Description 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.
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 ΞΎ
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.
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