The Exoteric Square of Opposition: The Sixth World Congress on the Square of Opposition

Preface
Contents
Contributors
The Square of Opposition: Past, Present, and Future
1 The Square of Opposition: A Diagram and a Theory
2 The Sixth World Congress on the Square of Opposition: Crete 2018
3 The Square of Opposition: An Ongoing Open Project
References
Division of Entities and Foundations of Reality: Aristotle's Ontological Square
1 Introduction
2 Terminology, Definitions, Translations
3 Division of Entities: Aristotle's Ontological Square
4 Lowe's Ontological Square
5 A Proposal of Adaptation of Lowe's Scheme
6 Foundations: Two-District Ontology
7 Instances and Properties
8 Some Remarks on the Relevance of the Essence in Aristotle's Ontological System
9 Metaphysics Mu 10: Instances and Universals
10 On Properties
11 Aristotle's Polemical Targets: The One over Many Argument and the Third Man Argument
12 Conclusions
References
Logical Oppositions in Avicenna's Hypothetical Logic
1 Introduction
2 The Quantified Hypothetical Propositions
3 The Quantified Hypothetical Propositions with Quantified Clauses
4 How to Formalize these Propositions
5 The Logical Relations Between these Propositions
6 The Octagons of Oppositions with these Propositions
7 Combining the Octagons Two by Two
8 Conclusion
Bibliography
Incommensurability and Inapplicability of the Squares of Opposition
1 Background
2 The Controversy of Squares
2.1 Transformation of Traditional to Revised Square
2.2 Some Problems
2.3 Summary
3 The Paradigms of Squares
3.1 Kuhnian Notions
3.2 Possible Worlds
3.3 Summary
4 The Incommensurability of Squares
4.1 Taxonomic Incommensurability
4.2 Methodological Incommensurability
4.3 Summary
5 The Inapplicability of Squares
5.1 Prof. Wascot vs. Prof. Palton
5.2 Prof. Mind vs. Prof. Head
5.3 Tomato vs. Tomato
5.4 HEisenberg vs. HeIsenberg
5.5 Summary
6 Conclusion
References
The Square of Opposition as a Framework for Stephen Langton's Theological Solutions
1 Introduction
2 Langton's Logical Structures in Questions on God's Will
1 Langton's logical structures in questions on God's will
3 Towards the Square
4 The Significance for Theology
5 The Hexagon of Will
6 Conclusions
References
The Limits of the Square: Hegel's Opposition to Diagrams in Its Historical Context
1 Introduction
2 The Historical Context
2.1 Where to Find the Square?
2.2 Why Is the Square So Rare?
3 Hegel and Hegelianism
3.1 Hegelianism
3.2 Hegel
3.2.1 Hegel's Concept of the Concept
3.2.2 Hegel's Critique of Diagrams in Logic
4 Conclusion
References
Augustus De Morgan's Unpublished Octagon of Opposition
1 De Morgan, Symbols, Syllogistics, and Diagrams?
2 The Senate House Library (SHL) Diagrams
2.1 Provenance and Context
2.2 The Manuscript Materials
2.2.1 The Figures
2.2.2 The Comments
3 Reading Sense Into the Manuscript Materials
3.1 Notation
3.2 Propositions
3.3 Relations Between Terms and Propositions
3.3.1 “Contraries,” “Subcontraries,” “Supercontraries”
3.3.2 “Subidentical,” “Subcomplement,” “Subtotal”
3.3.3 Aristotelian Relations
4 Logical Analysis of De Morgan's Octagon of Opposition
5 Conclusion
References
A Bitstring Semantics for Calculus CL
1 Introduction
2 Bitstring Semantics
2.1 Identity Without Existence
2.2 Boolean Ontology
2.3 Term Semantics
2.4 Truth and Falsity
2.5 Hamming Measurements
2.6 Boolean Quantifications
2.7 Objections and Answers
3 Basic Principles of CL
3.1 Basics
3.2 Classes
3.3 Arrows
4 Testing Inferences with CL4
4.1 Examples
4.2 Application
5 Conclusion and Outlook
References
Logical Diagrams, Visualization Criteria, and Boolean Algebras
1 Introduction
2 Visualizing Logical Structure
3 The Logical Hexagon
3.1 Inclusion and Logical Consequence
3.2 Partitions and Opposition
3.3 Permutations
4 The Logical Tetrakis Hexahedron
4.1 Inclusion and Logical Consequence
4.2 Partitions and Opposition
4.3 Permutations
5 Hexagons inside the Tetrakis Hexahedron?
6 Conclusion
References
Turnstile Figures of Opposition
1 The Hexagon of Opposition and the Turnstile
2 Tautological Figures of Opposition
2.1 Two Pretty Different Contrariety Triangles
2.2 Symbolic Representation of the Tautological Triangle
2.3 Turnstile Tautological Hexagons
3 Hexagons of Opposition for Consequence Relations
Bibliography
The Naturalness of Jacques Lacan's Logic
1 Introduction
2 The Four Formulas of Sexuation
3 Lacan's Formulas in Comparison to the Traditional Square of Opposition
4 A “Source” of Lacan's Logic?
5 Le Gaufey's Lacanian Logical Square
5.1 Why Do the Positions of the Four Formulas of Sexuation change as Soon as They Are Put into a Logical Square?
5.2 What Is the Difference Between Lacan's Logic and the Traditional Logic?
6 The Naturalness of Lacan's Logic
References
On Modal Opposition Within Some Modal Discussive Logics
1 Introduction
2 Basics
2.1 Standard Modal Formulas
2.2 Discussive Logic
2.2.1 Translation from Ford into Form
2.2.2 Historical Reminder
2.3 The Discussive Logic D2 as a Set of Discussive Formulas
3 Discussive Modal Logic
3.1 Extension of Jaśkowski's Translation
3.2 Semantics for the Modal Discussive Logic
4 Another Modal Extension of D2
4.1 Semantics for General/Public Discussive Modalities
5 Square of Opposition
References
On the Transformations of the Square of Opposition from the Point of View of Institution Model Theory
1 Introduction
2 Squares of Opposition
3 Preliminaries
3.1 Essentials of Category Theory
3.2 Essentials of Institution Theory
3.3 Examples of Institutions
3.4 Morphisms
4 Institution-Theoretic Square of Opposition
4.1 The Aristotelian Relations of Sentences in Institution-Theoretic Setting
4.2 The Example of PL: Square of Opposition
4.3 After the Action of Morphisms
5 Institution-Theoretic Treatment of the Square of Opposition
5.1 Galois Connection
5.2 Aristotelian Relations and the Galois Connection
5.3 The Dual Square of Opposition
6 Conclusions
References
Color-Coded Epistemic Modes in a Jungian Hexagon of Opposition
1 Introduction
2 Modern Color Theory
3 Logic Structures and Color Theory
4 Epistemic Color-Coded Cognitive Modes
5 Yellow: Invisible Carriers in Charge
6 Cyan Science: As in Heaven Not on Earth
7 Purple-Violet: Suggestive Instincts-Insights
8 Final Remarks
References
Videos and Simulations
Many-Valued Logical Hexagons in a 3-Oppositional Trisimplex
1 The Context of This Study
2 Some Numerical Sheaves on a Topological Space with One Non-trivial Open Subset
3 The Aristotelian 32-semantics Reconsidered in This World of Sheaves
4 A Generic Trisimplex of Sheaves
5 Many-Valued Logical Hexagons in Our Trisimplex of Sheaves
6 The ``Jewel Nonagon'' as the Trisimplex
References
Tri-simplicial Contradiction: The “Pascalian 3D Simplex” for the Oppositional Tri-segment
1 The Context of This Study
1.1 The Controversy on the Foundations of Logical Negation (2003)
1.2 The Mathematics of Opposition: There Is an Oppositional Geometry
1.3 A General Extension of OG: The “Oppositional Poly-Simplexes”
1.4 Angot-Pellissier's Sheaf-Theoretical Method for Poly-Simplexes
1.5 Our Proposal of a “Pascalian” Extra Tool for the Poly-Simplexes
1.6 Flashback: The Primitive Idea of Oppositional Tri-segment (2009)
2 Studying with These New Tools the Oppositional Tri-segment
2.1 Oppositional Sub-sheaves of the Tri-segment: Which Are Vertices?
2.2 The Oppositional Relations Between the Sub-sheaves: Edges!
2.3 The Geometrical Problem: Having a Strange Pentadic Structure
2.4 The Pascalian 3D Simplex and Its “2D Section for Tri-segments”
2.5 A Point About “Points”: The Oppositional Colors of the Six Vertices
2.6 Back to the Geometrical Quest of the Oppositional Structure
3 More on the Inner Geometry of the Oppositional Tri-segment
3.1 There Are Three Possible Vertex Horizons Inside the Tri-segment
3.2 Three Possible Inner Circuits of the Oppositional Tri-segment
3.3 Which Is “the Best” Global Representation of the Tri-segment?
3.4 Inner Jungle: Possible Hybrid Substructures of the Tri-segment
3.5 How to Put “Semantic Values” on the Vertices of the Tri-segment?
3.6 Which Possible “Truth-Valuations” of the Global Tri-segment?
4 There Is Logical Geometry Inside the Oppositional Tri-segment!
4.1 The Implication Geometry's 32-Semantics/Lattice of the Tri-simplex
4.2 Calculating the 21 Implicative Relations (Edges) of the Tri-segment
4.3 The Global “Implication Geometry” of the Tri-segment
4.4 Overview of Some Inner Structures of the Implicative Tri-segment
4.5 Is an Aristotelian Tri-segment Possible? Yes! Meaningful? Very!
4.6 “Logical Geometry” or “Poly-Simplicial Oppositional Geometry”?
5 Consequences/Applications of the Tri-segment: Some Remarks
5.1 Some General Remarks on What Has Been Seen so Far in This Study
5.2 The Tri-segment and Many-Valued Logics: Some Remarks
5.3 The Tri-segment and Paraconsistent Logics: Some Remarks
5.4 The Tri-segment and Quantum Logics: Some Remarks
5.5 The Tri-segment and Hegelian-Marxian Dialectics: Some Remarks
5.6 The Tri-segment and Psychoanalysis: Some Remarks
6 Conclusion
References