Philosophical Logic: Current Trends in Asia: Proceedings of AWPL-TPLC 2016 (Logic in Asia: Studia Logica Library)

Annotation This volume brings together a group of logic-minded philosophers and philosophically oriented logicians, mainly from Asia, to address a variety of logical and philosophical topics of current interest, offering a representative cross-section of the philosophical logic landscape in early 21st-century Asia. It surveys a variety of fields, including modal logic, epistemic logic, formal semantics, decidability and mereology. The book proposes new approaches and constructs more powerful frameworks, such as cover theory, an algebraic approach to cut-elimination, and a Boolean approach to causal discovery, to name but a few. Readers may find a wide range of applications of these original works in current research of philosophical logic, especially in the structural and conceptual analysis of some significant semantic properties and formal systems. The variety of topics and issues discussed here will appeal to readers from a broad spectrum of disciplines, ranging from mathematical/philosophical logic, computing science, cognitive science and artificial intelligence, to linguistics, game theory and beyond Preface 6 Contents 13 Contributors 15 Representing and Completing Lattices by Propositions of Cover Systems 17 1 Introduction 17 2 Closure Operators and Cover Systems 18 3 Finite Covers for Representing a Lattice 21 4 Distributive Lattices 23 5 Heyting Algebras 25 6 Ideal Completion 27 7 Covers for the MacNeille Completion 28 8 Ortholattices 30 9 Stricter Cover Systems for Heyting Algebras 32 References 33 A Uniform Algebraic Approach to Cut Elimination via Semi-completeness 35 1 Maehara's Approach Revisited 35 2 Semi-completeness 37 3 Quasi-homomorphisms and Downward Saturations 40 4 A Semi-complete Sequent System for Intuitionistic Logic 42 5 Quasi-homomorphisms and Quasi-embeddings 44 6 Semi-completeness of Sequent Systems for Substructural Predicate Logics 49 7 A Semantical Proof of Cut Elimination for Intuitionistic Predicate Logic 51 8 General Quasi-homomorphisms on Expanded Algebraic Structures 54 References 58 Ancient Indian Logic, Paksa and Analogy 60 1 Introduction 60 2 The Paksa Formalisation 62 3 Unary Pure Inductive Logic 63 3.1 The Principle of Atom Exchangeability, Ax 64 3.2 The Principle of Constant Exchangeability, Ex 65 4 The Main Theorems 66 5 Weakening Atom Exchangeability 68 5.1 The Principle of Predicate Exchangeability, Px 68 5.2 The Strong Negation Principle, SN 68 6 Conclusion 69 References 73 Provability and Decidability of Arithmetical Sentences 74 1 Introduction 74 2 Notations and Preliminaries 76 3 Hilbertian Properties 79 4 Arithmetical Sentences 81 References 92 On the Minimization Principle in the Boolean Approach to Causal Discovery 94 1 Introduction 94 2 The Boolean Approach to Causal Discovery 96 3 Counterexamples to the Minimization Principle 100 4 The Purpose of Minimization 104 5 Conclusion 107 References 109 Contentual and Formal Aspects of Gentzen's Consistency Proofs 110 1 Introduction 110 2 Contentual and Formal Correctness Proofs 113 3 Finite Notations for Infinitary Derivations 118 4 Contentual and Formal Aspects of Gentzen's 1936 Proof 126 5 Some Remarks 142 5.1 Gentzen's Finitist Interpretation of Transfinite Propositions 142 5.2 Uniform Interpretation of Gentzen's Three Proofs 144 5.3 No-Counterexample Interpretation 145 6 Conclusion 150 References 151 Saving Supervaluationism from the Challenge of Higher-Order Vagueness Argument 153 1 Vagueness and Higher-Order Vagueness 153 2 The Naïve Strategies to Vagueness and Higher-Order Vagueness Challenge 156 3 Supervaluationism and Williamson's Criticisms 158 4 Admissibility as Dynamic 161 References 166 Cut Free Labelled Sequent Calculus for Dynamic Logic of Relation Changers 167 1 Introduction 167 2 Dynamic Logic of Relation Changers (DLRC) 169 2.1 Syntax and Kripke Semantics 169 2.2 Hilbert-Style Axiomatization HDLRC 171 3 Labelled Sequent Calculus GDLRC 173 3.1 Labelled Formalism 173 3.2 All Theorems of HDLRC are Derivable in GDLRC 176 3.3 Cut Elimination of GDLRC 182 4 Soundness of GDLRC for Kripke Semantics 189 5 Conclusion 192 References 193 On Second Order Propositional Intuitionistic Logics 195 1 Introduction 195 2 Sequent Calculi 197 3 Deleting Disjunction 199 4 Kripke Model 200 5 Soundness of Sequent Calculi 201 6 Sequent Tree Calculi and Their Completeness 202 7 From Sequent Trees to Formulas 207 8 Conclusions and Future Study 208 References 210 Classical Model Existence Theorem in Subclassical Predicate Logics. II 211 1 Introduction 211 2 Classical Model Existence Theorem in Extensions of BCI 212 3 Herbrand-Henkin Style Proof Systems for CME in BCI-Based Predicate Logics 218 4 Hintikka Style Proof Systems for CME in BCI-Based Predicate Logics 224 5 Concluding Remarks 226 References 226 On Incorporating Reasoning Time into Epistemic Logic 227 1 Introduction 227 2 From MEL to tMEL 229 3 Logic of tS4K 234 4 Logic of tS4infty 236 5 Comparisons 238 6 Some Discussions 240 6.1 Moore's Paradox 240 6.2 Moorean Sentences 242 References 244 Proof-Theoretic Embedding from Visser's Basic Propositional Logic to Modal Logic K4 via Non-labelled Sequent Calculi 246 1 Introduction 246 2 Visser's Basic Propositional Logic 249 3 G3-Style Non-labelled Sequent Calculus for BPL 249 4 Admissibility of Cut 252 4.1 Soundness and Completeness 260 5 Proof-Theoretic Embedding from BPL to Modal Logic K4 261 5.1 G3-Style Non-labelled Sequent Calculus for K4 261 5.2 Embedding Theorem 262 6 Extending Our Proof-Theoretic Embedding 267 7 Further Direction 268 References 269 Varieties of Parthood 271 1 Introduction 271 2 Transitivity 273 3 Overlap-Inclusion and Strong Supplementation 274 3.1 Overlap 274 3.2 Overlap-Inclusion: General 275 3.3 Overlap-Inclusion of a Transitive Relation 275 3.4 Strong Supplementation 276 3.5 Reflexivity 277 3.6 Fusion 279 4 Quasi-isomorphism 280 4.1 Reduction of a Transitive Relation 282 4.2 Features Preserved Across Reductions 284 4.3 The Existence of Reductions and Quasi-isomorphisms 285 4.4 Unique Fusions and Classical Mereology 286 4.5 Groundedness 287 5 Merging Domains 290 5.1 Non-interference 290 5.2 Articulation and Fusion 291 5.3 Discussion 293 6 Completions and General Structure 294 6.1 Discussion 296 6.2 Expansion Without Groundedness 297 References 298 Infinite ``Atomic'' Mereological Structures 299 1 Introduction 299 2 The Case of GEM+(A) 301 3 The Case of Classical Mereology with the Atomicity Axiom 305 4 Concluding Remarks 307 References 308 Front Matter ....Pages i-xvi Representing and Completing Lattices by Propositions of Cover Systems (Robert Goldblatt)....Pages 1-18 A Uniform Algebraic Approach to Cut Elimination via Semi-completeness (Hiroakira Ono)....Pages 19-43 Ancient Indian Logic, Pakṣa and Analogy (Jeffrey B. Paris, Alena Vencovská)....Pages 45-58 Provability and Decidability of Arithmetical Sentences (Shih Ping Tung)....Pages 59-78 On the Minimization Principle in the Boolean Approach to Causal Discovery (Jiji Zhang)....Pages 79-94 Contentual and Formal Aspects of Gentzen’s Consistency Proofs (Ryota Akiyoshi, Yuta Takahashi)....Pages 95-137 Saving Supervaluationism from the Challenge of Higher-Order Vagueness Argument (Hao-Cheng Fu)....Pages 139-152 Cut Free Labelled Sequent Calculus for Dynamic Logic of Relation Changers (Ryo Hatano, Katsuhiko Sano, Satoshi Tojo)....Pages 153-180 On Second Order Propositional Intuitionistic Logics (Ryo Kashima)....Pages 181-196 Classical Model Existence Theorem in Subclassical Predicate Logics. II (Jui-Lin Lee)....Pages 197-212 On Incorporating Reasoning Time into Epistemic Logic (Ren-June Wang)....Pages 213-231 Proof-Theoretic Embedding from Visser’s Basic Propositional Logic to Modal Logic K4 via Non-labelled Sequent Calculi (Sakiko Yamasaki, Katsuhiko Sano)....Pages 233-257 Varieties of Parthood (Paul Hovda)....Pages 259-286 Infinite “Atomic” Mereological Structures (Hsing-Chien Tsai)....Pages 287-296 This volume brings together a group of logic-minded philosophers and philosophically oriented logicians, mainly from Asia, to address a variety of logical and philosophical topics of current interest, offering a representative cross-section of the philosophical logic landscape in early 21st-century Asia. It surveys a variety of fields, including modal logic, epistemic logic, formal semantics, decidability and mereology. The book proposes new approaches and constructs more powerful frameworks, such as cover theory, an algebraic approach to cut-elimination, and a Boolean approach to causal discovery, to name but a few. Readers may find a wide range of applications of these original works in current research of philosophical logic, especially in the structural and conceptual analysis of some significant semantic properties and formal systems. The variety of topics and issues discussed here will appeal to readers from a broad spectrum of disciplines, ranging from mathematical/phil osophical logic, computing science, cognitive science and artificial intelligence, to linguistics, game theory and beyond