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A companion to modal logic. (English) Zbl 0625.03005

University Paperbacks, 867. London-New York: Methuen. XVII, 203 p. (1984).
This book is best seen as a sequel to the authors’ earlier book: An introduction to modal logic (1968; Zbl 0205.005). It concentrates on recent developments of modal logic which concern questions about general properties of modal systems. The choice of systems is confined to those with one necessity operator, and among those only so called ‘normal’ systems.
A model of a modal system is a triple \(<W,R,V>\), in which W is a set of ‘worlds’, R is an ‘accessibility’ relation defined over W, and V is a truth-value assignment. A modal system is characterized by a certain class of models if the theorems of the system are precisely the formulas valid in every model in that class. One of the first results in the book is that every normal modal system is characterized by a single model called the canonical model. A large number of modal systems are characterized by ‘translating’ their axioms into formulas specifying certain conditions for the relation R.
A frame is the \(<W,R>\) part of a model. A modal system is characterized by a certain class of frames if its theorems are precisely the formulas that are valid on every frame in that class. Although every normal system is characterized by a class of models, e.g. its canonical model, it is proved that there are normal modal systems which are not characterized by any class of frames at all. Several further results generated by the distinction between models and frames are proved.
A system if said to have the finite model property if it is characterized by a class of finite models. This property is investigated for a number of system and, among other things, it is shown that there are normal systems which lack the finite model property.
[See also the review of the authors’ paper below (Zbl 0625.03006).]
Reviewer: P.Gärdenfors

MSC:

03B45 Modal logic (including the logic of norms)
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
03C52 Properties of classes of models
03C35 Categoricity and completeness of theories