R-Calculus, II: Many-Valued Logics

Description

This second volume of the book series shows R-calculus is a combination of one monotonic tableau proof system and one non-monotonic one. The R-calculus is a Gentzen-type deduction system which is non-monotonic, and is a concrete belief revision operator which is proved to satisfy the AGM postulates and the DP postulates. It discusses the algebraical and logical properties of tableau proof systems and R-calculi in many-valued logics. This book offers a rich blend of theory and practice. It is suitable for students, researchers and practitioners in the field of logic. Also it is very useful for all those who are interested in data, digitization and correctness and consistency of information, in modal logics, non monotonic logics, decidable/undecidable logics, logic programming, description logics, default logics and semantic inheritance networks.   Preface to the Series Preface Contents 1 Introduction 1.1 R-Calculus 1.2 Many-Valued Logics 1.3 Contents in the First Volume 1.4 Contents in This Volume 1.5 Notations References 2 R-Calculus for PL 2.1 Basic Definitions 2.2 Monotonic Tableau Proof Systems 2.2.1 Tableau Proof System Tf 2.2.2 Tableau Proof System Tt 2.3 Nonmonotonic Tableau Proof Systems 2.3.1 Tableau Proof System St 2.3.2 Tableau Proof System Sf 2.4 R-Calculi 2.4.1 R-Calculus Rt 2.4.2 R-Calculus Rf 2.5 Projecting R-Calculi to Tableau Proof Systems 2.6 Notes References 3 R-Calculus for Description Logic 3.1 Basic Definitions 3.2 Monotonic Tableau Proof Systems 3.2.1 Tableau Proof System Tt 3.2.2 Tableau Proof System Tf 3.3 Nonmonotonic Tableau Proof Systems 3.3.1 Tableau Proof System St 3.3.2 Tableau Proof System Sf 3.4 R-Calculi 3.4.1 R-Calculus Rt 3.4.2 R-Calculus Rf 3.5 Projecting R-Calculi to Tableau Proof Systems References 4 R-Calculus for L3-Valued PL 4.1 Basic Definitions 4.2 Monotonic Tableau Proof Systems 4.2.1 Tableau Proof System Tt 4.2.2 Tableau Proof System Tm 4.2.3 Tableau Proof System Tf 4.3 Nonmonotonic Tableau Proof Systems 4.3.1 Tableau Proof System St 4.3.2 Tableau Proof System Sm 4.3.3 Tableau Proof System Sf 4.4 R-Calculi 4.4.1 R-Calculus Rt 4.4.2 R-Calculus Rm 4.4.3 R-Calculus Rf 4.5 Satisfiability and Unsatisfiability 4.5.1 t-Satisfiability and t-Unsatisfiability 4.5.2 m-Satisfiability and m-Unsatisfiability 4.5.3 f-Satisfiability and f-Unsatisfiability 4.6 Projecting R-Calculi to Tableau Proof Systems 4.7 Notes References 5 R-Calculus for L3-Valued PL, II 5.1 Monotonic Tableau Proof Systems 5.1.1 Tableau Proof System Tt 5.1.2 Tableau Proof System Tm 5.1.3 Tableau Proof System Tf 5.2 Nonmonotonic Tableau Proof Systems 5.2.1 Tableau Proof System St 5.2.2 Tableau Proof System Sm 5.2.3 Tableau Proof System Sf 5.3 R-Calculi 5.3.1 R-Calculus Rt 5.3.2 R-Calculus Rm 5.3.3 R-Calculus Rf 5.4 Validity and Invalidity 5.4.1 t-Invalidity and t-Validity 5.4.2 m-Invalidity and m-Validity 5.4.3 f-Invalidity and f-Validity 5.5 Projecting R-Calculi to Tableau Proof Systems References 6 R-Calculus for B22-Valued PL 6.1 Basic Definitions 6.2 Monotonic Tableau Proof Systems 6.2.1 Tableau Proof System Tt 6.2.2 Tableau Proof System T 6.2.3 Tableau Proof System Tperp 6.2.4 Tableau Proof System Tf 6.3 Nonmonotonic Tableau Proof Systems 6.3.1 Tableau Proof System St 6.3.2 Tableau Proof System S 6.3.3 Tableau Proof System Sperp 6.3.4 Tableau Proof System Sf 6.4 R-Calculi 6.4.1 R-Calculus Rt 6.4.2 R-Calculus R 6.4.3 R-Calculus Rperp 6.4.4 R-Calculus Rf 6.5 Projecting R-Calculi to Tableau Proof Systems 6.6 Notes References 7 R-Calculus for B22-Valued PL,II 7.1 Monotonic Tableau Proof Systems Tast1ast2 7.1.1 Tableau Proof System Tt 7.1.2 Tableau Proof System Ttperp 7.2 Tableau Proof Systems Tast1ast2 7.2.1 Tableau Proof System Tt 7.2.2 Tableau Proof System Ttperp 7.3 Nonmonotonic Tableau Proof Systems 7.3.1 Tableau Proof System St 7.3.2 Tableau Proof System Stperp 7.3.3 Tableau Proof System St 7.3.4 Tableau Proof System Stperp 7.4 R-Calculi 7.4.1 R-Calculus Rt 7.4.2 R-Calculus Rtperp 7.4.3 R-Calculus Rf 7.4.4 R-Calculus Rfperp 7.5 Projecting R-Calculi to Tableau Proof Systems 7.6 Notes References 8 Co-R-Calculus for PL 8.1 Co-R-calculi in Propositional Logic 8.1.1 Co-R-Calculus Ut 8.1.2 Co-R-Calculus Uf 8.2 Co-R-Calculi in L3-Valued PL 8.2.1 Co-R-Calculus Ut 8.2.2 Co-R-Calculus Um 8.2.3 Co-R-Calculus Uf 8.3 Co-R-Calculi in B22-Valued Propositional Logic 8.3.1 Co-R-Calculus Ut 8.4 Notes References 9 Multisequents and Hypersequents 9.1 Tableau Proof Systems 9.1.1 Tableau-Typed Proof System Tt 9.1.2 Tableau Proof System Tt 9.2 Sequents in L3-Valued Propositional Logic 9.2.1 Gentzen Deduction System for ΔΣ 9.2.2 Gentzen Deduction System for ΘΞ 9.2.3 Gentzen Deduction System for ΓΠ 9.3 Multisequents in L3-Valued PL 9.3.1 Multisequents 9.3.2 Co-Multisequents 9.4 Hypersequents in L3-Valued PL 9.5 Notes References 10 Product of Two R-Calculi 10.1 Tableau Proof Systems in Modalized PL 10.1.1 Monotonic Tableau Proof Systems 10.1.2 Nonmonotonic Tableau Proof Systems 10.2 Product of B2-Valued PLs 10.2.1 Tableau Proof System P4t 10.2.2 Tableau Proof System P4t 10.2.3 Tableau Proof System Qt 10.3 Product of Two R-Calculi 10.3.1 R-Calculi Rt2 and Rf2 10.3.2 R-Calculus U4t 10.4 Notes References 11 Sum of Two R-Calculi 11.1 The Sum with One Common Element 11.1.1 B2[f,m]oplusB2[m,t] 11.1.2 Operators on Tableau Proof Systems 11.1.3 Sum of Tableau Proof Systems 11.1.4 R-Calculi 11.1.5 R-Calculi in PL 11.1.6 R-Calculi in L3-Valued PL 11.2 The Sum Without Common Elements 11.2.1 L4-Valued PL 11.2.2 Equivalences 11.2.3 Tableau Proof System Tt4 11.2.4 Tableau Proof System T4 11.2.5 Tableau Proof System Tperp4 11.2.6 Tableau Proof System Tf4 11.2.7 Sum of Tableau Proof Systems: Tt4=sim2(Tt2)oplusTt2 11.2.8 Sum of Tableau Proof Systems: Tt4=Tperp2oplusTt2 11.2.9 Sum of Tableau Proof Systems: St4=Sperp2oplusSt2 11.2.10 Sum of R-Calculi: Rt4equivRperp2oplusRt2 11.3 Notes References