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Deductive Systems in Traditional and Modern Logic
| Deductive Systems in Traditional and Modern Logic |
| Autore | Wybraniec-Skardowska Urszula |
| Pubbl/distr/stampa | Basel, Switzerland, : MDPI - Multidisciplinary Digital Publishing Institute, 2020 |
| Descrizione fisica | 1 electronic resource (298 p.) |
| Soggetto topico |
Research & information: general
Mathematics & science |
| Soggetto non controllato |
quine
logic ontology multiple conclusion rule disjunction property metadisjunction axiomatizations of arithmetic of natural and integers numbers second-order theories Peano's axioms Wilkosz's axioms axioms of integer arithmetic modeled on Peano and Wilkosz axioms equivalent axiomatizations metalogic categoricity independence consistency logic of typical and atypical instances (LTA) logic of determination of objects (LDO) quasi topology structure (QTS) concept object typical object atypical object lattice filter ideal discussive logics the smallest discussive logic discussive operators seriality accessibility relation Kotas' method modal logic deontic logic ontology of situations semantics of law formal theory of law Wittgenstein Wolniewicz non-Fregean logic identity connective sentential calculus with identity situational semantics deduction (dual) tableau Gentzen system deductive refutability refutation systems hybrid deduction-refutation rules derivative hybrid rules soundness completeness natural deduction meta-proof theory synthetic tableaux principle of bivalence cut first-order theory universal axiom Peano's axiomatics of natural numbers Leśniewski's elementary ontology Frege's predication scheme Frege's Zahl-Anzahl distinction term logic Franz Brentano Lewis Carroll logic trees logic diagrams paraconsistent logic paraconsistency Sette's calculus the law of explosion the principle of ex contradictione sequitur quodlibet semantic tree distribution Aristotle's logic syllogistic Jan Łukasiewicz axiomatic system axiomatic refutation temporal logic intuitionistic logic minimal system knowledge sequent-type calculi nonmonotonic logics default logic rejection systems Kripke models logics of evidence and truth |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Record Nr. | UNINA-9910557449803321 |
Wybraniec-Skardowska Urszula
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| Basel, Switzerland, : MDPI - Multidisciplinary Digital Publishing Institute, 2020 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
V.A. Yankov on non-classical logics, history and philosophy of mathematics / / edited by Alex Citkin, Ioannis M. Vandoulakis
| V.A. Yankov on non-classical logics, history and philosophy of mathematics / / edited by Alex Citkin, Ioannis M. Vandoulakis |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (319 pages) |
| Disciplina | 780 |
| Collana | Outstanding Contributions to Logic |
| Soggetto topico |
Mathematics - Philosophy
Proposició (Lògica) Filosofia de la matemàtica |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-06843-2 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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. |
| Record Nr. | UNISA-996499871403316 |
| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
V.A. Yankov on non-classical logics, history and philosophy of mathematics / / edited by Alex Citkin, Ioannis M. Vandoulakis
| V.A. Yankov on non-classical logics, history and philosophy of mathematics / / edited by Alex Citkin, Ioannis M. Vandoulakis |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (319 pages) |
| Disciplina | 780 |
| Collana | Outstanding Contributions to Logic |
| Soggetto topico |
Mathematics - Philosophy
Proposició (Lògica) Filosofia de la matemàtica |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-06843-2 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| 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. |
| Record Nr. | UNINA-9910629274503321 |
| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||