Mathematics, Logic, and their Philosophies: Essays in Honour of Mohammad Ardeshir (Logic, Epistemology, and the Unity of Science, 49)

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the contemporary philosophy of mathematics and logic. The third category contains papers on Avicenna’s logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community. Preface Acknowledgements The Complete List of Mohammad Ardeshir’s Publication 1. Publications 1.1. Books 1.2. Articles 1.3. Articles in Persian Contents 1 Equality and Equivalence, Intuitionistically 1.1 Introduction 1.2 Intuitionistic Model Theory 1.3 Equality May Be Undecidable 1.4 Spreads 1.5 Spreads with a Decidable Equality 1.6 Spreads with Exactly One Undecidable Point 1.7 More and More Undecidable Points: The Toy Spreads 1.8 Finite and Infinite Sums of Toy Spreads 1.8.1 A Main Result 1.8.2 Finitary Spreads Suffice 1.8.3 Comparison with an Older Theorem 1.9 The Vitali Equivalence Relation 1.10 A First Vitali Variation 1.11 More and More Vitali Relations 1.12 Equality and Equivalence 1.13 Notations and Conventions References 2 Binary Modal Companions for Subintuitionistic Logics 2.1 Introduction 2.2 Neighborhood Semantics for Modal and Subintuitionistic Logics 2.2.1 Neighborhood Semantics for Modal Logic 2.2.2 Neighborhood Semantics for Subintuitionistic Logics 2.3 A Complete Basic System for Strict Implication 2.4 Modal Companions 2.5 Translations 2.5.1 Translations Between E2Imp and EN 2.5.2 Translations Between Extensions of E2Imp U and EN 2.5.3 Translations, Axiomatizations and Standard Modal Logics 2.6 Conclusion References 3 Extension and Interpretability 3.1 Introduction 3.2 Preliminaries 3.2.1 Signatures, Formulas, Theories 3.2.2 Translations and Interpretations 3.2.3 The Structure 3.2.4 Some Salient Notions 3.3 Characterisations 3.3.1 The Logic of a Theory, Admissibility and Interpretability 3.3.2 Weak Interpretability and Local Cointerpretability 3.3.3 (Locally) Faithful Interpretability 3.3.4 Local Tolerance 3.4 Disjoint Sum is a Capital Operation 3.5 Faithful Interpretability and Locally Faithful Interpretability 3.5.1 Decidability 3.5.2 The Finite Model Property 3.5.3 Separating Examples 3.5.4 The Forward Property 3.6 Arithmetic 3.6.1 Incomparable Theories 3.6.2 Characterisations 3.6.3 Local (In)tolerance 3.7 Sequential Theories 3.8 Appendix: Basics 3.8.1 Theories and Provability 3.8.2 Translations 3.8.3 Relative Interpretations 3.8.4 Global and Local Interpretability 3.8.5 Piecewise Translations and Interpretations 3.8.6 Five Categories 3.8.7 Sums 3.8.8 Adding Pieces 3.9 Appendix: Sequential Theories References 4 Residuated Expansions of Lattice-Ordered Structures 4.1 Introduction 4.2 Expansions of Lattice-Ordered Groupoids into Residuated Ones 4.3 Left Residuated Expansions of Lattices with Implication 4.4 Links Between Lattices with Implication and Lattice-Ordered Groupoids 4.5 Representation and Left Residuated Expansion of Bounded Distributive lattice-Ordered Structures 4.6 Left Residuated Weak Heyting Algebras 4.7 Conservative Extensions, Finite Embeddability Property and Amalgamation Property 4.8 Concluding Remarks References 5 Everyone Knows that Everyone Knows 5.1 Introduction 5.2 Definitions 5.3 Examples 5.4 The Protocols CMO and PIG 5.4.1 The Protocol CMO 5.4.2 The Protocol PIG References 6 Fuzzy Generalised Quantifiers for Natural Language in Categorical Compositional Distributional Semantics 6.1 Introduction 6.2 Dedication 6.3 Generalised Quantifiers in Natural Language 6.4 Category Theoretic Definitions 6.5 Category of Sets and Many Valued Relations 6.6 Fuzzy Sets and Fuzzy Quantifiers 6.7 Fuzzy Quantified Sentences in V-Rel 6.8 Conclusions and Future Work References 7 Implication via Spacetime 7.1 Introduction 7.2 Preliminaries 7.3 Intuitionism via Quantales 7.4 Abstract Implications 7.4.1 Constructing New Implications from the Old 7.5 Non-commutative Spacetimes 7.6 Representation Theorems 7.7 Logics of Spacetime 7.8 Kripke Models 7.9 Sub-intuitionistic Logics References 8 Bounded Distributive Lattices with Two Subordinations 8.1 Introduction 8.2 Preliminaries 8.3 Subordination Relations and Quasi-modal Operators on Distributive Lattices 8.3.1 Two Maps on the Power Set of a Subordination Lattice Determined by the Subordination Relation 8.3.2 The Two Relations on the Set of Prime Filters of a Lattice Determined by a Subordination 8.4 Some Kinds of Bi-Subordination Lattices 8.5 Duality for Subordination Lattices and Bi-Subordination Lattices 8.6 Positive Bi-Subordination Lattices References 9 Hard Provability Logics 9.1 Dedication 9.2 Introduction 9.3 Definitions and Preliminaries 9.3.1 Preliminaries from Arithmetic 9.3.2 The NNIL Formulae and Related Topics 9.3.3 Intuitionistic Modal Kripke Semantics 9.4 Reduction of Arithmetical Completenesses 9.4.1 Two Special Cases 9.5 Relative Σ1-provability Logics for HA 9.5.1 Kripke Semantics 9.5.2 Arithmetical Interpretations 9.5.3 Arithmetical Completeness 9.5.4 Reductions 9.6 Relative Σ1-provability Logics for HA* 9.7 Relative Provability Logics for PA 9.7.1 Reducing PLΣ1 (PA,mathbbN) to PLΣ1 (HA,mathbbN) 9.7.2 Kripke Semantics 9.7.3 Arithmetical Completeness 9.7.4 Reductions 9.8 Relative Provability Logics for PA* 9.8.1 Kripke Semantics 9.8.2 Reductions 9.9 Conclusion References 10 On PBZast–Lattices 10.1 Introduction 10.2 Preliminaries 10.3 A Study of Some Subvarieties 10.3.1 The F8 Problem 10.3.2 Covers in the Lattice of Subvarieties of mathbbPBZLast 10.3.3 Subdirect Products and Varieties of PBZast–lattices 10.4 Comparison with Other Structures 10.4.1 Distributive Lattices with Two Unary Operations 10.4.2 Modal Algebras References 11 From Intuitionism to Many-Valued Logics Through Kripke Models 11.1 Introduction and Preliminaries 11.2 ω-Many Values for Intuitionistic Propositional Logic 11.3 Propositional Connectives Inside Gödel-Dummett Logic References 12 Non-conditional Contracting Connectives 12.1 Introduction 12.2 Setting the Stage 12.3 Blamey's Transplication 12.4 The OCO Conditional and P-Logical Consequence 12.5 Rogerson and Butchart's Conditional 12.6 A Variant of Rogerson and Butchart's Conditional 12.7 Conclusions References 13 Deflationary Reference and Referential Indeterminacy 13.1 The Incompatibility Thesis 13.2 The Argument from Disquotation 13.3 The Argument from Explanation 13.3.1 The Formulation of the Argument 13.3.2 Incompatibility 13.3.3 Explanation 13.4 Concluding Remarks References 14 The Curious Neglect of Geometry in Modern Philosophies of Mathematics 14.1 Origins of Paradigm Change 14.2 Foundationalist Schools 14.3 Geometry as a Mode of Mathematical Thought References 15 De-Modalizing the Language 15.1 Relativizing to the Background Theory 15.2 Objections and Replies 15.2.1 The Case of “Un-Actualized Physical Possibilities” 15.2.2 Objection: The Open-Minded Physicist 15.2.3 Objection: Practical Versus in-Principle 15.2.4 Objection: Physicists’ Use of Modalities 15.2.5 The Failure of Factivity 15.2.6 Objection: All Laws Are Necessary? 15.3 Ways of Relativizing Modalities 15.3.1 À la Montague et Anderson 15.3.2 À la Hale-Leech References 16 On Descriptional Propositions in Ibn Sīnā: Elements for a Logical Analysis 16.1 Introduction 16.2 On What and How 16.3 Substantial and Descriptional Propositions 16.4 Time Parameters 16.4.1 Preliminaries on Temporal Reference in CTT 16.4.2 Descriptional Propositions Relativized by Saturation 16.4.3 Descriptional Propositions Relativized by Enrichment 16.5 Existence With and Without Existence Predicate 16.6 Conclusion References 17 Avicenna on Syllogisms Composed of Opposite Premises 17.1 Introduction 17.2 Aristotle on Syllogisms Composed of Opposite Premises 17.3 Avicenna on Opposite Premises 17.3.1 Truth and Opposite Premises 17.3.2 Syllogisms from Opposite Premises 17.4 Conclusion: Paraconsistency and Syllogistic References 18 Is Avicenna an Empiricist? 18.1 Introduction 18.2 Avicenna’s Language 18.3 Two Examples 18.4 What Is Empiricism? 18.5 Empiricism Reinterpreted 18.6 Is Avicenna an Origin-Empiricist? 18.6.1 Sense Perception and Imagination Assist the Intellect 18.6.2 Perceiving the Intelligibles Needs the Mediation of the Sensible Forms 18.6.3 Lack of Sensation Implies Lack of Knowledge 18.7 Limitations to Avicenna’s Origin-Empiricism 18.7.1 Celestial Bodies 18.7.2 Unseen Things 18.7.3 Immaterial Substances 18.8 Is Avicenna a Content-Empiricist? 18.9 Open Questions 18.9.1 Is Avicenna a Rationalist? 18.9.2 Is Avicenna an Abstractionist? 18.9.3 Is Avicenna a Compatibilist? 18.10 Conclusion Bibliography Author Index Subject Index