Around and Beyond the Square of Opposition Jean-Yves Béziau and Dale Jacquette

STUDIES IN UNIVERSAL LOGIC

2012, DOI: 10.1007/978-3-0348-0379-3

Series Editor

Jean-Yves Béziau (Federal University of Rio de Janeiro and Brazilian Research Council,

Rio de Janeiro, Brazil)

Editorial Board Members

Hajnal Andréka (Hungarian Academy of Sciences, Budapest, Hungary)

Mark Burgin (University of California, Los Angeles, USA)

Razvan Diaconescu ˘ (Romanian Academy, Bucharest, Romania)

Josep Maria Font (University of Barcelona, Barcelona, Spain)

Andreas Herzig (Centre National de la Recherche Scientifique, Toulouse, France)

Arnold Koslow (City University of New York, New York, USA)

Jui-Lin Lee (National Formosa University, Huwei Township, Taiwan)

Larissa Maksimova (Russian Academy of Sciences, Novosibirsk, Russia)

Grzegorz Malinowski (University of Łód´z, Łód´z, Poland)

Darko Sarenac (Colorado State University, Fort Collins, USA)

Peter Schröder-Heister (University Tübingen, Tübingen, Germany)

Vladimir Vasyukov (Russian Academy of Sciences, Moscow, Russia)

Preface

The theory of inferences and oppositions among categorical propositions, based on Aristotelian term logic, is pictured in a striking square diagram. The graphic representation

of contradictories, contraries, subcontraries and subalterns intended as a foundation for

syllogistic logic can be understood and applied in many different ways with interesting

implications for various disciplines, notably including epistemology, linguistics, mathematics, psychology. The square can also be generalized in other two-dimensional and

multi-dimensional graphic depictions of logical and other relations, extending in breath

and depth the original Aristotelian theory. The square of opposition is accordingly a very

attractive theme which has persisted down through the centuries with no signs of disappearing or even diminishing in fascination. For the last 10 years, there has been a new

growing interest for the square due to new discoveries and challenging interpretations.

This book presents a collection of previously unpublished papers by well-regarded specialists on the theory and interpretation of the concept and application of the square of

opposition from all over the world. We thank all the authors who have contributed a paper

to this book, and the referees who have analyzed, commented on, and made invaluable

recommendations for improving the essays.

Jean-Yves Béziau

Dale Jacquette

Contents

Part I Historical and Critical Aspects of the Square

The New Rising of the Square of Opposition . . . . . . . . . . . . . . . . . . .  3

Jean-Yves Béziau

Logical Oppositions in Arabic Logic: Avicenna and Averroes . . . . . . . . . .  21

Saloua Chatti

Boethius on the Square of Opposition . . . . . . . . . . . . . . . . . . . . . . .  41

Manuel Correia

Leibniz, Modal Logic and Possible World Semantics: The Apulean Square as

a Procrustean Bed for His Modal Metaphysics . . . . . . . . . . . . . . . .  53

Jean-Pascal Alcantara

Thinking Outside the Square of Opposition Box . . . . . . . . . . . . . . . . .  73

Dale Jacquette

John Buridan’s Theory of Consequence and His Octagons of Opposition . . .  93

Stephen Read

Why the Fregean “Square of Opposition” Matters for Epistemology . . . . . .  111

Raffaela Giovagnoli

Part II Philosophical Discussion Around the Square of Opposition

Two Concepts of Opposition, Multiple Squares . . . . . . . . . . . . . . . . . .  119

John T. Kearns

Does a Leaking O-Corner Save the Square? . . . . . . . . . . . . . . . . . . . .  129

Pieter A.M. Seuren

The Right Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  139

Hartley Slater

Oppositions and Opposites . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  147

Fabien Schang

viiviii Contents

Pluralism in Logic: The Square of Opposition, Leibniz’ Principle of Sufficient

Reason and Markov’s Principle . . . . . . . . . . . . . . . . . . . . . . . .  175

Antonino Drago

Part III The Square of Opposition and Non-classical Logics

The Square of Opposition in Orthomodular Logic . . . . . . . . . . . . . . . .  193

H. Freytes, C. de Ronde, and G. Domenech

No Group of Opposition for Constructive Logics: The Intuitionistic and

Linear Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  201

Baptiste Mélès

The Square of Opposition and Generalized Quantifiers . . . . . . . . . . . . .  219

Duilio D’Alfonso

Privations, Negations and the Square: Basic Elements of a Logic of Privations 229

Stamatios Gerogiorgakis

Fuzzy Syllogisms, Numerical Square, Triangle of Contraries, Inter-bivalence . 241

Ferdinando Cavaliere

Part IV Constructions Generalizing the Square of Opposition

General Patterns of Opposition Squares and 2n-gons . . . . . . . . . . . . . . .  263

Ka-fat Chow

The Cube Generalizing Aristotle’s Square in Logic of Determination of

Objects (LDO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  277

Jean-Pierre Desclés and Anca Pascu

Hypercubes of Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  293

Thierry Libert

Part V Applications of the Square of Opposition

How to Square Knowledge and Belief . . . . . . . . . . . . . . . . . . . . . . .  305

Wolfgang Lenzen

Structures of Oppositions in Public Announcement Logic . . . . . . . . . . . .  313

Lorenz Demey

Logical Opposition and Collective Decisions . . . . . . . . . . . . . . . . . . . .  341

Srecko Kova ´ cˇ

A Metamathematical Model for A/O Opposition in Scientific Inquiry . . . . .  357

Mark WeinsteinContributors

Jean-Pascal Alcantara Centre Georges Chevrier-UMR 5605, Université de Bourgogne,

Dijon, France

Jean-Yves Béziau CNPq – Brazilian Research Council, UFRJ – University of Brazil,

Rio de Janeiro, Brazil

Ferdinando Cavaliere Cesenatico, FC, Italy

Saloua Chatti Department of Philosophy, Faculté des Sciences Humaines et Sociales,

University of Tunis, Tunis, Tunisia

Ka-fat Chow The Hong Kong Polytechnic University, Hong Kong, China

Manuel Correia Facultad de Filosofía, Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

Duilio D’Alfonso University of Calabria, Arcavacata di Rende, CS, Italy

C. de Ronde Departamento de Filosofía “Dr. A Korn”, Universidad de Buenos AiresCONICET, Buenos Aires, Argentina; Center Leo Apostel and Foundations of the Exact

Sciences, Vrije Universiteit Brussel, Brussels, Belgium

Lorenz Demey Center for Logic and Analytical Philosophy, Institute of Philosophy, KU

Leuven – University of Leuven, Leuven, Belgium

Jean-Pierre Desclés LaLIC (Langues, Logiques, Informatique et Cognition), Université

de Paris Sorbonne, Paris, France

G. Domenech Instituto de Astronomía y Física del Espacio, Buenos Aires, Argentina

Antonino Drago University of Pisa, Pisa, Italy

H. Freytes Universita degli Studi di Cagliari, Cagliari, Italy; Instituto Argentino de

Matemática, Buenos Aires, Argentina

Stamatios Gerogiorgakis University of Erfurt, Erfurt, Germany

Raffaela Giovagnoli Pontifical Lateran University, Vatican City, Italy

Dale Jacquette University of Bern, Bern, Switzerland

ixx Contributors

John T. Kearns Department of Philosophy and Center for Cognitive Science, University

at Buffalo, the State University of New York, Buffalo, NY, USA

Srecko Kova ´ cˇ Institute of Philosophy, Zagreb, Croatia

Wolfgang Lenzen Dept. of Philosophy, University of Osnabrueck, Osnabrueck, Germany

Thierry Libert Département de mathématiques, Université Libre de Bruxelles (U.L.B.),

Brussels, Belgium

Baptiste Mélès Université Blaise Pascal, Clermont-Ferrand, France

Anca Pascu LaLIC, Université de Bretagne Occidentale Brest, Brest Cedex 03, France

Stephen Read Department of Philosophy, University of St Andrews, St Andrews, Scotland, UK

Fabien Schang LHSP Henri Poincaré (UMR7117), Université de Lorraine, Nancy,

France

Pieter A.M. Seuren Max Planck Institute for Psycholinguistics, Nijmegen, The Netherlands

Hartley Slater University of Western Australia, Crawley, WA, Australia

Mark Weinstein Department of Educational Foundations, Montclair State University,

Upper Montclair, NJ, USA