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This volume starts the second edition of this famous Handbook series which comprised 4 volumes in its first edition (see Zbl 0538.03001, Zbl 0572.03003, Zbl 0603.03001, Zbl 0869.03001 and Zbl 0869.03002), and is planned now for 18 (!) volumes.
The new volume 1 coincides with the old volume 1 in the (updated) chapters on “Elementary predicate logic” (by W. Hodges), “Higher-order logic” (by J. van Benthem and K. Doets), and “Algorithms and decision problems” (by D. van Dalen). Additionally it offers a chapter on “Systems between first-order and second-order logic” (by S. Shapiro), and another one on “Mathematics of logic programming” (by H. D. Ebbinghaus and J. Flum).
The chapters from the first edition are slightly changed and updated. The new Shapiro-chapter has its focus on model-theoretically determined extensions of first-order logic which do not reach the full expressive power of second-order logic, including infinitary logics and substitutional semantics. And the Ebbinghaus/Flum-chapter discusses the resolution calculus including complexity results.
The book is completed with a (combined) index of notions, names, and symbols, and it starts with a rather concise table of contents. In the first edition, each chapter started with a detailed table of its contents: these are lacking now, unfortunately.
Despite the fact that the last decade has seen a lot of Handbooks for the computer science and AI fields, this well-established Handbook will remain to be a standard source not only for logicians in general, but also for mathematicians, computer scientists, and philosophers.
The articles of this volume will be reviewed individually.
Indexed articles:
Hodges, Wilfrid, Elementary predicate logic, 1-129 [Zbl 1003.03511]
Shapiro, Stewart, Systems between first-order and second-order logics, 131-187 [Zbl 1003.03512]
van Benthem, Johan; Doets, Kees, Higher-order logic, 189-243 [Zbl 1003.03513]
van Dalen, Dirk, Algorithms and decision problems: A crash course in recursion theory, 245-311 [Zbl 1003.03533]
Ebbinghaus, H.-D.; Flum, J., Mathematics of logic programming, 313-370 [Zbl 1003.03529]