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Table of Contents
Intro; Preface; Contents; Editors and Contributors; Part I Life and Work of Ewa Orłowska; 1 Everything is a Relation: A Preview; 1.1 Early Works of Ewa Orłowska on Deduction Systems and their Mechanization; 1.2 Knowledge Representation Systems and Information Logics; 1.3 Non-classical Logics in a Relational Framework; 1.4 Dual Tableau Systems; 1.5 Discrete Dualities; 1.6 Contributed Chapters; References; 2 Autobiography; References; 3 Engaged in Relations: A Trialogue; 4 Bibliography of Ewa Orłowska; Part II Deduction and Relational Semantics; 5 Tableaus and Dual Tableaus; 5.1 Introduction
5.2 Classical Propositional Logic and the Basic Ideas5.2.1 Classical Propositional Dual Tableaus; 5.2.2 Classical Propositional Tableaus; 5.2.3 Soundness; 5.2.4 Completeness; 5.2.5 What Dual Tableaus ``Really'' Look Like; 5.3 Intuitionistic Logic; 5.3.1 Intuitionistic Dual Tableaus; 5.3.2 Soundness; 5.3.3 Completeness; 5.3.4 Intuitionistic Tableaus; 5.3.5 Logical Consequence; 5.4 Conclusion; References; 6 Reasoning on Relations, Modalities, and Sets; 6.1 Preliminaries; 6.1.1 Syntax and Semantics of calLtimes; 6.1.2 Maddux's General Translation Technique
6.1.3 An Elementary Theory of Aggregates6.1.4 Rasiowa-Sikorski Systems; 6.2 A Rasiowa-Sikorski Deduction System for Elementary Set Theory; 6.2.1 The Translation from Mono-Modal Logic to Rest; 6.2.2 Modal Derivability Through Rest; 6.3 Equational Set Theory; 6.3.1 A Relational Counterpart of Est; 6.3.2 Specifying Full-Fledged Set Theories in calLtimes; 6.4 Automated Equational Reasoning; 6.4.1 First-Order Theorem Prover Serving Relational Reasoning; 6.4.2 Relational Deduction for Non-classical Logics; 6.5 Conclusions; References
7 Dual Tableau-Based Decision Procedures for Fragments of the Logic of Binary Relations7.1 Introduction; 7.2 The Relational Logic RL(mathbbC) and its Dual Tableau; 7.3 Useful Notions and Properties; 7.3.1 Parsing Trees; 7.3.2 Normal Forms and Term Components; 7.4 Dual Tableau-Based Decision Procedures for Fragments of the Logic of Binary Relations; 7.4.1 The Relational Fragments RLPOS, RLNEG, and RLNP; 7.4.2 Relational Fragments Corresponding to the Modal Logic K, to Some Multimodal Logics, and Description Logics
7.4.3 Fragments Characterized by Some Restrictions in the Construction of Terms of Type (R S); 7.5 Conclusions; References; 8 Logics for Order-of-Magnitude Qualitative Reasoning: Formalizing Negligibility; 8.1 Introduction; 8.2 Different Approaches to Negligibility; 8.2.1 Unidirectional Negligibility; 8.2.2 Bidirectional Negligibility; 8.2.3 Distance-Based Negligibility; 8.2.4 PDL-Based Negligibility; 8.2.5 Considering Proximity Intervals; 8.3 Relational Approach to Order-of-Magnitude Reasoning; 8.4 Conclusions; References; 9 Signed Dual Tableaux for Kleene Answer Set Programs
5.2 Classical Propositional Logic and the Basic Ideas5.2.1 Classical Propositional Dual Tableaus; 5.2.2 Classical Propositional Tableaus; 5.2.3 Soundness; 5.2.4 Completeness; 5.2.5 What Dual Tableaus ``Really'' Look Like; 5.3 Intuitionistic Logic; 5.3.1 Intuitionistic Dual Tableaus; 5.3.2 Soundness; 5.3.3 Completeness; 5.3.4 Intuitionistic Tableaus; 5.3.5 Logical Consequence; 5.4 Conclusion; References; 6 Reasoning on Relations, Modalities, and Sets; 6.1 Preliminaries; 6.1.1 Syntax and Semantics of calLtimes; 6.1.2 Maddux's General Translation Technique
6.1.3 An Elementary Theory of Aggregates6.1.4 Rasiowa-Sikorski Systems; 6.2 A Rasiowa-Sikorski Deduction System for Elementary Set Theory; 6.2.1 The Translation from Mono-Modal Logic to Rest; 6.2.2 Modal Derivability Through Rest; 6.3 Equational Set Theory; 6.3.1 A Relational Counterpart of Est; 6.3.2 Specifying Full-Fledged Set Theories in calLtimes; 6.4 Automated Equational Reasoning; 6.4.1 First-Order Theorem Prover Serving Relational Reasoning; 6.4.2 Relational Deduction for Non-classical Logics; 6.5 Conclusions; References
7 Dual Tableau-Based Decision Procedures for Fragments of the Logic of Binary Relations7.1 Introduction; 7.2 The Relational Logic RL(mathbbC) and its Dual Tableau; 7.3 Useful Notions and Properties; 7.3.1 Parsing Trees; 7.3.2 Normal Forms and Term Components; 7.4 Dual Tableau-Based Decision Procedures for Fragments of the Logic of Binary Relations; 7.4.1 The Relational Fragments RLPOS, RLNEG, and RLNP; 7.4.2 Relational Fragments Corresponding to the Modal Logic K, to Some Multimodal Logics, and Description Logics
7.4.3 Fragments Characterized by Some Restrictions in the Construction of Terms of Type (R S); 7.5 Conclusions; References; 8 Logics for Order-of-Magnitude Qualitative Reasoning: Formalizing Negligibility; 8.1 Introduction; 8.2 Different Approaches to Negligibility; 8.2.1 Unidirectional Negligibility; 8.2.2 Bidirectional Negligibility; 8.2.3 Distance-Based Negligibility; 8.2.4 PDL-Based Negligibility; 8.2.5 Considering Proximity Intervals; 8.3 Relational Approach to Order-of-Magnitude Reasoning; 8.4 Conclusions; References; 9 Signed Dual Tableaux for Kleene Answer Set Programs