The square of opposition is a diagram related to a theory of opposi-tions 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 quantifi-cation, 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 Oppo-sition, 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.

Jean-Yves Beziau is Professor of Logic at the department of philosophy of the Federal University of Rio de Janeiro and titular member of the Brazilian Academy of Philosophy. He has a PhD in Philosophy and a PhD in Mathematical logic. He has developed the field of universal logic, organizing world events on this topic and he has renewed the studies on the square of opposition organizing also world events on 13 this topic.

Table of Contents Jean-Yves Béziau / Stamatios Gerogiorgakis The Many Dimensions of the Square of Opposition Introduction 9 1 HISTORICAL AND CRITICAL ASPECTS OF THE SQUARE Paul J.E. Dekker Heraclitean Oppositions 19 Juliette Lemaire Is Aristotle the Father of the Square of Opposition? 33 Stamatios Gerogiorgakis A Vindication of a secundum-quid-et-simpliciter Solution of the Paradox of Epimenides by Way of Mereological Hexagons 71 Enrique Alvarez and Manuel Correia Conversion and Opposition: Traditional and Theoretical Formulations 87 Régis Angot-Pellissier Taoist Logical Hexagon. The Philosophical Meaning of 3-opposition and Weak 2-opposition in Cosmology 107 2 THE SQUARE OF OPPOSITION AND NON-CLASSICAL LOGICS Ka-fat Chow Opposition Inferences and Generalized Quantifiers 155 Claudio Pizzi Contingency Logics and Modal Squares of Opposition 201 Christan de Ronde, Hector Freytes and Graciela Domenech Quantum Mechanics and the Interpretation of the Orthomodular Square of Opposition 221 Luca Tranchini and Michael Arndt A Constructive View of the Square of Oppositions 241 Fatemah Ayatollah Zadeh Shirazi 2n-polygon of Opposition and 2n-tuples Satisfying It 275 3 APPLICATIONS OF THE SQUARE OF OPPOSITION François Nicolas The Hexagon of Opposition in Music 299 Jean-Yves Béziau Opposition and Order 321 Jonas R. Becker Arenhart and Décio Krause Oppositions and Quantum Mechanics: Superposition and Identity 337 Katarzyna Gan-Krzywoszyńska and Piotr Leśniewski An Erotetic Hexagon: Oppositions as a Basis for the Logic of Questions 355 Fabio Tfouni Interdiction and Silence: A Traditional Reading of the Square of Opposition 377 Index Abstracts / Authors’ Short Biographies