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

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)

Logic: Techniques of Formal Reasoning, 2/e is an introductory volume that teaches students to recognize and construct correct deductions. It takes students through all logical steps--from premise to conclusion--and presents appropriate symbols and terms, while giving examples to clarify principles. Logic, 2/e uses models to establish the invalidity of arguments, and includes exercise sets throughout, ranging from easy to challenging. Solutions are provided to selected exercises, and historical remarks discuss major contributions to the theories covered.

The present text book is intended as an introduction to elementary logic. Its content, structure, and manner have been determined in large measure - perhaps 'caused' is the better word- by certain desiderata about which the reader should be informed at the outset. The leading idea is that even an introductory treatment of logic may profitably be fashioned around a rigorous framework.

Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement (...) calculus, descriptions, and equality. Succeeding chapters explore abstract set theory—with examinations of class membership as well as relations and functions—cardinal and ordinal arithmetic, and the axiom of choice. An invaluable reference book for all mathematicians, this text is suitable for advanced undergraduates and graduate students. Numerous exercises make it particularly appropriate for classroom use. (shrink)

This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.

Coherent, well organized text familiarizes readers with complete theory of logical inference and its applications to math and the empirical sciences. Part I deals with formal principles of inference and definition; Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Last section introduces numerous examples of axiomatically formulated theories in both discussion and exercises. Ideal for undergraduates; no background in math or philosophy required.

Elementary Applied Symbolic Logic was first published by Prentice-Hall in 1976. It went through two editions with them, then had a successful classroom run of 25 years by various publishers, before it finally went out of print in 2001.I am reviving it here, because during its run it acquired a reputation as an outstanding textbook for getting students to understand symbolic logic.I immodestly believe it is the best textbook ever written on the subject.------------This is a book on applied symbolic logic. (...) It provides the bridge between statements and arguments in English, and their formal counterparts in symbolic logic. Extensive exercises are given, illustrating how different natural-language concepts can correspond to the same symbolism, and how English sentences may be translated into formulae. Translation is heavily emphasized.It is intended to make learning symbolic logic (relatively) easy, by starting out with very basics and progressing from there a step at a time, building on what came before. I tried to make it as close to a self-teaching text as I could manage. It has two major divisions: Propositional Logic and Quantifier Logic.The first starts with propositions and truth-values, then truth-tables for evaluating the status of statements and arguments. It then moves to natural deduction, with rules for making inferences and transformations. Procedures are given for proving both validity and invalidity.Exercises increase in complexity as things move along. Solutions to selected exercises are included at the back of the book.Quantifier Logic starts with Monadic predicate logic, involving only single-place predicates ("properties"). It starts with singular statements and propositional functions, then moves to statements containing a single universal or existential quantifier, then to statements and arguments involving multiple quantifiers. It covers inferences using quantificational inference and transformation rules, and gives methods of invalidity proof.Its second half goes into polyadic predicates ("relations") of various degrees, moves on to identity, and finally to definite descriptions.Appendices on various related and supplementary topics are included at the end. The original appendix on Completeness and Consistency was complicated and confusing. It has been deleted, and replaced with an addendum at the end. (shrink)

This comprehensive introduction presents the fundamentals of symbolic logic clearly, systematically, and in a straightforward style accessible to readers. Each chapter, or unit, is divided into easily comprehended small bites that enable learners to master the material step-by-step, rather than being overwhelmed by masses of information covered too quickly. The book provides extremely detailed explanations of procedures and techniques, and was written in the conviction that anyone can thoroughly master its content. A four-part organization covers sentential logic, monadic predicate logic, (...) relational predicate logic, and extra credit units that glimpse into alternative methods of logic and more advanced topics. (shrink)

In this study on deduction, the authors argue that people reason by imagining the relevant state of affairs, ie building an internal model of it, formulating a tentative conclusion based on this model and then searching for alternative models.

INTRODUCTION MATHEMATICAL logic differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, ...

"One of the most careful and intensive among the introductory texts that can be used with a wide range of students. It builds remarkably sophisticated technical skills, a good sense of the nature of a formal system, and a solid and extensive background for more advanced work in logic.... The emphasis throughout is on natural deduction derivations, and the text's deductive systems are its greatest strength. Lemmon's unusual procedure of presenting derivations before truth tables is very effective." --Sarah Stebbins, The (...) Journal of Symbolic Logic. (shrink)

Die Leitgedanken meiner Untersuchungen über die Grundlagen der Mathematik, die ich - anknüpfend an frühere Ansätze - seit 1917 in Besprechungen mit P. BERNAYS wieder aufgenommen habe, sind von mir an verschiedenen Stellen eingehend dargelegt worden. Diesen Untersuchungen, an denen auch W. ACKERMANN beteiligt ist, haben sich seither noch verschiedene Mathematiker angeschlossen. Der hier in seinem ersten Teil vorliegende, von BERNAYS abgefaßte und noch fortzusetzende Lehrgang bezweckt eine Darstellung der Theorie nach ihren heutigen Ergebnissen. Dieser Ergebnisstand weist zugleich die Richtung (...) für die weitere Forschung in der Beweistheorie auf das Endziel hin, unsere üblichen Methoden der Mathematik samt und sonders als widerspruchsfrei zu erkennen. Im Hinblick auf dieses Ziel möchte ich hervorheben, daß die zeit weilig aufgekommene Meinung, aus gewissen neueren Ergebnissen von GÖDEL folge die Undurchführbarkeit meiner Beweistheorie, als irrtüm lich erwiesen ist. Jenes Ergebnis zeigt in der Tat auch nur, daß man für die weitergehenden Widerspruchsfreiheitsbeweise den finiten Stand punkt in einer schärferen Weise ausnutzen muß, als dieses bei der Be trachtung der elementaren Formallsmen erforderlich ist. Göttingen, im März 1934 HILBERT Vorwort zur ersten Auflage Eine Darstellung der Beweistheorie, welche aus dem HILBERTschen Ansatz zur Behandlung der mathematisch-logischen Grundlagenpro bleme erwachsen ist, wurde schon seit längerem von HILBERT ange kündigt. (shrink)

From the reviews: "A good textbook can improve a lecture course enormously, especially when the material of the lecture includes many technical details. Van Dalen's book, the success and popularity of which may be suspected from this steady interest in it, contains a thorough introduction to elementary classical logic in a relaxed way, suitable for mathematics students who just want to get to know logic. The presentation always points out the connections of logic to other parts of mathematics. The reader (...) immediately see the logic is "just another branch of mathematics" and not something more sacred." Acta Scientiarum Mathematicarum, Hungary. (shrink)

Logic: Techniques of Formal Reasoning, 2/e is an introductory volume that teaches students to recognize and construct correct deductions. It takes students through all logical steps--from premise to conclusion--and presents appropriate symbols and terms, while giving examples to clarify principles. Logic, 2/e uses models to establish the invalidity of arguments, and includes exercise sets throughout, ranging from easy to challenging. Solutions are provided to selected exercises, and historical remarks discuss major contributions to the theories covered.

Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.

This outstanding book is a leading text for symbolic or formal logic courses. All techniques and concepts are presented with clear, comprehensive explanations and numerous, carefully constructed examples. Its flexible organization (all chapters are complete and self-contained) allows instructors the freedom to cover the topics they want in the order they choose.

Provides comprehensive coverage of logical structure as well as the techniques of formal reasoning.

This article examines Quine's original proposal for a natural deduction calculus including an existential specification rule, it argues that it introduces a new paradigm of natural deduction alternative to Gentzen's but has some substantial defects. As an alternative the article puts forward a system of sequent natural deduction.