V.A. Yankov on non-classical logics, history and philosophy of mathematics / / edited by Alex Citkin, Ioannis M. Vandoulakis

• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (319 pages)
• Disciplina: 780
• Soggetto topico: Mathematics - Philosophy; Proposició (Lògica); Filosofia de la matemàtica
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): CitkinAlex; VandoulakisIoannis M.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: V.A. Yankov on non-classical logics, history and philosophy of mathematics
• ISBN: 3-031-06843-2
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996499871403316
• Lo trovi qui: Univ. di Salerno
• Serie: Outstanding Contributions to Logic