V.A. Yankov on Non-Classical Logics, History and Philosophy of Mathematics (Outstanding Contributions to Logic, 24)

This book is dedicated to V.A. Yankov’s seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic. The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov’s results and their applications in algebraic logic, the theory of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics. The book also contains an exposition of Yankov’s revolutionary approach to constructive proof theory. The editors also include Yankov’s contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics. Preface Contents Contributors 1 Short Autobiography Complete Bibliography of Vadim Yankov Part I Non-Classical Logics 2 V. Yankov's Contributions to Propositional Logic 2.1 Introduction 2.2 Classes of Logics and Their Respective Algebraic Semantics 2.2.1 Calculi and Their Logics 2.2.2 Algebraic Semantics 2.2.3 Lattices sans serif upper D e d Subscript upper CDedC and sans serif upper L i n d Subscript left parenthesis upper C comma k right parenthesisLind(C,k) 2.3 Yankov's Characteristic Formulas 2.3.1 Formulas and Homomorphisms 2.3.2 Characteristic Formulas 2.3.3 Splitting 2.3.4 Quasiorder 2.4 Applications of Characteristic Formulas 2.4.1 Antichains 2.5 Extensions of upper CC-Logics 2.5.1 Properties of Algebras bold upper A Subscript iAi 2.5.2 Proofs of Lemmas 2.6 Calculus of the Weak Law of Excluded Middle 2.6.1 Semantics of sans serif upper K upper CKC 2.6.2 sans serif upper K upper CKC from the Splitting Standpoint 2.6.3 Proof of Theorem2.5 2.7 Some Si-Calculi 2.8 Realizable Formulas 2.9 Some Properties of Positive Logic 2.9.1 Infinite Sequence of Independent Formulas 2.9.2 Strongly Descending Infinite Sequence of Formulas 2.9.3 Strongly Ascending Infinite Sequence of Formulas 2.10 Conclusions References 3 Dialogues and Proofs; Yankov's Contribution to Proof Theory 3.1 Introduction 3.2 Consistency Proofs 3.3 Yankov's Approach 3.4 The Calculus 3.5 The Dialogue Method 3.6 Bar Induction 3.7 Proofs 3.8 Concluding Remarks References 4 Jankov Formulas and Axiomatization Techniques for Intermediate Logics 4.1 Introduction 4.2 Intermediate Logics and Their Semantics 4.2.1 Intermediate Logics 4.2.2 Heyting Algebras 4.2.3 Kripke Frames and Esakia Spaces 4.3 Jankov Formulas 4.3.1 Jankov Lemma 4.3.2 Splitting Theorem 4.3.3 Cardinality of the Lattice of Intermediate Logics 4.4 Canonical Formulas 4.4.1 Subframe Canonical Formulas 4.4.2 Negation-Free Subframe Canonical Formulas 4.4.3 Stable Canonical Formulas 4.5 Canonical Formulas Dually 4.5.1 Subframe Canonical Formulas Dually 4.5.2 Stable Canonical Formulas Dually 4.6 Subframe and Cofinal Subframe Formulas 4.7 Stable Formulas 4.7.1 Stable Formulas 4.7.2 Cofinal Stable Rules and Formulas 4.8 Subframization and Stabilization 4.8.1 Subframization 4.8.2 Stabilization References 5 Yankov Characteristic Formulas (An Algebraic Account) 5.1 Introduction 5.2 Background 5.2.1 Basic Definitions 5.2.2 Finitely Presentable Algebras 5.2.3 Splitting 5.3 Independent Sets of Splitting Identities 5.3.1 Quasi-order 5.3.2 Antichains 5.4 Independent Bases 5.4.1 Subvarieties Defined by Splitting Identities 5.4.2 Independent Bases in the Varieties Enjoying the Fsi-Spl Property 5.4.3 Finite Bases in the Varieties Enjoying the Fsi-Spl Property 5.4.4 Reduced Bases 5.5 Varieties with a TD Term 5.5.1 Definition of the TD Term 5.5.2 Definition and Properties of Characteristic Identities 5.5.3 Independent Bases in Subvarieties Generated by Finite Algebras 5.5.4 A Note on Iterated Splitting 5.6 Final Remarks 5.6.1 From Characteristic Identities to Characteristic Rules 5.6.2 From Characteristic Quasi-identities to Characteristic Implications 5.6.3 From Algebras to Complete Algebras 5.6.4 From Finite Algebras to Infinite Algebras References 6 The Invariance Modality 6.1 Introduction 6.2 Preliminaries 6.2.1 Transformational and Invariance Models 6.3 Classical Models and Ultrapowers 6.4 Strong Completeness Theorems 6.4.1 Invariance Models 6.5 Conclusions References 7 The Lattice NExtS41 as Composed of Replicas of NExtInt, and Beyond 7.1 Introduction 7.2 Preliminaries 7.3 The Interval [M0,S1] 7.4 The Interval [S4,S5] 7.5 The Interval [S4,Grz] 7.6 Sublattices mathcalS, mathcalR, and mathcalT 7.7 Mathematical Remarks 7.8 Philosophical Remarks 7.9 Appendix References 8 An Application of the Yankov Characteristic Formulas 8.1 Introduction 8.2 Intuitionistic Propositional Logic 8.3 Heyting Algebras and Yankov's Characteristic Formulas 8.4 Medvedev Logic 8.5 Propositional Logic of Realizability 8.6 Realizability and Medvedev Logic References 9 A Note on Disjunction and Existence Properties in Predicate Extensions of Intuitionistic Logic—An Application of Jankov Formulas to Predicate Logics 9.1 Introduction 9.2 Preliminaries 9.3 Modified Jankov Formulas—Learning Jankov's Technique 9.3.1 Heyting Algebras and Jankov Formulas 9.3.2 Modified Jankov Formulas for PEI's Without EP 9.4 Modified Jankov Formulas Preserve DP—Learning Minari's and Nakamura's Idea 9.4.1 Kripke Frame Semantics 9.4.2 Pointed Joins of Kripke-Frame Models 9.5 Strongly Independent Sequence of Modified Jankov Formulas—Jankov's Method for Predicate Logics 9.5.1 Special Algebraic Kripke Sheaves 9.5.2 Toolkit for normal upper OmegaΩ-Brooms 9.5.3 Proofs of Lemma9.9 and the Main Theorem 9.6 Concluding Remarks References Part II History and Philosophy of Mathematics 10 On V. A. Yankov's Contribution to the History of Foundations of Mathematics 10.1 Introduction 10.2 Logic and Foundations of Mathematics in Russia and the Soviet Union and the Rise of Constructive Mathematics 10.3 Yankov's Contribution to the History of Constructive Mathematics 10.4 Markov's Philosophy of Constructive Mathematics 10.4.1 Mathematical Objects 10.4.2 The Infinite 10.4.3 Mathematical Existence 10.4.4 Normal Algorithms 10.4.5 Church Thesis 10.4.6 The Concept of Number and the Continuum 10.4.7 Constructive Mathematics is a Technological Science 10.5 Yankov on Esenin-Vol'pin's Ultra-Intuitionism 10.5.1 On the Concept of Natural Numbers and ``Factual (Practical) Realizability'' 10.5.2 On the Ultra-Intuitionistic Program of Foundations of Mathematics 10.5.3 Esenin-Vol'pin's Works on Modal and Deontic Logics 10.6 Conclusion References 11 On V. A. Yankov's Existential Interpretation of the Early Greek Philosophy. The Case of Heraclitus 11.1 Introduction 11.2 A General Outline of V.A. Yankov's Interpretation of Early Greek Philosophy 11.3 On the Ontological Essence of Early Greek Philosophy 11.4 On the Existential Ideas in the Early Greek Philosophy 11.5 On the History of Existential Interpretations of the Early Greek Philosophy 11.6 The Complexity of the Interpretation of Heraclitus 11.7 V.A. Yankov on the Traditional Interpretation of Heraclitus 11.8 Yankov's Predecessors About Heraclitus' Existential Ideas 11.9 The Existential Dimension of the Doctrine of Logos 11.10 Conclusion References 12 On V. A. Yankov's Hypothesis of the Rise of Greek Mathematics 12.1 On Yankov's Motivation to Study the Rise of Rational Thinking 12.2 Outline of Yankov's Hypothesis of the Rise of Greek Mathematics 12.3 An appreciation of Yankov's Hypothesis 12.4 In Lieu of a Conclusion References Index