Suisse, aux États-Unis (UCLA et Stanford), en Pologne et développé la logique universelle. Éditeur-en-chef de la revue Logica Universalis et de la collection Studies in Universal Logic (Springer), il est actuellement professeur à l’Université Fédérale de Rio de Janeiro et membre de l’Académie brésilienne de Philosophie. SOMMAIRE -/- PRÉFACE L’arbitraire du signe face à la puissance du symbole Jean-Yves BÉZIAU La logique et la théorie de la notation (sémiotique) de Peirce (Traduit de l’anglais par Jean-Marie Chevalier) Irving H. ANELLIS Langage symbolique de Genèse 2-3 Lytta BASSET -/- Mécanique quantique : quelle réalité derrière les symboles ? Hans BECK -/- Quels langages et images pour représenter le corps humain ? Sarah CARVALLO Des jeux symboliques aux rituels collectifs. Quelques apports de la psychologie du développement à l’étude du symbolisme Fabrice CLÉMENT Les panneaux de signalisation (Traduit de l’anglais par Fabien Shang) Robert DEWAR Remarques sur l’émergence des activités symboliques Jean LASSÈGUE Les illustrations du "Songe de Poliphile" (1499). Notule sur les hiéroglyphes de Francesca Colonna Pierre-Alain MARIAUX Signes de vie Jeremy NARBY Visualising relations in society and economics. Otto Neuraths Isotype-method against the background of his economic thought Elisabeth NEMETH Algèbre et logique symboliques : arbitraire du signe et langage formel Marie-José DURAND – Amirouche MOKTEFI Les symboles mathématiques, signes du Ciel Jean-Claude PONT La mathématique : un langage mathématique ? Alain M. ROBERT. (shrink)
After explaining the interdisciplinary aspect of the series of events organized around the square of opposition since 2007, we discuss papers related to the 4th World Congress on the Square of Opposition which was organized in the Vatican at the Pontifical Lateran University in 2014. We distinguish three categories of work: those dealing with the evolution and development of the theory of opposition, those using the square as a metalogical tool to give a better understanding of various systems of logic (...) and those related with applications of the theory of opposition to conceptual analysis and pedagogy. (shrink)
We assess the celebration of the 1st World Logic Day which recently took place all over the world. We then answer the question Why a World Logic Day? in two steps. First we explain why promoting logic, emphasizing its fundamental importance and its relations with many other fields. Secondly we examine the sense of a one-day celebration: how this can help reinforcing logic day-to-day and why logic deserves it. We make a comparison with other existing one-day celebrations. We end by (...) presenting and commenting the logo of the World Logic Day. (shrink)
Universal Logic is not a new logic, but a general theory of logics, considered as mathematical structures. The name was introduced about ten years ago, but the subject is as old as the beginning of modern logic: Alfred Tarski and other Polish logicians such as Adolf Lindenbaum developed a general theory of logics at the end of the 1920s based on consequence operations and logical matrices. The subject was revived after the flowering of thousands of new logics during the last (...) thirty years: there was a need for a systematic theory of logics to put some order in this chaotic multiplicity. This book contains recent works on universal logic by first-class researchers from all around the world. The book is full of new and challenging ideas that will guide the future of this exciting subject. It will be of interest for people who want to better understand what logic is. Tools and concepts are provided here for those who want to study classes of already existing logics or want to design and build new ones. (shrink)
The square of opposition is a diagram related to a theory of oppositions that goes back to Aristotle. Both the diagram and the theory have been discussed throughout the history of logic. Initially, the diagram was employed to present the Aristotelian theory of quantification, but extensions and criticisms of this theory have resulted in various other diagrams. The strength of the theory is that it is at the same time fairly simple and quite rich. The theory of oppositions has recently (...) become a topic of intense interest due to the development of a general geometry of opposition (polygons and polyhedra) with many applications. A congress on the square with an interdisciplinary character has been organized on a regular basis (Montreux 2007, Corsica 2010, Beirut 2012, Vatican 2014, Rapa Nui 2016). The volume at hand is a sequel to two successful books: The Square of Opposition - A General Framework of Cognition, ed. by J.-Y. Béziau & G. Payette, as well as Around and beyond the Square of Opposition, ed. by J.-Y. Béziau & D. Jacquette, and, like those, a collection of selected peer-reviewed papers. The idea of this new volume is to maintain a good equilibrium between history, technical developments and applications. The volume is likely to attract a wide spectrum of readers, mathematicians, philosophers, linguists, psychologists and computer scientists, who may range from undergraduate students to advanced researchers. (shrink)
This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The notion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacement theorem and algebraization of logic are studied (including MacLane-Curry’s theorem and a discussion about Curry’s algebras). We also examine how these concepts are related to such notions as (...) semantics, truth-functionality and bivalence. We argue that a logic, which is simple, can deserve the name logic and that the opposite view is connected with a reductionist perspective (reduction of logic to algebra). (shrink)
Many-valued logics are standardly defined by logical matrices. They are truth-functional. In this paper non truth-functional many-valued semantics are presented, in a philosophical and mathematical perspective.
Papers... "selected from a larger number of contributions most of them based on talks presented at the First World Congress on the Square of Opposition organized in Montreux in June 2007"--Preface, p. 12.
Modern science has qualified human beings as homo sapiens. Is there a serious scientific theory backing this nomenclature? And can we proclaim ourselves as wise? The classical rational animals characterization has apparently the same syntactic form but it is not working exactly in the same way. Moreover the semantics behind is more appropriate, encompassing a pivotal ambiguity. In the second part of the paper, we further delve into this ambiguity, relating rationality with three fundamental features of these creatures: ability to (...) laugh, sexuality and transformation. (shrink)
For several years I have been developing a general theory of logics that I have called Universal Logic. In this article I will try to describe how I was led to this theory and how I have progressively conceived it, starting my researches about ten years ago in Paris in paraconsistent logic and the broadening my horizons, pursuing my researches in Brazil, Poland and the USA.
After mentioning the cogent connection between pure semantics and the particular set theoretical framework in which it is formulated, some issues regarding the conceptual status of semantics itself, as well as its relationship to logic, are concisely raised.
Neste artigo, discutimos em que sentido a verdade é considerada como um objeto matemático na lógica proposicional. Depois de esclarecer como este conceito é usado na lógica clássica, através das noções de tabela de verdade, de função de verdade, de bivaloração, examinamos algumas generalizações desse conceito nas lógicas não clássicas: semânticas matriciais multi-valoradas com três ou quatro valores, semântica bivalente não veritativa, semânticas dos mundos possiveis de Kripke. DOI:10.5007/1808-1711.2010v14n1p31.
This paper introduces the special issue on Logic and Philosophy of Religion of the journal Sophia: International Journal of Philosophy and Traditions (Springer). The issue contains the following articles: Logic and Philosophy of Religion, by Ricardo Sousa Silvestre and Jean-Yvez Béziau; The End of Eternity, by Jamie Carlin Watson; The Vagueness of the Muse—The Logic of Peirce’s Humble Argument for the Reality of God, by Cassiano Terra Rodrigues; Misunderstanding the Talk(s) of the Divine: Theodicy in the Wittgensteinian Tradition, by Ondřej (...) Beran; On the Concept of Theodicy, by Ricardo Sousa Silvestre; The Logical Problem of the Trinity and the Strong Theory of Relative Identity, by Daniel Molto; Thomas Aquinas on Logic, Being, and Power, and Contemporary Problems for Divine Omnipotence, by Errin D. Clark. -/- . (shrink)
In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, distributivity does not arise if we use the usual notion of extension between consequence relations. (...) A detailed discussion about this phenomenon, as well as some elucidation for it, is given. (shrink)
We answer Slater's argument according to which paraconsistent logic is a result of a verbal confusion between «contradictories» and «subcontraries». We show that if such notions are understood within classical logic, the argument is invalid, due to the fact that most paraconsistent logics cannot be translated into classical logic. However we prove that if such notions are understood from the point of view of a particular logic, a contradictory forming function in this logic is necessarily a classical negation. In view (...) of this result, Slater's argument sounds rather tautological. (shrink)