Gathered together here are the fundamental texts of the great classical period in modern logic. A complete translation of Gottlob Frege's Begriffsschrift--which opened a great epoch in the history of logic by fully presenting propositional calculus and quantification theory--begins the volume, which concludes with papers by Herbrand and by Gödel.

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Godel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic. The 1930s was a time of creativity and ferment in the subject, when the notion of computable moved from the realm of (...) philosophical speculation to the realm of science. This was accomplished by the work of Kurt Gode1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of computable. When they all turned out to be equivalent, there was a collective realization that this was indeed the right notion. Kleene played a key role in this process. One could say that he was there at the beginning of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Godel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.". (shrink)

The first edition of the Handbook of Philosophical Logic (four volumes) was published in the period 1983-1989 and has proven to be an invaluable reference work ...

"A valuable collection both for original source material as well as historical formulations of current problems."-- The Review of Metaphysics "Much more than a mere collection of papers . . . a valuable addition to the literature."-- Mathematics of Computation An anthology of fundamental papers on undecidability and unsolvability by major figures in the field, this classic reference opens with Godel's landmark 1931 paper demonstrating that systems of logic cannot admit proofs of all true assertions of arithmetic. Subsequent papers by (...) Godel, Church, Turing, and Post single out the class of recursive functions as computable by finite algorithms. Additional papers by Church, Turing, and Post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by Kleene and Post includes initiation of the classification theory of unsolvable problems. Suitable for graduate and undergraduate courses. 1965 ed. (shrink)

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part II supplements (...) the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, ranging from small (...) time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

§1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions ; and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his (...) thesis and in his subsequent attempts to convince himself of Church's Thesis, developed a general theory of the behavior of the recursive functions. He continued to develop this theory and extend it to new situations throughout his mathematical career. Indeed, all of the research which he did had a close relationship to recursive functions.Church's Thesis arose in an accidental way. In his investigations of a system of logic which he had invented, Church became interested in a class of functions which he called the λ-definable functions. Initially, Church knew that the successor function and the addition function were λ-definable, but not much else. During 1932, Kleene gradually showed1 that this class of functions was quite extensive; and these results became an important part of his thesis 1935a. (shrink)

This book is intended to be used as a textbook by students of mathematics, and also within limitations as a reference work.

This book is well known for its proof that many mathematical systems - including lattice theory and closure algebras - are undecidable. It consists of three treatises from one of the greatest logicians of all time: "A General Method in Proofs of Undecidability," "Undecidability and Essential Undecidability in Mathematics," and "Undecidability of the Elementary Theory of Groups.".

The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.

Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel's incompleteness theorems, but also a large number of optional topics, from Turing's theory of computability to Ramsey's theorem. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a (...) simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems. This updated edition is also accompanied by a website as well as an instructor's manual. (shrink)

In the general theory of logic built up by Whitehead and Russell to furnish a basis for all mathematics there is a certain subtheory which is unique in its simplicity and precision; and though all other portions of the work have their roots in this subtheory, it itself is completely independent of them. Whereas the complete theory requires for the enunciation of its propositions real and apparent variables, which represent both individuals and propositional functions of different kinds, and as a (...) result necessitates- the introduction of the cumbersome theory of types, this subtheory uses only real variables, and these real variables represent but one kind of entity—which the authors have chosen to call elementary propositions. The most general statements are formed by merely combining these variables by means of the two primitive propositional functions of propositions Negation and Disjunction; and the entire theory is concerned with the process of asserting those combinations which it regards as true propositions, employing for this purpose a few general rules which tell how to assert new combinations from old, and a certain number of primitive assertions from which to begin. (shrink)

Classical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to place, for example, second-order logic, type theory, modal and dynamic logics and MSL itself. The aim is two fold: only one theorem-prover is needed; proofs of the metaproperties of the different existing calculi can be avoided by borrowing them from (...) MSL. To make the book accessible to readers from different disciplines, whilst maintaining precision, the author has supplied detailed step-by-step proofs, avoiding difficult arguments, and continually motivating the material with examples. Consequently this can be used as a reference, for self-teaching or for first-year graduate courses. (shrink)

In 1934 Alonzo Church, Kurt Gödei, S. C. Kleene, and J. B. Rosser were all to be found in Princeton, New Jersey. In 1936 Church founded The Journal of Symbolic Logic. Shortly thereafter Alan Turing arrived for a two year visit. The United States had become a world center for cutting-edge research in mathematical logic. In this brief survey1 we shall examine some of the writings of American logicians during the 1920s, a period of important beginnings and remarkable insights as (...) well as of confused gropings.The publication of Whitehead and Russell's monumental Principia Mathematica [18] during the years 1910-1913 provided the basis for much of the research that was to follow. It also provided the basis for confusion that remained a factor during the period we are discussing. In 1908, Henri Poincaré, a famous skeptic where mathematical logic was concerned, wrote pointedly :It is difficult to admit that the word if acquires, when written ⊃, a virtue it did not possess when written if.Principia provided no very convincing answer to Poincaré. Indeed the fact that the authors of Principia saw fit to place their first two “primitive propositions”*1.1: Anything implied by a true proposition is true.*1.2: ⊢ p ⋁ p ⊃ punder one and the same heading suggest that they had thought of what they were doing as just such a translation as Poincare had derided. (shrink)

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs (...) of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

8.3 The consistency proof -- 8.4 Applications of the consistency proof -- 8.5 Second-order arithmetic -- Problems -- Chapter 9: Set Theory -- 9.1 Axioms for sets -- 9.2 Development of set theory -- 9.3 Ordinals -- 9.4 Cardinals -- 9.5 Interpretations of set theory -- 9.6 Constructible sets -- 9.7 The axiom of constructibility -- 9.8 Forcing -- 9.9 The independence proofs -- 9.10 Large cardinals -- Problems -- Appendix The Word Problem -- Index.

§1. Introduction. This paper deals with aspects of my doctoral dissertation which contributed to the early development of model theory. What was of use to later workers was less the results of my thesis, than the method by which I proved the completeness of first-order logic—a result established by Kurt Gödel in his doctoral thesis 18 years before.The ideas that fed my discovery of this proof were mostly those I found in the teachings and writings of Alonzo Church. This may (...) seem curious, as his work in logic, and his teaching, gave great emphasis to the constructive character of mathematical logic, while the model theory to which I contributed is filled with theorems about very large classes of mathematical structures, whose proofs often by-pass constructive methods.Another curious thing about my discovery of a new proof of Gödel's completeness theorem, is that it arrived in the midst of my efforts to prove an entirely different result. Such “accidental” discoveries arise in many parts of scientific work. Perhaps there are regularities in the conditions under which such “accidents” occur which would interest some historians, so I shall try to describe in some detail the accident which befell me.A mathematical discovery is an idea, or a complex of ideas, which have been found and set forth under certain circumstances. The process of discovery consists in selecting certain input ideas and somehow combining and transforming them to produce the new output ideas. The process that produces a particular discovery may thus be represented by a diagram such as one sees in many parts of science; a “black box” with lines coming in from the left to represent the input ideas, and lines going out to the right representing the output. To describe that discovery one must explain what occurs inside the box, i.e., how the outputs were obtained from the inputs. (shrink)