Il s’agit de comprendre dans cet article l’opposition formulée par Gilles-Gaston Granger entre deux types de négation : la négation « radicale », d’un côté, et les négations « appliquées » de l’autre. Nous examinerons les propriétés de cette opposition, ainsi que les enseignements à en tirer sur la philosophie de la logique de Granger. Puis nous proposerons une théorie constructive des valeurs logiques considérées comme des objets structurés, consolidant à la fois l’unité de la théorie logique de Granger et (...) son pluralisme philosophique. (shrink)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

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)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then introduce (...) two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

We discuss the history of the revival of the theory of opposition, with its emerging paradigms of research, and the related events that are organized in this perspective, including the latest one in Leuven in 2022.

Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for assigning bitstrings (...) to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before. (shrink)

Some Carrollian posthumous manuscripts reveal, in addition to his famous ‘logical diagrams’, two mysterious ‘logical charts’. The first chart, a strange network making out of fourteen logical sentences a large 2D ‘triangle’ containing three smaller ones, has been shown equivalent—modulo the rediscovery of a fourth smaller triangle implicit in Carroll's global picture—to a 3D tetrahedron, the four triangular faces of which are the 3+1 Carrollian complex triangles. As it happens, such an until now very mysterious 3D logical shape—slightly deformed—has been (...) rediscovered, independently from Carroll and much later, by a logician , a mathematician and a linguist studying the geometry of the ‘opposition relations’, that is, the mathematical generalisations of the ‘logical square’. We show that inside what is called equivalently ‘n-opposition theory’, ‘oppositional geometry’ or ‘logical geometry’, Carroll's first chart corresponds exactly, duly reshap.. (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)

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)

Wittgenstein took the _Tractatus Logico-Philosophicus_ to be eventually invalidated by logical atomism. Our main thesis is that it can be revalidated, provided that we subtract the thesis 2.02 (“The object is simple.”) from it: atoms are not simple objects but, rather, bits of information the objects are made of. Starting from an introductory discussion about what is meant by a ‘logic of colors’, an explanatory framework is then proposed in the form of a partition semantics. The philosophical problem of Wittgenstein’s (...) color-statements is reformulated accordingly, thus overcoming the trouble with logical atomism. A Boolean treatment is advanced to solve the initial difficulty and leads to two final results: Wittgenstein’s logic of colors requires a mereological understanding of the relations between colors in order to make sense; the analysis of psychophysical concepts like colors casts some additional doubt about the analytic–synthetic distinction, to the point of treating relations between colors as analytic a posteriori judgments. (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.

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)