Jean-Yves Béziau

The Power of the Hexagon

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.

Aspects of Paraconsistent Logic

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Paraconsistent logic from a modal viewpoint

Abstract In this paper we study paraconsistent negation as a modal operator, considering the fact that the classical negation of necessity has a paraconsistent behavior. We examine this operator on the one hand in the modal logic S5 and on the other hand in some new four-valued modal logics.

Around and beyond the square of opposition

1 Historical and Critical Aspects of the Square.- 2 Philosophical Discussions around the Square of Opposition.- 3 The Square of Opposition and Non-Classical Logics.- 4 Constructions Generalizing the Square of Opposition.- 5 Applications of the Square of Opposition.

Classical Negation can be Expressed by One of its Halves

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 dene a classical negation into K=2 and that the classical propositional 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. 1

Semantic computations of truth based on associations already learned

Abstract This article sets forth a detailed theoretical proposal of how the truth of ordinary empirical statements, often atomic in form, is computed. The method of computation draws on psychological concepts such as those of associative networks and spreading activation, rather that the concepts of philosophical or logical theories of truth. Axioms for a restricted class of cases are given, as well as some detailed examples.

What is many-valued logic?

Firstly we examine the definition of many-valued logic within the framework of (logical) matrix theory. Secondly we discuss the general result, challenging the existence of many-valued logic, according to which every logic may be seen as two-valued. Thirdly we analyze the principle of bivalence and show that it appears at a deeper level than one usually thinks.

The paraconsistent logic Z. A possible solution to Jaśkowski’s problem

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.

Disentangling Contradiction from Contrariety via Incompatibility

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.

The New Rising of the Square of Opposition

In this paper I relate the story about the new rising of the square of opposition: how I got in touch with it and started to develop new ideas and to organize world congresses on the topic with subsequent publications. My first contact with the square was in connection with Slater’s criticisms of paraconsistent logic. Then by looking for an intuitive basis for paraconsistent negation, I was led to reconstruct S5 as a paraconsistent logic considering ¬□ as a paraconsistent negation. Making the connection between ¬□ and the O-corner of the square of opposition, I developed a paraconsistent star and hexagon of opposition and then a polyhedron of opposition, as a general framework to understand relations between modalities en negations. I also proposed the generalization of the theory of oppositions to polytomy. After having developed all this work I have begun to promote interdisciplinary world events on the square of opposition.