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.
Author(s): Mojtaba Mojtahedi; Shahid Rahman; Mohammad Saleh Zarepour
Publisher: Springer
Year: 2021
Language: English
Pages: 483
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
