Foundations of Mathematical Logic (Dover Books on Mathematics) [Paperback]

Haskell B. Curry (Author)

Book Description

Publication Date: November 18, 2010 | Series: Dover Books on Mathematics

Written by a pioneer of mathematical logic, this comprehensive graduate-level text explores the constructive theory of first-order predicate calculus. It covers formal methods — including algorithms and epitheory — and offers a brief treatment of Markov's approach to algorithms. It also explains elementary facts about lattices and similar algebraic systems. 1963 edition.

Product Details

• Paperback: 416 pages
• Publisher: Dover Publications; 2 Revised edition (November 18, 2010)
• Language: English
• ISBN-10: 0486634620
• ISBN-13: 978-0486634623
• Product Dimensions: 8.1 x 5.6 x 0.6 inches
• Shipping Weight: 14.9 ounces (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #126,130 in Books (See Top 100 in Books)

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45 of 47 people found the following review helpful:

4.0 out of 5 stars Still an interesting read...., July 9, 2001

By 

Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: Foundations of Mathematical Logic (Dover Books on Mathematics) (Paperback)

Those interested in mathematical logic will appreciate this book written by one of the main contributors to the field in the twentieth century. The technique of "currying" in higher order logic is named after the author, wherein unary functions can be used to emulate functions with many parameters. The book was first published in 1963, reprinted in 1977, and so is not a up-to-date treatment of mathematical logic, but it could still be used as an historical supplement to a course in this subject. The reader should be aware though the terminology employed by the author is very idiosyncratic and therefore it may not reflect what is currently used in the literature.

The first chapter of the book could be considered an introduction to the philosophy of logic and mathematics. The author though views "philosophical logic" as the study of the principles of valid reasoning, and this is to be distinguished from "mathematical logic", wherein mathematical systems are constructed to study (formally) the principles of valid reasoning. One can also according to the author view logic as a theory in itself, and many "models" of it can be studied, in much the same way as many different models of geometry can be considered. The author also discusses very succinctly the logical paradoxes, and the different schools of thought in mathematics, such as Platonism, intuitionism, and formalism. The author clearly advocates the formalist school of thought in this book.

In chapter 2, the author gets more into the details of formal reasoning, the field of semiotics is outlined, and the author first begins defining the grammar and symbols for the upcoming discussion. A theory is defined as a class of statements, and consistency and decidability of theories is defined. The idea of a deductive theory is also defined, and the author defines the notion of such a theory being complete. The notions of consistency, decidability, and completeness are the familiar ones now entrenched in current textbooks on mathematical logic. A formal system, according to the author, is a theory in which the parameters of the statements of the theory are introduced as unspecified objects, and the statements of the theory make assertions on the properties of the parameters and their relations. The author considers syntactical systems, wherein the formal objects are taken from some object language, and what he calls Ob systems, which are essentially the systems considered in modern mathematical logic.The author employs the familiar Godel numbering scheme to numerically represent formal objects. The notion of algorithm is brought in here as an effective procedure to manipulate the formal objects of a system.

The next chapter is basically an introduction to the analysis of what would now be called the metalanguage of a formal system. This analysis is done in terms of what the author calls epistatements and epitheorems. Examples of these epitheorems include the Godel incompleteness theorem and the Skolem-Lowenheim theorem. The author introduces and classifies variables, and defines free and bound variables. A brief introduction to the lambda calculus and combinatory logic is given.

Then in chapter 4, the author discusses logical systems which are relational but with no bound variables. These are called logical algebras by the author, and the reader will encounter the famous truth tables and lattices in this chapter. A discussion of the Heyting algebra is given in the notes to the chapter. The reader interested in the more exotic types of algebraic logic, such as quantum logic, could benefit greatly from the reading of this chapter.

The logic of propositional calculus in terms of algebraic logic is discussed in chapter 5. Called propositional algebras by the author, the author proves the deduction theorem for such systems in this chapter. Interestingly, the L systems introduced by Gentzen are also discussed in this chapter. Although there are much better overviews of Gentzen's work in the current literature, a reader may still profit from a perusing of this chapter. L-systems where negation is added is then the subject of the next chapter.

Quantification in formal systems is taken up in chapter 7, considered both in the usual predicate calculus and in L systems. Prenex normal forms, the Herbrand-Gentzen theorem, and the completeness theorem are discussed in fairly good detail, albeit with old-fashioned notation.

The last chapter covers the interesting concept of modal logic. First considered by Aristotle, the author discusses it in the context of L systems, with the presentation being the shortest in the book.