Raymond M. Smullyan First-order logic Corrected republication of XL 237. Dover Publications, New York 1995, xii + 158 pp. Summary: Raymond M. Smullyan First-order logic Corrected republication of XL 237. Dover Publications, New York 1995, xii + 158 pp. In reconciling the contrary viewpoints of syntax and semantics, GĻodel's com- pleteness theorem lies at the heart of mathematical logic. Over the years numer- ous approaches to proving completeness have been explored, the efforts justified by the theorem's primary importance. Smullyan's First-order logic, a corrected reprinting of the 1968 original (XL 237), explains many of these approaches in detail. For example, the method of analytic tableaux runs roughly as follows. Let be a sentence of predicate logic, and suppose we want to prove that either is satisfiable or its negation is provable. Try to satisfy by building a tree: to satisfy one needs to satisfy both and ; to satisfy one can branch and try to satisfy either one of the two; to satisfy x (x) one tries to satisfy (c) for a suitable constant c; and so on. If every branch of the tree yields a contradiction at some finite stage, we have the desired proof of Ž. On the other hand, if the construction has been done carefully, an infinite branch yields what is called a Hintikka set. These sets, though not maximal, are saturated downwards (e.g. if S then S and S, though not necessarily conversely), and from such a set it is easy to build a model of . Such constructions are the essence of Smullyan's book. Part I covers propo- Collections: Multidisciplinary Databases and Resources; Mathematics