First Steps in Modal Logic Review

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First Steps in Modal Logic ReviewThis text aims to be a mathematically rigorous exposition of modal logic for primarily computer scientists or mathematicians and secondarily philosophers. The focus is semantical, using modal languages to help understand labeled transition structures (i.e. generalized Kripke structures), rather than vice versa.
A minor quibble: the notation is annoyingly nonstandard. I found it easy to get used to and in fact it may aid in distinguishing between the accessibility predicate (of the language) and the accessibility relation (of the structure). This is likely never a problem however. Another minor annoyance was the use of 'Th(x)' for the provability predicate of Peano arithmetic (in the Lob Logic chapter) and its use in previous chapters for the theory of a class of models. Finally, at rare times a formula will fall off the page and is hence unreadable.
A major aggravation: the book has no index! Yet there is a full bibliography and answers to some exercises. Fortunately the ToC is fairly detailed, but it is certainly no substitute for an index.
On the good side, I found the layout and pace very friendly. The proofs are easy to follow and the writing style is clear. The writing style is, however, condescending at times to both the reader and others working in the field. (The Preface begins by knocking essentially all the texts on the market at the time. Such insults recur throughout the text.) The reader is sometimes treated as if he were an idiot.
What's covered? The usual normal modal logics, some of which lack either the finite model property (fmp) or canonicity. These "negative" results are a nice addition to an introductory text because they are often left out of others. Likewise for bisimulations also covered in this text. Some temporal polymodal systems are introduced to motivate the reason for going general with labeled transition structures rather than the usual monomodal structures. There is also an interesting polymodal system SLL of Segerberg (which is not canonical but has the fmp) having its own chapter.
An underlying theme is proving correspondence results, canonicity, fmp, and decidability. For this end bisimulations, morphisms, and filtrations are used. A generalized version of the (i,j,m,n)-convergence/confluence property is proved, which is straightforward but somewhat neat. Another feature that sets the book apart from most others is its inclusion of modal algebras that come surprisingly right at the beginning. Unfortunately their inclusion is poorly motivated.
The lack of index should come as an immediate turn-off. Also a lot of the material (including exercises) is unmotivated (in a severe way). A chapter on general frames would've been nice too. I would recommend this as an undergraduate text for computer scientists who do not intend it to be a reference text.First Steps in Modal Logic Overview

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