Book Description

Logic is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof. Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, The Calculi of Lambda-Conversion (1941), established an invaluable tool that computer scientists still use today.

Even beyond the accomplishment of that book, however, his second Princeton book, Introduction to Mathematical Logic, defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic.

Church was one of the principal founders of the Association for Symbolic Logic; he founded the Journal of Symbolic Logic in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.

Customer Reviews (2)

a classic, but mostly useful as a historical reference
I give this book 5 stars out of respect for its enormous contribution to mathematical logic; for no doubt many of the authors of the more modern math-logic texts were greatly influenced by this book. But with that said, all of the material here is a proper subset of other current books which present the material much more clearly and using better notation. Examples include Burris' "Logic for Mathematics and Computer Science", Ebbinhaus' "Intro. To Math Logic", and Gallier's "Logic for Computer Scientists".

One of the classics
This book, which first appeared in print as an issue in Annals of Mathematics in 1944, is now a classic in mathematical logic, and is still worth perusing in spite of the out-dated notation. The author outlines comprehensively the propositional calculus and predicate calculus. Although the book is mostly formal in its style, the author does introduce the reader to some elementary notions in logic, and some brief commentary on what would now be classified as philosophical logic. He defines logic as the analysis of propositions and their proof according to their form and not their content. He notes also that inductive logic and the theory of partial confirmation should also be included as part of mathematical logic. There are exercises throughout the book, and so it could conceivably be used as a textbook, in spite of its publication date. The book could better be used as a historical supplement to a course in mathematical logic or one in the philosophy of logic.

In the introduction to the book the author defines the terms and concepts he will use in the book, with a discussion of proper names, constants and variables, functions, and sentences. He adopts the Fregian point of view that sentences are names of a particular kind. His discussion of this is rather vague however, for he does not give enough clarification of the difference between an "assertive" use of a sentence and its "non-assertive" use. Readers will have to do further reading on Frege in order to understand this distinction more clearly, but essentially what Church is saying here is that sentences are names with truth values. The existential and universal quantifiers are introduced as well. And here the author also introduces the concepts of object language and metalanguage, along with a discussion of the axiomatic method. The author distinguishes between informal and formal axiomatic methods. The modern notions of syntax and semantics are given a nice treatment here, and the di

scussion is more in-depth than one might get in more modern texts on mathematical logic.

Chapter 1 is a detailed overview of propositional logic, being the usual formal system with three symbols, one constant, an infinite number of variables, rules on how to form well-formed formulas, and the rules of inference. The deduction theorem is proved in detail along with a discussion of the decision problem for propositional logic, with the famous truth tables due to W. Quine introduced here. The notions of consistency and completeness are briefly discussed.

The discussion of the propositional calculus is continued in the next chapter where a new system of propositional calculus is obtained by dropping the constants from the first one and adding another symbol (negation). The two systems are shown to be equivalent to each other using a particular well-formed formula in the second one to replace the constant in the first. Other systems of propositional calculus are also introduced here, using the idea of primitive connectives such as disjunction, along with various rules of inference. Church also outlines an interesting propositional calculus due to J.G.P.Nicod, which assumes only one primitive connective, one axiom, and only one rule of inference (besides substitution). The author also introduces partial systems of propositional calculus, with the goal of showing just what must be added to these systems to obtain the full propositional calculus. He discusses the highly interesting and thought-provoking intuitionistic propositional calculus, due to A. Heyting, which is a formalization of the famous mathematical intuitionism of L.E.J. Brouwer. The system he discusses is a variant of Heyting's and he gives references to the positive solution of the decision problem for this system. The author ends the chapter with a brief discussion of how to construct a propositional calculus by employing axiom schemata.

The author then moves on to what he has termed functional calculi of first order beginning in the next chapter. Called predicate calculi in today's parlance, the author first defines the pure functional calculus of first order, and shows that the theorems of the propositional calculus also follow when considered as part of this system. Free and bound variables are defined, and Church proves explicitly the consistency of this system, and the deduction theorem. The important construction of a prenex normal form of a well-formed formula is discussed, and the author shows that every well-formed formula of the functional calculus is equivalent to some well-formed formula in prenex normal form.

In chapter 4, the author gives an alternative formulation of pure functional calculus of first order, wherein rules of substitution are used and axiom schemata are replaced by instances, making the number of axioms finite. The Skolem normal form of a well-formed formula is defined, which sets up a discussion of satisfiability and validity. The author then proves the Godel completeness theorem, which states that every valid well-formed formula is a theorem. This is followed by a very well written discussion of the Skolem-Lowenheim theorem, and an overview of the decision problem in functional (predicate) calculus.

In the last chapter of the book the author considers functional (predicate) calculi of second order, which is distinguished from the first order case by allowing the variables to range over what its predicates and subjects represent. In second-order functional calculus, propositional and predicate variables can have bound occurrences. The author discusses the elimination problem and consistency for second-order predicate calculus, and gives a proof of the (Henkin) completeness theorem. A fairly detailed discussion of a logical system for elementary number theory is given, but the treatment involves notation that is somewhat clumsy and the discussion is difficult to follow. ... Read more

Book Description

This volume began as a remembrance of Alonzo Church while he was still with us and is now finally complete. It contains papers by many well-known scholars, most of whom have been directly influenced by Church's own work. Often the emphasis is on foundational issues in logic, mathematics, computation, and philosophy - as was the case with Church's contributions, now universally recognized as having been of profound fundamental significance in those areas.

Book Description
This digital document, covering the life and work of Alonzo Church, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 1165 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
• Family members
• Education
• Professional associations and honors
• Employment
• Writings, including books and periodicals
• A description of the author's work
• References to further readings about the author