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**Modal logic.**
*(English)*
Zbl 0871.03007

Oxford Logic Guides. 35. Oxford: Clarendon Press. xv, 605 p. (1997).

This book presents a rich resource for modern mathematical modal logic, useful both as an advanced textbook and as a source for up-to-date results. Begining with fundamental concepts applied to standard systems, it proceeds with ever-increasing abstraction and generality to sophisticated problems of very current research. The exposition is clear, and numerous examples illustrate the concepts and theorems as they are introduced. The modal logics investigated are all extensions of the propositional calculus \(K(=\) classical logic + the modal postulate \(\square(p_0\to p_1)\to (\square p_0\to\square p_1)\) with the rules modus ponens, substitution, and necessitation: given \(\varphi\), infer \(\square\varphi)\), including its quasi-normal extensions (closed under mp and sub) as well as its normal extensions (closed also under nec). In addition, superintuitionistic logics (non-modal extensions of Heyting’s intuitionistic propositional calculus) are given parallel treatments. Higher-order systems are not discussed.

Unlike other books on modal logic, the emphasis here is on large classes of logics rather than specific systems (except for illustration). These are presented semantically, first in terms of classes of Kripke frames and models, then, to achieve greater generality, in terms of more abstract algebraic structures. These methods are applied to study a number of logical and metalogical properties, such as completeness, decidability and undecidability, finite axiomatizability, finite approximability (the finite frame property), etc.

The book is divided into five parts. Part I (Ch. 1-4) introduces the fundamental concepts and methods of Kripke semantics as standardly applied to various central modal and superintuitionistic logics. Part II (Ch. 5-6) extends those methods to establish completeness and decidability properties for a wide class of logics, and also to establish a number of negative results, logics lacking finite approximability, canonicity, compactness, even Kripke completeness. Since not all modal and superintuitionistic logics are Kripke complete (i.e., characterized by a class of Kripke frames), Part III (Ch. 7-9) develops more adequate semantics, first with modal and pseudo-Boolean algebras, and then with general frames. (By the results of Stone and of Jónsson and Tarski, general frames are shown to be relational representations of corresponding algebras, thus combining the power of algebras with some of the ‘geometrical’ perspicuousness of Kripke frames.) Part IV (Ch. 10-15) applies these structures to investigate various properties such as those mentioned above and other lattice-theoretic and metalogical properties, e.g., Post completeness, interpolation, the disjunction property, etc. Part V (Ch. 16-18) raises a number of algorithmic problems regarding complexity and decidability, including not only the decidability of the logics but also the decidability of their properties. Each chapter ends with a set of exercises, often results established in recent years, and also a set of historical notes giving the backgrounds of the results and methods presented in that chapter. The book concludes with an extensive bibliography, including references to much recent research.

Unlike other books on modal logic, the emphasis here is on large classes of logics rather than specific systems (except for illustration). These are presented semantically, first in terms of classes of Kripke frames and models, then, to achieve greater generality, in terms of more abstract algebraic structures. These methods are applied to study a number of logical and metalogical properties, such as completeness, decidability and undecidability, finite axiomatizability, finite approximability (the finite frame property), etc.

The book is divided into five parts. Part I (Ch. 1-4) introduces the fundamental concepts and methods of Kripke semantics as standardly applied to various central modal and superintuitionistic logics. Part II (Ch. 5-6) extends those methods to establish completeness and decidability properties for a wide class of logics, and also to establish a number of negative results, logics lacking finite approximability, canonicity, compactness, even Kripke completeness. Since not all modal and superintuitionistic logics are Kripke complete (i.e., characterized by a class of Kripke frames), Part III (Ch. 7-9) develops more adequate semantics, first with modal and pseudo-Boolean algebras, and then with general frames. (By the results of Stone and of Jónsson and Tarski, general frames are shown to be relational representations of corresponding algebras, thus combining the power of algebras with some of the ‘geometrical’ perspicuousness of Kripke frames.) Part IV (Ch. 10-15) applies these structures to investigate various properties such as those mentioned above and other lattice-theoretic and metalogical properties, e.g., Post completeness, interpolation, the disjunction property, etc. Part V (Ch. 16-18) raises a number of algorithmic problems regarding complexity and decidability, including not only the decidability of the logics but also the decidability of their properties. Each chapter ends with a set of exercises, often results established in recent years, and also a set of historical notes giving the backgrounds of the results and methods presented in that chapter. The book concludes with an extensive bibliography, including references to much recent research.

Reviewer: L.F.Goble (Salem)

### MSC:

03B45 | Modal logic (including the logic of norms) |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

03B55 | Intermediate logics |