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Modal logic for philosophers. (English) Zbl 1109.03001
Cambridge: Cambridge University Press (ISBN 0-521-86367-8/hbk; 0-521-68229-0/pbk). xv, 455 p. (2006).
This is a very well written introduction into propositional and first-order modal logic. Its qualification “for philosophers” mainly comes from the fact that the author is very careful in explaining the use of logic as a tool to formalize and discuss problems from applications. And it is the field of philosophy, in a wide sense of that word, which offers the main examples for motivations of formal developments and for application examples.
Propositional and first-order modal logics are developed to very similar extents, both covering about half of the text. In both cases the author offers a natural deduction calculus as well as a tableau calculus, and he gives a method to transfer a closed tableau into a natural deduction proof. Additionally the standard Kripke-style semantics are introduced, and soundness and completeness proofs given, of course for all the main systems “between” K and S5. The semantic entailment relation, however, is considered only for finite lists of premisses, so compactness considerations do not enter the field of topics.
In the first-order case four different versions of Kripke-style semantics are considered, each one for the whole bunch of propositional systems under consideration, always yielding the same entailment relation. Great care is taken to discuss rigid as well as non-rigid terms and thus the problem of transworld identity, and also to discuss the constant-domain versus varying-domain approaches. To cover a wide variety of possibilities, the background for non-modal first-order logic is chosen to be a free one.
As two particularly important applications of this whole machinery the author discusses an interesting modal reading of definite descriptions, as well as the usual de re / de dicto distinction, in the latter case with reference to the notation of \(\lambda\)-abstraction.
This book is a very valuable enlargement of the textbook literature, particularly for the field of first-order modal logics. And it is not only suitable for philosophers; also mathematicians and computer scientists may use it with benefit. It correctly defines all notions, makes clear claims, and proves them in detail.

03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
03B45 Modal logic (including the logic of norms)