In Possibilities and Paradox, J.C. Beall and Bas Van Fraassen team up to present a rigorous introduction to modal and many-valued logic. After presenting a brief motivation for the development of modal logic (chapter 1), the authors provide a discussion of set theory (chapter 2), logical semantics (chapter 3), the tableau or "tree" method of proof (chapter 4), normal modal logics (chapter 5), and intuitionistic and non-normal modal logics (chapter 6). These chapters prepare the reader for future topics, including many-valued logics (chapter 7), the Sorites and Liar paradoxes (chapter 8), and supervaluations and continuum-valued logics (chapter 9). The final section is devoted to metatheory and includes further tools for understanding and working with, among other things, functions, sentences, and soundness and completeness proofs.
For the student with only a basic logic course or two under his or her belt, Possibilities and Paradox is going to be a tough read. Despite its brevity (233 total pages), the book is extremely information dense--the philosophical equivalent of Grape Nuts cereal, if you will. Virtually every sentence of every chapter contains an essential point or concept that merits a careful reading, re-reading, re-re-reading, and perhaps two or three readings on top of that before an exam if it is a required text, as it was in my case. While this makes the book exemplary when it comes to an efficient presentation of the material, since it is an introductory text it would have benefited from at least three additional features. First, I would have liked to see the authors provide more of a "big picture" framework for approaching the mechanics of logic- what a good logical language should do, why we'd want a syntax to work a certain way, etc. The authors do a fairly good job of this in the later chapters, but bit more attention to this in the first three or four would make the book even more accessible to students. Second, a chapter or two on proofs would be especially helpful to students without a strong background in this aspect of logic; something like a brief, modal and many-valued version of Daniel Velleman's, How to Prove It. Finally, a chapter or two on quantification would be a nice addition to future editions of the book.
One last thing (note well): as of the writing of this review, a helpful list of corrections to various errata is available on professor Van Fraassen's website.
