# Kripke’s Worlds

## An Introduction to Modal Logics via Tableaux

• Olivier Gasquet
• Andreas Herzig
• Bilal Said
• François Schwarzentruber
Book

Part of the Studies in Universal Logic book series (SUL)

1. Front Matter
Pages I-XV
2. Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber
Pages 1-21
3. Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber
Pages 23-51
4. Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber
Pages 53-85
5. Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber
Pages 87-123
6. Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber
Pages 125-146
7. Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber
Pages 147-156
8. Olivier Gasquet, Andreas Herzig, Bilal Said, François Schwarzentruber
Pages 157-189
9. Back Matter
Pages 191-198

### Introduction

Possible worlds models were introduced by Saul Kripke in the early 1960s. Basically, a possible worlds 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 a 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.

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### Keywords

epistemic logics modal logics tableau method temporal logics

#### Authors and affiliations

• Olivier Gasquet
• 1
• Andreas Herzig
• 2
• Bilal Said
• 3
• François Schwarzentruber
• 4
1. 1.Institut de Recherche en Informatique de Toulouse (IRIT)Université Paul SabatierToulouseFrance
2. 2.Institut de Recherche en Informatique de Toulouse (IRIT)Université Paul SabatierToulouseFrance
3. 3.Institut de Recherche en Informatique de Toulouse (IRIT)Université Paul SabatierToulouseFrance
4. 4.Institut de Recherche en Informatique de Toulouse (IRIT)Université Paul SabatierToulouseFrance

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