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2023 | Book

Introduction Read chapter Read first chapter

R-Calculus, IV: Propositional Logic

Authors: Wei Li, Yuefei Sui

Publisher: Springer Nature Singapore

Book Series : Perspectives in Formal Induction, Revision and Evolution

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This fourth volume of the book series combines propositional logic and R-calculus for a new point of view to consider belief revision. It gives the R-calculi for propositional logic, description logics, propositional modal logic, logic programming, ⇝-propositional logic, semantic networks, and three-valued logic, etc.. Applications of R-calculus in logic of supersequents are also given.

This book offers a rich blend of theory and practice. It is suitable for students, researchers and practitioners in the field of logic.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
Propositional logic is basic, based on which we developed first-order logic and modal logic, which compose of classical logics.
Wei Li, Yuefei Sui
Chapter 2. R-Calculus for Simplified Propositional Logics
Abstract
Assume that the logical language of propositional logic contains three logical symbols: \(\lnot , \wedge , \vee .\) There is a classical Gentzen deduction system \(\textbf{G}\) for propositional logic which is sound and complete with respect to classical semantics (Li 2010; Mendelson 1964; Takeuti 1987).
Wei Li, Yuefei Sui
Chapter 3. R-Calculi for Tableau/Gentzen Deduction Systems
Abstract
A logic consists of a logical language, syntax and semantics, where the logical language specifies what symbols can be used in the logic, and the symbols are decomposed into two classes: logical and nonlogical, where the logical symbols are the ones used in each language of the logic, and the nonlogical symbols are those which are different for different logical languages of the logic; the syntax specifies what strings of symbols are meaningful (formulas) in logic, and the semantics specifies the truth-values of formulas under an assignment (or a model).
Wei Li, Yuefei Sui
Chapter 4. R-Calculi
Abstract
Let \(Q_1,Q_2\in \{\textbf{A},\textbf{E}\}\). We consider \(\textbf{G}^{Q_1Q_2}\)-valid sequents and \(\textbf{G}_{Q_1Q_2}\)-valid co-sequents, Gentzen deduction systems \(\textbf{G}^{Q_1Q_2}, \textbf{G}_{Q_1Q_2}\), and corresponding R-calculi \(\textbf{R}^{Q_1Q_2},\textbf{R}_{Q_1Q_2}\).
Wei Li, Yuefei Sui
Chapter 5. R-Calculi
Abstract
We consider sequents of form \(\textbf{G}^{Q_1iQ_2j}\) and co-sequents of form \(\textbf{G}_{Q_1iQ_2j}\) (Li 2010; Takeuti and Barwise 1987) and corresponding R-calculi \(\textbf{R}^{Q_1iQ_2j}\) and \(\textbf{R}_{Q_1iQ_2j}\) (Li (2007)), where \(Q_1, Q_2\in \mathbf{A ,\textbf{E} }\) and \(i,j\in {0,1}.\)
Wei Li, Yuefei Sui
Chapter 6. R-Calculi:
Abstract
Let \(Q_1,Q_2\in \{\textbf{A},\textbf{E}\}, i,j\in \{0, 1\},\) and \(Y_1,Y_2\in \{\textbf{R},\textbf{Q},\textbf{P}\}.\)
Wei Li, Yuefei Sui
Chapter 7. R-Calculi for Supersequents
Abstract
A sequent \(\Gamma \Rightarrow \Delta \) is valid if for any assignment v, v satisfying each formula in \(\Delta \) implies v satisfying some formula in \(\Gamma ,\) equivalently, either v satisfies the negation of some formula in \(\Delta \) or v satisfies some formula in \(\Gamma .\)
Wei Li, Yuefei Sui
Chapter 8. R-Calculi for -Propositional Logic
Abstract
By taking \(\lnot \) as a logical connective, in traditional Getzen deduction system, we have the following deduction rules
Wei Li, Yuefei Sui
Metadata
Title
R-Calculus, IV: Propositional Logic
Authors
Wei Li
Yuefei Sui
Copyright Year
2023
Publisher
Springer Nature Singapore
Electronic ISBN
978-981-19-8633-8
Print ISBN
978-981-19-8632-1
DOI
https://doi.org/10.1007/978-981-19-8633-8

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