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The Square of Opposition

A General Framework for Cognition

Jean-Yves Beziau and Gillman Payette

The square of opposition is a simple geometrical figure expressing some fundamental ideas about cognition. It is based on Aristotle’s philosophy and has been fascinating people for two thousand years. The three notions of opposition presented in the square can be applied to analyze and understand such diverse subjects as reasoning about mathematical objects, perceptions of reality, speech acts, moral reasoning and reasoning about possibility.
This book presents recent research papers dealing with the history and philosophy of the square, new diagrammatic and mathematical developments arising from it, and its applications to the fields of linguistics, psychology and argumentation.
It also includes a DVD composed of events from the first world congress on the square of opposition held in June 2007 in Montreux, featuring some of the speeches and presentations of the participants, like the professors Pascal Engel, Laurence Horn, Terence Parsons, Jan Wolenski. Further, the DVD contains extracts of a square jazz show which was composed and presented for this occasion and extracts of the movie «The Square of Salomé». The movie, which was produced for this event, is a remake of the famous biblical story using the square to display the relations between the main characters.
Contents: Jean-Yves Beziau/Gillman Payette: Preface – Volker Peckhaus: Algebra of Logic, Quantification Theory, and the Square of Opposition – Yurii Khomskii: William of Sherwood, Singular Propositions and the Hexagon of Opposition – John N. Martin: Existential Commitment and the Cartesian Semantics of the Port-Royal Logic – Juan M. Campos-Benítez: The Medieval Modal Octagon and the S5 Lewis Modal System – Alessio Moretti: From the «Logical Square» to the «Logical Poly-Simplexes» – Jonathan Westphal: The Application of Vector Theory to Syllogistic Logic – Sumanta Sarathi Sharma: Interpreting Squares of Oppositions with the Help of Diagrams – Dag Westerståhl: Classical vs. Modern Squares of Opposition, and Beyond – Pieter A. M. Seuren: From Logical Intuitions to Natural Logic – Fabien Schang: Questions and Answers about Oppositions – Peter Schroeder-Heister: Definitional Reasoning in Proof-Theoretic Semantics and the Square of Opposition – Régis Angot-Pellissier: 2-Opposition and the Topological Hexagon – Corina Strössner/Niko Strobach: Veridications and their Square of Opposition – Laurence R. Horn: Histoire d’*O: Lexical Pragmatics and the Geometry of Opposition – Sémir Badir: How the Semiotic Square Came – Ivana Bianchi/Ugo Savardi: The Cognitive Dimensions of Contrariety – Sieghard Beller: Human Deontic Reasoning in the Deontic Square – Antoine Seilles/Jean Sallantin: n-Opposition Theory to Structure Debates.