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Table of Contents
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
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