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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-

  

Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University

 

Collections: Multidisciplinary Databases and Resources; Mathematics