Studies in Universal Logic

Kripke’s Worlds

An Introduction to Modal Logics via Tableaux

Authors: Gasquet, O., Herzig, A., Said, B., Schwarzentruber, F.

  • Aims at filling the gap between existing introductory and advanced textbooks
  • Introduces to the most important modal logics with multiple modalities from the perspective of the associated reasoning tasks
  • Concentrates on the most general and powerful reasoning method for modal logics: tableaux systems
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  • ISBN 978-3-7643-8504-0
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  • ISBN 978-3-7643-8503-3
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About this book

Possible worlds models were introduced by Saul Kripke in the early 1960s. Basically, a possible world's model is nothing but a graph with labelled nodes and labelled edges. Such graphs provide semantics for various modal logics (alethic, temporal, epistemic and doxastic, dynamic, deontic, description logics) and also turned out useful for other nonclassical logics (intuitionistic, conditional, several paraconsistent and relevant logics). All these logics have been studied intensively in philosophical and mathematical logic and in computer science, and have been applied increasingly in domains such as program semantics, artificial intelligence, and more recently in the semantic web. Additionally, all these logics were also studied proof theoretically. The proof systems for modal logics come in various styles: Hilbert style, natural deduction, sequents, and resolution. However, it is fair to say that the most uniform and most successful such systems are tableaux systems. Given logic and a formula, they allow one to check whether there is a model in that logic. This basically amounts to trying to build a model for the formula by building a tree.

This book follows a more general approach by trying to build a graph, the advantage being that a graph is closer to a Kripke model than a tree. It provides a step-by-step introduction to possible worlds semantics (and by that to modal and other nonclassical logics) via the tableaux method. It is accompanied by a piece of software called LoTREC (www.irit.fr/Lotrec). LoTREC allows to check whether a given formula is true at a given world of a given model and to check whether a given formula is satisfiable in a given logic. The latter can be done immediately if the tableau system for that logic has already been implemented in LoTREC. If this is not yet the case LoTREC offers the possibility to implement a tableau system in a relatively easy way via a simple, graph-based, interactive language.

Reviews

From the reviews:

“This is an excellent book to use –– either as a stand-alone text or with another textbook –– for an introductory undergraduate course in logic addressed to majors in the humanities, social sciences, computer science, or mathematics. … A basic unifying theme of the book is to construct models of possible worlds and to check formula satisfiability using graph-theoretic tableaux systems.” (Russell Jay Hendel, MAA Reviews, February, 2014)

“This book provides an accessible introduction to modal logics indeed. … The book is well written and quite informative … . It can be used as an easy-going introduction for all who are interested in automated reasoning and need some formal tools for playing with modal logics.” (Andrzej Indrzejczak, zbMATH, Vol. 1280, 2014)


Table of contents (7 chapters)

  • Modelling with Graphs

    Gasquet, Olivier (et al.)

    Pages 1-21

  • Talking About Graphs

    Gasquet, Olivier (et al.)

    Pages 23-51

  • The Basics of the Model Construction Method

    Gasquet, Olivier (et al.)

    Pages 53-85

  • Logics with Simple Constraints on Models

    Gasquet, Olivier (et al.)

    Pages 87-123

  • Logics with Transitive Accessibility Relations

    Gasquet, Olivier (et al.)

    Pages 125-146

Buy this book

eBook $54.99
price for USA (gross)
  • ISBN 978-3-7643-8504-0
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $69.99
price for USA
  • ISBN 978-3-7643-8503-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Kripke’s Worlds
Book Subtitle
An Introduction to Modal Logics via Tableaux
Authors
Series Title
Studies in Universal Logic
Copyright
2014
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-7643-8504-0
DOI
10.1007/978-3-7643-8504-0
Softcover ISBN
978-3-7643-8503-3
Series ISSN
2297-0282
Edition Number
1
Number of Pages
XV, 198
Number of Illustrations and Tables
73 b/w illustrations
Topics