Metalogic: An Introduction to the Metatheory of Standard First Order Logic [Book Review]

Review of Metaphysics 25 (1):127-127 (1971)
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Abstract

In his preface, Hunter explains that this volume is intended to provide for non-mathematicians an introduction to the most important results of modern mathematical logic. The reader will find here the work of Post, Skolem, Gödel, Church, Henkin, and others, presented in a terse and closely-knit style. Though acknowledging the trend toward natural deduction systems, Hunter sticks to more classical axiomatic systems on the grounds that the proofs of metatheorems are simplified by that choice. He begins with a formal system of propositional logic, for which he develops proofs of completeness, consistency, and decidability. Systems of first order predicate logic are then dealt with, and Church's and Gödel's results on undecidability and incompleteness are presented. There is a brief, useful discussion indicating what parts of first order predicate logic are decidable. It is assumed that the reader has some familiarity with elementary formal logic: Hunter develops the set-theoretical results he requires. The compactness of the presentation is facilitated by the use of earlier results for propositional logic to obtain certain results for predicate logic. Especially helpful is Hunter's technique of prefacing each of the longer and more difficult proofs with a section explaining the key ideas and the general method of proof; summaries are also often provided in conclusion. Because of the orientation of the book, a more extensive discussion of the significance of the results obtained and the philosophical issues which have been raised concerning some of them, would have been welcome. Nonetheless, this is a good, concise introduction for those who are not yet prepared to tackle the more technical literature. A bibliography is provided for those interested in pursuing the subject in greater detail.--E. M. F.

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