Introduction to Mathematical Logic Reprint Edition

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.

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".

Alonzo Church was one of the giants of 20th-century mathematical logic: most people with an acquaintance with "Church's Thesis" regarding effective computability, or the lambda calculus he invented, already knew that. He was also a giant of the exposition and propagation of logical results: as editor of the *Journal of Symbolic Logic* for decades he wrote countless influential reviews of new logical texts and made the exciting results of other greats like Goedel and Tarski available to a wider mathematical audience. Jointly, these propositions imply a third -- Church's *Introduction to Mathematical Logic* must be an undying classic of metamathematics. It is. Though people looking for select recent results in model theory or proof theory may turn up their noses at a text last revised in 1956, people with wider logical interests will benefit immensely by considering Church's way of going about things. The book begins with a lengthy introduction to logical semantics, precisely spelling out the concepts of "logistic" that make up the methods of modern mathematical logic; this section is an excellent guide to the logical "commons", principles all modern formal thinkers make use of (generally without offering neophytes much of a clue as to the proper operation of the operators).

There is then another lengthy section on propositional logic, which includes many things contemporary texts leave out -- the calculi of logical or 'strict' implication that led to modern relevance logics and the concept of propositional quantification, important for modern "deflationary" accounts of truth, are two 'obscure' items Church does a good job covering. The section on the "functional calculi of first order" is the one least likely to impress: Church uses an older method of proving Goedel's Completeness theorem for predicate logic, not the Henkin method that has become standard --though Henkin's proof of the partial completeness of second-order logic appears in the last chapter -- and foregoes more than a mention of his own proof of the undecidability of predicate calculus (though the decidability of monadic predicate calculus and other fragments of first-order logic is covered). Gentzen systems, which have become the bedrock of proof theory, are explored minimally, and generally speaking if you don't know what you are looking for you might not find it in this somewhat unwieldy tome. However, if you do have a sense of logic and want to deepen your understanding, a glance through this is obligatory.