Abstract

A comprehensive introduction to modal logic is long overdue and this one has many virtues. It is clearly written and should be accessible to any student who has at least one semester of basic logic and is willing to read carefully and think abstractly. The first part, on modal propositional logic, begins with a summary account of classical propositional logic, the axiomatization of Principia Mathematica being the basis for the development of modal logics throughout the book. The transition to modal logic is nicely motivated by a clear presentation of intuitive notions of modality and the requirements to be included in a modal logic. Thereafter three standard systems of modal propositional logic are developed axiomatically, Feys' system T and Lewis' systems S4 and S5. The semantics for these systems is then developed in the manner of Kripke with some terminological modifications. The explanation of accessibility relations between possible worlds is made especially clear through a helpful analogy with certain sorts of games. Decision procedures and completeness proofs are then developed. A similar pattern of exposition is given to modal predicate logic in Part II, the difference being that Henkin-type methods are used in the completeness proofs. Because of the many philosophical problems raised by modal predicate logic, Part II contains more philosophical discussion than Part I. A discussion of identity and descriptions in modal predicate logic is also included. The third and final part is a survey of modal systems beginning with the Lewis' systems S1-S5 and going on to systems weaker and stronger than the Lewis' systems and others which are independent of them. Systems with alternative primitives and axiomatic bases are also discussed. A final chapter discusses the relation of Boolean algebras to modal logics and brief appendices deal with natural deduction systems of modal logic, systems of entailment, alternative notations and the semantics of Kripke and Hintikka among other topics. There are exercises at the end of each chapter. This summary suggests the merits of this introduction, its clear exposition and the enormous amount of material it brings together and summarizes.--R. H. K.