R-Calculus, II: Many-Valued Logics

✍ Scribed by Wei Li, Yuefei Sui

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 281
• Series: Perspectives in Formal Induction, Revision and Evolution
• Category: Library

✦ Synopsis

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.  

✦ Table of Contents

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