# Tools and Techniques in Modal Logic

**By**

- M. Kracht, Freie Universität Berlin, II. Mathematisches Institut, Berlin, Germany

This book treats modal logic as a theory, with several subtheories, such as completeness theory, correspondence theory, duality theory and transfer theory and is intended as a course in modal logic for students who have had prior contact with modal logic and who wish to study it more deeply. It presupposes training in mathematical or logic. Very little specific knowledge is presupposed, most results which are needed are proved in this book.

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### Book information

- Published: June 1999
- Imprint: NORTH-HOLLAND
- ISBN: 978-0-444-50055-7

### Reviews

....As a whole, the book deserves attention from specialists in the field....The author has succeeded in showing that the use of mathematical (algebraic and topological) tools provides an excellent means for deeper understanding of the theory.

Bulletin of the Symbolic Logic

### Table of Contents

About this Book. Overview. **Part 1. The Fundamentals.** Chapter 1. Algebra, Logic and Deduction. Basic facts and structures. Propositional languages. Algebraic constructions. General logic. Completeness of matrix semantics. Properties of logics. Boolean logic. Some notes on computation and complexity. Chapter 2. Fundamentals of Modal Logic I. Syntax of modal logics. Modal algebras. Kripke-Frames and frames. Frame constructions I. Some important modal logics. Decidability and finite modal property. Normal forms. The Lindenbaum-Tarski construction. The lattices of normal and quasi-normal logics. Chapter 3. Fundamentals of Modal Logic II. Local and global consequense relations. Completeness, correspondence and persistence. Frame constructions II. Weakly transitive logics I. Subframe logics. Constructive reduction. Interpolation and beth theorems. Tableau calculi and interpolation. Modal consequence relations. **Part 2. The General Theory of Modal Logic.** Chapter 4. Universal Algebra and Duality Theory. More on products. Varieties, logics and equationally definable classes. Weakly transitive logics II. Stone representation and duality. Adjoint functors and natural transformations. Generalized frames and modal duality theory. Frame constructions III. Free algebras, canonical frames and descriptive frames. Algebraic characterizations of interpolation. Chater 5. Definability and Correspondence. Motivation. The languages of description. Frame correspondence - an example. The basic calculus of internal descriptions. Sahlqvist's theorem. Elementary Sahlqvist conditions. Preservation classes. Some results from model theory. Chapter 6. Reducing Polymodal Logic to Monomodal Logic. Interpretations and simulations. Some preliminary results. The fundamental construction. A general theorem for consistency reduction. More preservation results. Thomason simulations. Properties of the simulation. Simulation and transfer - some generalizations. Chapter 7. Lattices of Modal Logics. The relevance of studying lattices of logics. Splittings and other lattice concepts. Irreducible and prime logics. Duality theory for upper continuous lattices. Some consequences of the duality theory. Properties of logical calculi and related lattice properties. Splittings of the lattices of modal logics and completeness. Blok's alternative. The lattice of tense logics. **Part 3. Case Studies.** Chapter 8. Extensions of **K4**. The global structure of &xgr;**K4**. The structure of finitely generated **K4**-frames. The selection procedure. Refutation patterns. Embeddability patterns and the elementarity of logics. Logics of finite width I. Logics of finite width II. Bounded properties and precomplete logics above **S4**. Logics of finite tightness. Chapter 9. Logics of Bounded Alternativity. The logics containing **K.alt**1. Polymodal logics with quasi-functional operators. Colourings and decolourings. Decidability of logics. Decidability of properties of logics I. Decidability of properties of logics II. Chapter 10. Dynamic Logic. **PDL** - A calculus of compound modalities. Axiomatizing **PDL** The finite model property. Regular languages. An evaluation procedure. The unanswered question. The logic of finite computations. Index. Bibliography.