Gert H. Muller The growth of the number of publications in almost all scientific areas, as in the area of (mathematical) logic, is taken as a sign of our scientifically minded culture, but it also has a terrifying aspect. In addition, given the rapidly growing sophistica tion, specialization and hence subdivision of logic, researchers, students and teachers may have a hard time getting an overview of the existing literature, partic ularly if they do not have an extensive (...) library available in their neighbourhood: they simply do not even know what to ask for! More specifically, if someone vaguely knows that something vaguely connected with his interests exists some where in the literature, he may not be able to find it even by searching through the publications scattered in the review journals. Answering this challenge was and is the central motivation for compiling this Bibliography. The Bibliography comprises (presently) the following six volumes (listed with the corresponding Editors): I. Classical Logic W. Rautenberg II. Non-classical Logics W. Rautenberg III. Model Theory H. -D. Ebbinghaus IV. Recursion Theory P. G. Hinman V. Set Theory A. R. Blass VI. Proof Theory; Constructive Mathematics J. E. Kister; D. van Dalen & A. S. Troelstra. (shrink)

Classical logic is traditionally introduced by itself, but that makes it seem arbitrary and unnatural. This text introduces classical alongside several nonclassical logics (relevant, constructive, quantative, paraconsistent).

Provability, Computability and Reflection.

"This volume comprises the proceedings of the First All-Berlin Workshop on Nonclassical Logics and Information Processing, held at the Free University of Berlin, November 9-10, 1990. The scope of the ten papers in the volume is broad, covering various different subfields of logic - particularly nonclassical logic - and its applications in artificial intelligence. The papers are grouped according to the four major topics that emerged at the meeting: modal systems, logic programming, nonmonotonic logics, and (...) proof theory. The classification is only a rough guide since the four areas overlap considerably."--PUBLISHER'S WEBSITE. (shrink)

Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and (...) were revised and updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

The Liar paradox arguably shows that a coherent and self-applicable notion of truth is governed by nonclassical logic. It then seems natural to conclude that classical logic is inadequate for defining a truth theory. In this article, we argue that this is not the case. In the spirit of Reinhardt (Math Logic Formal Syst 94:227, 1985; J Philos Logic 15:219–251, 1986), and in analogy with Hilbert’s program for the foundation of classical mathematics, we will articulate (...) an instrumentalist justification for the use classical logic: it will be argued that classical reasoning is a _useful but dispensable instrument_, which can yield philosophically adequate truth theories. (shrink)

This is the first introductory textbook on non-classical propositional logics.

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, (...) and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

The subject of Labelled Non-Classical Logics is the development and investigation of a framework for the modular and uniform presentation and implementation of non-classical logics, in particular modal and relevance logics. Logics are presented as labelled deduction systems, which are proved to be sound and complete with respect to the corresponding Kripke-style semantics. We investigate the proof theory of our systems, and show them to possess structural properties such as normalization and the subformula property, which we exploit not only to (...) establish advantages and limitations of our approach with respect to related ones, but also to give, by means of a substructural analysis, a new proof-theoretic method for investigating decidability and complexity of (some of) the logics we consider. All of our deduction systems have been implemented in the generic theorem prover Isabelle, thus providing a simple and natural environment for interactive proof development. Labelled Non-Classical Logics is essential reading for researchers and practitioners interested in the theory and applications of non-classical logics. (shrink)

Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity. The proof theory portion presents (...) classical propositional logic and first-order logic using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time. Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings. Gives an exceptionally broad view of logic. Treats traditional logic in a modern format. Presents relevance logic with applications. Provides an ideal text for a variety of one-semester upper-level undergraduate courses. (shrink)

Provability, Computability and Reflection.

Logical Options introduces the extensions and alternatives to classical logic which are most discussed in the philosophical literature: many-sorted logic, second-order logic, modal logics, intuitionistic logic, three-valued logic, fuzzy logic, and free logic. Each logic is introduced with a brief description of some aspect of its philosophical significance, and wherever possible semantic and proof methods are employed to facilitate comparison of the various systems. The book is designed to be useful for philosophy (...) students and professional philosophers who have learned some classical first-order logic and would like to learn about other logics important to their philosophical work. (shrink)

The European Workshop on Logics in Artificial Intelligence was held at the Centre for Mathematics and Computer Science in Amsterdam, September 10-14, 1990. This volume includes the 29 papers selected and presented at the workshop together with 7 invited papers. The main themes are: - Logic programming and automated theorem proving, - Computational semantics for natural language, - Applications of non-classical logics, - Partial and dynamic logics.

The new area of logic and computation is now undergoing rapid development. This has affected the social pattern of research in the area. A new topic may rise very quickly with a significant body of research around it. The community, however, cannot wait the traditional two years for a book to appear. This has given greater importance to thematic collections of papers, centred around a topic and addressing it from several points of view, usually as a result of a (...) workshop, summer school, or just a scientific initiative. Such a collection may not be as coherent as a book by one or two authors yet it is more focused than a collection of key papers on a certain topic. It is best thought of as a thematic collection, a study in the area of logic and computation. The new series Studies in Logic and Computation is intended to provide a home for such thematic collections. Substructural logics are nonclassical logics, which arose in response to problems in foundations of mathematics and logic, theoretical computer science, mathematical linguistics, and category theory. They include intuitionistic logic, relevant logic, BCK logic, linear logic, and Lambek's calculus of syntactic categories. Substructural logics differ from classical logics, and from each other, in their presuppositions about Gentzen's structural rules, although their presuppositions about the deductive role of logical constants are invariant. Substructural logics have been a subject of study for logicians during the last sixty years. Specialists have often worked in isolation, however, largely unaware of the contributions of others. This book brings together new papers by some of the most eminent authorities in these varioustraditions to produce a unified view of substructural logics. (shrink)

_Logic: The Basics_ is an accessible introduction to several core areas of logic. The first part of the book features a self-contained introduction to the standard topics in classical logic, such as: · mathematical preliminaries · propositional logic · quantified logic · English and standard ‘symbolic translations’ · tableau procedures. Alongside comprehensive coverage of the standard topics, this thoroughly revised second edition also introduces several philosophically important nonclassical logics, free logics, and modal logics, and (...) gives the reader an idea of how they can take their knowledge further. With its wealth of exercises, _Logic: The Basics _is a useful textbook for courses ranging from the introductory level to the early graduate level, and also as a reference for students and researchers in philosophical logic. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they (...) can take their knowledge further and the philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

The Joint International Conference on Logic Programming, sponsored by the Association for Logic Programming, is a major forum for presentations of research, applications, and implementations in this important area of computer science. Logic programming is one of the most promising steps toward declarative programming and forms the theoretical basis of the programming language Prolog and its various extensions. Logic programming is also fundamental to work in artificial intelligence, where it has been used for nonmonotonic and commonsense (...) reasoning, expert systems implementation, deductive databases, and applications such as computer-aided manufacturing. Krzysztof R. Apt is project leader at the Centre for Mathematics and Computer Science in Amsterdam and part-time Professor in Computer Science at the University of Amsterdam. Topics covered: Theory Foundations. Programming Languages. Implementation. Programming Methodologies and Tools. Applications. Deductive Databases. Artificial Intelligence. Parallelism. (shrink)

Standard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using anonclassical substructural logicas the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with anoncontractive logicin the background. This sheds light on which modal principles (...) are invariant under changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics. (shrink)

The First International Workshop brings together researchers from the theoretical ends of the logic programming and artificial intelligence communities to discuss their mutual interests. Logic programming deals with the use of models of mathematical logic as a way of programming computers, where theoretical AI deals with abstract issues in modeling and representing human knowledge and beliefs. One common ground is nonmonotonic reasoning, a family of logics that includes room for the kinds of variations that can be (...) found in human reasoning. Topics covered: Stable Semantics. Default Logic. AutoEpistemic Logic. Truth Maintenance Systems. Implementation Issues. Diagnosis. Applications. Inheritance Reasoning. Logics of Belief. Inconsistency and Non-Monotonicity. (shrink)

Philosophical Logic is a clear and concise critical survey of nonclassical logics of philosophical interest written by one of the world's leading authorities on the subject. After giving an overview of classical logic, John Burgess introduces five central branches of nonclassical logic, focusing on the sometimes problematic relationship between formal apparatus and intuitive motivation. Requiring minimal background and arranged to make the more technical material optional, the book offers a choice between an overview and in-depth (...) study, and it balances the philosophical and technical aspects of the subject.The book emphasizes the relationship between models and the traditional goal of logic, the evaluation of arguments, and critically examines apparatus and assumptions that often are taken for granted. Philosophical Logic provides an unusually thorough treatment of conditional logic, unifying probabilistic and model-theoretic approaches. It underscores the variety of approaches that have been taken to relevantistic and related logics, and it stresses the problem of connecting formal systems to the motivating ideas behind intuitionistic mathematics. Each chapter ends with a brief guide to further reading.Philosophical Logic addresses students new to logic, philosophers working in other areas, and specialists in logic, providing both a sophisticated introduction and a new synthesis. (shrink)

Mathematics, even more than the other sciences, is often presented as essentially unique, as if it could not be any other way. And yet, prima facie alternative mathematics are all over the place, from non-Western mathematics to mathematics based on nonclassical logics. Taking inspiration from Robin Dembroff’s analysis of critical gender kinds, and from Andrew Aberdein and Stephen Read’s analysis of alternative logics, in this paper I will introduce a practice-centered framework for the study of alternative mathematics based on (...) the notion of critical math kind. After sketching a model of mainstream mathematics, I will provide examples of how deviation along several distinct dimensions can occur, and how deviations can vary in their gravity. I will then discuss how the framework can be used to think of questions concerning the alternativeness status and philosophical implications of alleged alternative mathematics, and help us in identifying alternatives that suit our purposes. (shrink)

Historically, nonclassical physics developed in three stages. First came a collection of ad hoc assumptions and then a cookbook of equations known as "quantum mechanics". The equations and their philosophical underpinnings were then collected into a model based on the mathematics of Hilbert space. From the Hilbert space model came the abstaction of "quantum logics". This book explores all three stages, but not in historical order. Instead, in an effort to illustrate how physics and abstract mathematics influence each other (...) we hop back and forth between a purely mathematical development of Hilbert space, and a physically motivated definition of a logic, partially linking the two throughout, and then bringing them together at the deepest level in the last two chapters. This book should be accessible to undergraduate and beginning graduate students in both mathematics and physics. The only strict prerequisites are calculus and linear algebra, but the level of mathematical sophistication assumes at least one or two intermediate courses, for example in mathematical analysis or advanced calculus. No background in physics is assumed. (shrink)

A history of logic -- Patterns of reasoning -- A language and its meaning -- A symbolic language -- 1850-1950 mathematical logic -- Modern symbolic logic -- Elements of set theory -- Sets, functions, relations -- Induction -- Turning machines -- Computability and decidability -- Propositional logic -- Syntax and proof systems -- Semantics of PL -- Soundness and completeness -- First order logic -- Syntax and proof systems of FOL -- Semantics of FOL (...) -- More semantics -- Soundness and completeness -- Why is first order logic "First Order"? (shrink)

En associant philosophie et logique, cet ouvrage apporte un éclairage sur l'évolution de la logique de l'époque d'Aristote à nos jours. Il présente les différents formalismes des logiques non-standard en expliquant tout d'abord les besoins de ces logiques puis leurs descriptions syntaxiques et sémantiques. A l'aide d'exercices, il offre un panorama de ces logiques qui sont complexes à " utiliser " pour modéliser des connaissances permettant ainsi aux lecteurs d'appréhender ce domaine.

The papers presented in this volume examine topics of central interest in contemporary philosophy of logic. They include reflections on the nature of logic and its relevance for philosophy today, and explore in depth developments in informal logic and the relation of informal to symbolic logic, mathematical metatheory and the limiting metatheorems, modal logic, many-valued logic, relevance and paraconsistent logic, free logics, extensional v. intensional logics, the logic of fiction, epistemic (...) class='Hi'>logic, formal logical and semantic paradoxes, the concept of truth, the formal theory of entailment, objectual and substitutional interpretation of the quantifiers, infinity and domain constraints, the Löwenheim-Skolem theorem and Skolem paradox, vagueness, modal realism v. actualism, counterfactuals and the logic of causation, applications of logic and mathematics to the physical sciences, logically possible worlds and counterpart semantics, and the legacy of Hilbert’s program and logicism. The handbook is meant to be both a compendium of new work in symbolic logic and an authoritative resource for students and researchers, a book to be consulted for specific information about recent developments in logic and to be read with pleasure for its technical acumen and philosophical insights. Key Features - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic -Useful bibliographies in every chapter - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic - Useful bibliographies in every chapter. (shrink)

Mathematical logic is a branch of mathematics that takes axiom systems and mathematical proofs as its objects of study. This book shows how it can also provide a foundation for the development of information science and technology. The first five chapters systematically present the core topics of classical mathematical logic, including the syntax and models of first-order languages, formal inference systems, computability and representability, and Gödel’s theorems. The last five chapters present extensions and developments of (...) classical mathematical logic, particularly the concepts of version sequences of formal theories and their limits, the system of revision calculus, proschemes (formal descriptions of proof methods and strategies) and their properties, and the theory of inductive inference. All of these themes contribute to a formal theory of axiomatization and its application to the process of developing information technology and scientific theories. The book also describes the paradigm of three kinds of language environments for theories and it presents the basic properties required of a meta-language environment. Finally, the book brings these themes together by describing a workflow for scientific research in the information era in which formal methods, interactive software and human invention are all used to their advantage. The second edition of the book includes major revisions on the proof of the completeness theorem of the Gentzen system and new contents on the logic of scientific discovery, R-calculus without cut, and the operational semantics of program debugging. This book represents a valuable reference for graduate and undergraduate students and researchers in mathematics, information science and technology, and other relevant areas of natural sciences. Its first five chapters serve as an undergraduate text in mathematical logic and the last five chapters are addressed to graduate students in relevant disciplines. (shrink)

The papers presented in this volume examine topics of central interest in contemporary philosophy of logic. They include reflections on the nature of logic and its relevance for philosophy today, and explore in depth developments in informal logic and the relation of informal to symbolic logic, mathematical metatheory and the limiting metatheorems, modal logic, many-valued logic, relevance and paraconsistent logic, free logics, extensional v. intensional logics, the logic of fiction, epistemic (...) class='Hi'>logic, formal logical and semantic paradoxes, the concept of truth, the formal theory of entailment, objectual and substitutional interpretation of the quantifiers, infinity and domain constraints, the Löwenheim-Skolem theorem and Skolem paradox, vagueness, modal realism v. actualism, counterfactuals and the logic of causation, applications of logic and mathematics to the physical sciences, logically possible worlds and counterpart semantics, and the legacy of Hilbert’s program and logicism. The handbook is meant to be both a compendium of new work in symbolic logic and an authoritative resource for students and researchers, a book to be consulted for specific information about recent developments in logic and to be read with pleasure for its technical acumen and philosophical insights. Key Features - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic -Useful bibliographies in every chapter - Written by leading logicians and philosophers - Comprehensive authoritative coverage of all major areas of contemporary research in symbolic logic - Clear, in-depth expositions of technical detail - Progressive organization from general considerations to informal to symbolic logic to nonclassical logics - Presents current work in symbolic logic within a unified framework - Accessible to students, engaging for experts and professionals - Insightful philosophical discussions of all aspects of logic - Useful bibliographies in every chapter. (shrink)

This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The (...) most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. (shrink)

We present a procedure which allows us to recover classical and nonclassical logical structures as concrete logics associated with physical theories expressed by means of classical languages. This procedure consists in choosing, for a given theory ${{\mathcal{T}}}$ and classical language ${{\fancyscript{L}}}$ expressing ${{\mathcal{T}}, }$ an observative sublanguage L of ${{\fancyscript{L}}}$ with a notion of truth as correspondence, introducing in L a derived and theory-dependent notion of C-truth (true with certainty), defining a physical preorder $\prec$ induced by C-truth, and finally (...) selecting a set of sentences ϕ V that are verifiable (or testable) according to ${{\mathcal{T}}, }$ on which a weak complementation ⊥ is induced by ${{\mathcal{T}}. }$ The triple $(\phi_{V},\prec,^{\perp})$ is then the desired concrete logic. By applying this procedure we recover a classical logic and a standard quantum logic as concrete logics associated with classical and quantum mechanics, respectively. The latter result is obtained in a purely formal way, but it can be provided with a physical meaning by adopting a recent interpretation of quantum mechanics that reinterprets quantum probabilities as conditional on detection rather than absolute. Hence quantum logic can be considered as a mathematical structure formalizing the properties of the notion of verification in quantum physics. This conclusion supports the general idea that some nonclassical logics can coexist without conflicting with classical logic (global pluralism) because they formalize metalinguistic notions that do not coincide with the notion of truth as correspondence but are not alternative to it either. (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)

My purpose here is purely historical. It is not an attempt to resolve the question as to whether Russell did or did not countenance nonclassical logics, and if so, which nonclassical logics, and still less to demonstrate whether he himself contributed, in any manner, to the development of nonclassical logic. Rather, I want merely to explore and insofar as possible document, whether, and to what extent, if any, Russell interacted with the various, either the various candidates (...) or their, ideas that Dejnožka and others have proposed as potentially influential in Russell’s intellectual reactions to nonclassical logic or to the philosophical concepts that might contribute to his reactions to nonclassical logics. (shrink)

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

Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.

Recent years have seen the appearance of many English-language hand books of logic and numerous monographs on topical discoveries in the foundations of mathematics. These publications on the foundations of mathematics as a whole are rather difficult for the beginners or refer the reader to other handbooks and various piecemeal contribu tions and also sometimes to largely conceived "mathematical fol klore" of unpublished results. As distinct from these, the present book is as easy as possible systematic exposition of (...) the now classical results in the foundations of mathematics. Hence the book may be useful especially for those readers who want to have all the proofs carried out in full and all the concepts explained in detail. In this sense the book is self-contained. The reader's ability to guess is not assumed, and the author's ambition was to reduce the use of such words as evident and obvious in proofs to a minimum. This is why the book, it is believed, may be helpful in teaching or learning the foundation of mathematics in those situations in which the student cannot refer to a parallel lecture on the subject. This is also the reason that I do not insert in the book the last results and the most modem and fashionable approaches to the subject, which does not enrich the essential knowledge in founda tions but can discourage the beginner by their abstract form. A. G. (shrink)