Algebraic Perspectives on Substructural Logics (Trends in Logic Book 55)

This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics. Acknowledgements 6 Contents 7 Contributors 8 Editorial Introduction 10 1 Algebra, Residuation, and Logic 10 2 Substructural Logics and Quantum Logics 13 3 The AsubL Conferences 14 4 Outline of the Volume 15 References 17 Distributivity and Varlet Distributivity 19 1 Preliminaries 19 2 The Relationships 22 References 28 On Distributive Join Semilattices 29 1 Introduction 29 2 Preliminaries 30 3 Distributivity and Natural Deduction 31 4 Notions of Distributivity for Join Semilattices 35 4.1 GS-Distributivity 35 4.2 K-Distributivity 37 4.3 H-Distributivity 37 4.4 LR-Distributivity 41 4.5 B-Distributivity 42 4.6 Sn-Distributivity 43 5 Join Semilattices with Arrow 44 6 Conclusions 47 References 48 Implication in Weakly and Dually Weakly Orthomodular Lattices 49 References 63 Residuated Operators and Dedekind–MacNeille Completion 65 1 Introduction 66 2 Dedekind–MacNeille Completion 67 3 Completion of Pseudo-orthomodular Posets 71 References 79 PBZast–Lattices: Ordinal and Horizontal Sums 81 1 Introduction 81 2 Preliminaries 83 2.1 Universal Algebra and Lattice Theory 83 2.2 Ordinal and Horizontal Sums of Lattices 85 2.3 PBZast–Lattices 86 3 Dense Elements in PBZast–Lattices 90 4 Ordinal Sums in Antiortholattices 92 5 Horizontal Sums of PBZast–Lattices 97 5.1 General Properties 97 5.2 Congruences of Horizontal Sums 102 6 Varieties of PBZast–Lattices Generated by Horizontal Sums 106 References 112 EMV-Algebras—Extended MV-Algebras 114 1 Introduction 115 2 EMV-Algebras 117 2.1 MV-Algebras 118 2.2 Finite EMV-Algebras 118 2.3 Generalized Boolean Algebras 118 2.4 Rings of Subsets 118 2.5 EMV-Algebra mathcalN of Integers 119 2.6 EMV-Clan of Fuzzy Sets 119 2.7 Positive Cone of an ell-Group 120 2.8 Direct Product of EMV-Algebras 120 2.9 Sum of MV-Algebras 120 3 Basic Properties of EMV-Algebras 120 4 Basic Representation Theorem and Classes of EMV-Algebras 123 5 Categorical Equivalencies 126 6 States and State-Morphisms on EMV-Algebras 128 7 Loomis–Sikorski Theorem for σ-Complete EMV-Algebras 133 8 Free EMV-Algebras and Weakly Free EMV-Algebras 135 8.1 Free EMV-Algebras over Finite Set of Generators 137 8.2 Weakly Free EMV-Algebras with Infinite Generator 137 9 Conclusion 138 References 138 Quasi-Nelson; Or, Non-involutive Nelson Algebras 140 1 Introduction 141 2 Quasi-Nelson Residuated Lattices 144 3 Twist-Structures and Quasi-Nelson Algebras 148 4 Refining the Twist-Structure Representation 158 5 (0, 1)-Congruence Orderability 165 6 Conclusions and Future Work 172 References 174 Hyperdoctrines and the Ontology of Stratified Semantics 176 1 Introduction 176 2 Preliminaries 178 3 Stratified Semantics 178 4 RW Hyperdoctrines 181 5 Soundness for RW Hyperdoctrines 184 6 Correspondence Between Stratified Models and Hyperdoctrines 187 7 Ontology 191 References 200 This volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra et Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics Front Matter ....Pages i-x Editorial Introduction (Davide Fazio, Antonio Ledda, Francesco Paoli)....Pages 1-9 Distributivity and Varlet Distributivity (Paolo Aglianò)....Pages 11-20 On Distributive Join Semilattices (Rodolfo C. Ertola-Biraben, Francesc Esteva, Lluís Godo)....Pages 21-40 Implication in Weakly and Dually Weakly Orthomodular Lattices (Ivan Chajda, Helmut Länger)....Pages 41-56 Residuated Operators and Dedekind–MacNeille Completion (Ivan Chajda, Helmut Länger, Jan Paseka)....Pages 57-72 PBZ\(^{*}\)–Lattices: Ordinal and Horizontal Sums (Roberto Giuntini, Claudia Mureşan, Francesco Paoli)....Pages 73-105 EMV-Algebras—Extended MV-Algebras (Anatolij Dvurečenskij, Omid Zahiri)....Pages 107-132 Quasi-Nelson; Or, Non-involutive Nelson Algebras (Umberto Rivieccio, Matthew Spinks)....Pages 133-168 Hyperdoctrines and the Ontology of Stratified Semantics (Shay Allen Logan)....Pages 169-193