 
Summary: Raymond M. Smullyan Firstorder 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 Firstorder 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
