Dick de Jongh on Intuitionistic and Provability Logics

• Autore: Bezhanishvili Nick
• Pubblicazione: Cham : , : Springer International Publishing AG, , 2024; ©2024
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (386 pages)
• Disciplina: 160
• Altri autori: IemhoffRosalie YangFan
• Nota di contenuto: Intro -- Preface -- Contents -- Contributors -- 1 Notes on My Scientific Life -- References -- 2 Lewisian Fixed Points I: Two Incomparable Constructions -- 2.1 Introduction -- 2.1.1 Dick de Jongh and Explicit Fixed Points -- 2.1.2 Explicit Fixed Points in More Detail -- 2.1.3 Our Contributions -- 2.2 Basics -- 2.3 Semantics -- 2.3.1 Algebraic Semantics -- 2.3.2 Kripke Semantics -- 2.4 Arithmetical Interpretations -- 2.5 Extension Stability -- 2.6 Löb-Lewis Logic Meets de Jongh and Visser -- 2.7 Löb-Lewis Logic Meets de Jongh and Sambin -- 2.8 The Join of JV and JS -- 2.9 Four Salient Fixed Points: Subtheories of sans serif upper J sans serif upper S Superscript flatJS -- 2.9.1 The Principle sans serif upper W Superscript asteriskWast -- 2.9.2 The Principle sans serif upper W Subscript degreesW° -- 2.9.3 The Principle sans serif upper L Subscript degreesL° -- 2.9.4 The Principle sans serif 4 Subscript degrees4° -- 2.10 Correspondence and Non-conservativity -- 2.11 Explicit Fixed Points and the Beth Property -- 2.12 Conclusions and Future Work -- References -- 3 An Abstract Look at the Fixed-Point Theorem for Provability Logic -- 3.1 Introduction -- 3.2 The Fixed-Point Theorem for Provability Logic -- 3.3 De Jongh's Generalized Quantifier Analysis -- 3.4 Fixed-Points on Generalized Well-Founded Orders -- 3.5 Fixed-Point Theorems for Generalized Modal Languages -- 3.6 Fixed-Point Theorems for Non-persistent Modalities -- 3.7 Further Directions -- 3.8 Conclusion -- References -- 4 The normal upper Sigma 1Σ1-Provability Logic of HA Revisited -- 4.1 Preface -- 4.2 Introduction -- 4.3 Basic Definitions -- 4.4 backslash NNILNNIL Propositions -- 4.4.1 The backslash NNILNNIL-Algorithm -- 4.4.2 The Binary Relation backslash brt -- 4.4.3 backslash NNIL Superscript white medium squareNNIL Propositions -- 4.5 Leivant's Translation. ; 4.6 Gödel's Translation and Heyting's Normal Form -- 4.7 Axiom Schemas and Theories -- 4.8 Embedding of backslash llespiH'σ in backslash iglcaiGLCa -- References -- 5 An Overview of Verbrugge Semantics, a.k.a. Generalised Veltman Semantics -- 5.1 Introduction -- 5.1.1 The Beginnings -- 5.1.2 Relation to Meta-Mathematics -- 5.1.3 Abstract Semantics -- 5.1.4 Decidability and Complexity -- 5.1.5 Many Classical Results Carry Over to Interpretability Logics -- 5.1.6 Proof Theory -- 5.2 Logics for Interpretability -- 5.2.1 Modal Interpretability Logics -- 5.2.2 Arithmetical Semantics -- 5.2.3 Relational Semantics -- 5.3 Verbrugge Semantics -- 5.3.1 Replacing Worlds by Sets of Worlds -- 5.3.2 On Quasi-Transitivity -- 5.3.3 Veltman Semantics Versus Verbrugge Semantics -- 5.4 Verbrugge Semantics for Separating Systems -- 5.4.1 Principles and Veltman Models -- 5.4.2 Generalised Frame Conditions and Independence -- 5.5 Modal Completeness: Preliminaries -- 5.5.1 Overview of Approaches -- 5.5.2 Completeness w.r.t. Generalised Semantics -- 5.5.3 A Note on Verbrugge Semantics and Labelling -- 5.6 Modal Completeness of Various Systems -- 5.6.1 The Logic ILM -- 5.6.2 The Logic ILM0 -- 5.6.3 The Logics ILP, ILP0 and ILR -- 5.6.4 The Logics ILW and ILW* -- 5.6.5 The Logic ILWR -- 5.6.6 Logics Below IL -- 5.7 Bisimulations and Filtrations -- 5.7.1 Bisimulations -- 5.7.2 Filtrations and the Finite Model Property -- 5.8 Hierarchies and Frame Conditions -- 5.8.1 A Broad Series of Principles -- 5.8.2 Frame Conditions for VS -- References -- 6 Deciding Dependence in Logic and Algebra -- 6.1 Introduction -- 6.2 Equational Consequence and Free Algebras -- 6.3 An Algebraic Theory of Dependence -- 6.4 Deciding Dependence -- 6.5 Dependence and Minimal Provability -- 6.6 Open Problems -- References -- 7 About the Unification Types of Modal Logics -- 7.1 Introduction. ; 7.2 Modal Logics -- 7.2.1 Formulas -- 7.2.2 Modal Logics -- 7.2.3 Substitutions -- 7.3 Unifiability -- 7.3.1 Unifiers -- 7.3.2 Bases -- 7.3.3 Types of Formulas -- 7.3.4 Types of Logics -- 7.3.5 Reduced Formulas -- 7.4 Projective, Transparent, Directed and Reasonable Unification -- 7.4.1 Projective Unification -- 7.4.2 Transparent Unification -- 7.4.3 Directed Unification -- 7.4.4 Reasonable Unification -- 7.5 Some Unitary and Finitary Modal Logics -- 7.5.1 Preamble -- 7.5.2 Extensions of bold upper S Baseline 5S5 -- 7.5.3 Extensions of bold upper K Baseline 45K45 -- 7.5.4 Some Non-transitive Modal Logics -- 7.6 Some Nullary Modal Logics -- References -- 8 Proof Theory for Lax Logic -- 8.1 Introduction -- 8.2 Preliminaries -- 8.2.1 The Calculi sans serif upper G Baseline sans-serif 3 sans serif i sans serif upper L sans serif upper LG3iLL and sans serif upper G Baseline sans-serif 4 sans serif i sans serif upper L sans serif upper LG4iLL -- 8.2.2 Properties of Calculi -- 8.2.3 Structural Rules -- 8.3 Cut-Elimination -- 8.4 Equivalence of G3iLL and G4iLL -- 8.5 Interpolation -- 8.6 Uniform Interpolation -- 8.6.1 Uniform Interpolants for Formulas -- 8.6.2 Reductive Orders and Rank -- 8.6.3 Interpolant Assignments -- 8.6.4 Uniform Interpolants for Sequents -- 8.6.5 The Inductive Properties -- 8.6.6 Interpolant Assignment for PLL -- 8.6.7 Soundness of the Interpolant Assignment -- 8.6.8 Uniform Interpolation for Lax Logic -- 8.7 Conclusion -- References -- 9 Intermediate Logics in the Setting of Team Semantics -- 9.1 Introduction -- 9.2 Intuitionistic Logic and de Jongh Formulas -- 9.3 Team-Based Intuitionistic Logic and de Jongh Formulas -- 9.4 Generalized Team Intuitionistic Kripke Semantics -- 9.4.1 The Power Set Model and Intuitionistic Modal Logic -- 9.4.2 Generalized Team Intuitionistic Kripke Semantics. ; 9.4.3 Recovering the Standard tIPC, and Distributivity -- 9.5 Concluding Remarks and Open Problems -- References -- 10 Well Partial Orders -- 10.1 Introduction -- 10.2 My Personal View on Well Partial Orderings -- 10.3 Well Partial Orderings: Basic Material -- 10.3.1 Operations on Well Partial Orders -- 10.4 Finite Trees -- 10.5 Well Partial Orderings and Their Maximal Order Types -- 10.6 Friedman Style Miniaturizations and Their Phase Transitions -- 10.6.1 Proof of the Upper Bound -- 10.6.2 Proof of the Lower Bound -- References -- 11 Learning to Act and Observe in Partially Observable Domains -- 11.1 Introduction -- 11.2 Dynamic Epistemic Logic (DEL) -- 11.3 Transition Systems and Partially Observable Domains -- 11.4 Learning Explicit Domain Knowledge -- 11.4.1 Compatibility Domain -- 11.4.2 Behavioural Correctness and Learners -- 11.4.3 A Behaviourally Correct Learner of Explicit Knowledge -- 11.5 Extending Learning Beyond Explicit Knowledge -- 11.6 Learning Implicit Domain Knowledge -- 11.6.1 Implicit Knowledge and Behavioural Equivalence -- 11.6.2 Behavioural Equivalence Domain -- 11.6.3 Behavioural Correctness and Learnability -- 11.6.4 A Behaviourally Correct Learner of Implicit Knowledge -- 11.7 Related Work -- 11.8 Final Remarks and Future Work -- References -- 12 Axiomatizing Origami Planes -- 12.1 Introduction -- 12.2 Background on Axiomatic Geometry -- 12.3 Background on Interpretations -- 12.4 Background on Coordinatization -- 12.5 Metric Wu Planes and Pythagorean Fields -- 12.6 Undecidability -- 12.7 Logical Huzita-Justin Axioms -- 12.8 Ordered Metric Wu Planes and Pythagorean Fields -- 12.9 Euclidean and Vieta Fields and Origami Axioms -- 12.10 Discussion -- 12.11 Special Acknowledgements -- References -- 13 The Complete Bibliography of Dick de Jongh.
• Titolo autorizzato: Dick de Jongh on Intuitionistic and Provability Logics
• ISBN: 3-031-47921-1
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910878055403321
• Lo trovi qui: Univ. Federico II
• Serie: Outstanding Contributions to Logic Series