_Philosophical Approaches to the Foundations of Logic and Mathematics_ consists of eleven articles addressing various aspects of the "roots" of logic and mathematics, their basic concepts and the mechanisms that work in the practice of their use.

This is a review of Yves Nievergelt, Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Birkäuser Verlag, Boston, 2002, €90, pp. 480, ISBN 0-8176-4249-8, hardcover.

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 volume deals with the history of logic, the question of the nature of logic, the relation of logic and mathematics, modal or alternative logics (many-valued, relevant, paraconsistent logics) and their relations, including translatability, to classical logic in the Fregean and Russellian sense, and, more generally, the aim or aims of philosophy of logic and mathematics. Also explored are several problems concerning the concept of definition, non-designating terms, the interdependence of quantifiers, and the (...) idea of an assertion sign. The contributions concerned with Wittgenstein's investigations into the philosophy of logic and mathematics pursue issues relating to logical necessity, the undeniability of the law of the excluded middle, and the source of self-evidence, often characterized in the literature as the "rule-following considerations". Additionally, they examine Wittgenstein's attitudes towards the very idea of set-theory as a possible foundation for arithmetic. The volume also includes a number of contributions on specific issues concerning Wittgenstein's views on moral and religious judgements. (shrink)

In this paper I consider three mathematicians who allowed some role for menial processes in the foundations of their logical or mathematical theories. Boole regarded his Boolean algebra as a theory of mental acts; Cantor permitted processes of abstraction to play a role in his set theory; Brouwer took perception in time as a cornerstone of his intuitionist mathematics. Three appendices consider related topics.

Book review r A. (2004). "Book review: Yves nievergelt, foundations of ...

Mathematics is a formal science, and it has historically established itself as a paradigm of rigor and precision for other sciences. This is because, over the centuries, mathematics was considered...

The contributions to this volume are from participants of the international conference "Kreisel's Interests - On the Foundations of Logic and Mathematics", which took place from 13 to 14 2018 at the University of Salzburg in Salzburg, Austria. The contributions have been revised and partially extended. Among the contributors are Akihiro Kanamori, Göran Sundholm, Ulrich Kohlenbach, Charles Parsons, Daniel Isaacson, and Kenneth Derus. The contributions cover the discussions between Kreisel and Wittgenstein on philosophy of mathematics, Kreisel's (...) Dictum, proof theory, the discussions between Kreisel and Gödel on philosophy of mathematics, some bibliographical facts, and a collection of extracts from Kreisel's letters. (shrink)

Carnap's philosophy is examined from new viewpoints, including three important distinctions: (i) language as calculus vs language as universal medium; (ii) different senses of completeness: (iii) standard vs nonstandard interpretations of (higher-order) logic. (i) Carnap favored in 1930-34 the "formal mode of speech," a corollary to the universality assumption. He later gave it up partially but retained some of its ingredients, e.g., the one-domain assumption. (ii) Carnap's project of creating a universal self-referential language is encouraged by (ii) and by (...) the author's recent work. (iii) Carnap was aware of (iii) and occasionally used the standard interpretation, but was not entirely clear of the nature of the contrast. (shrink)

This volume comprises a selection of papers that were contributed to the 7th International Congress of Logic, Methodology and Philosophy of Science, which was held in Salzburg from the 11th - 16th July, 1983. There were 14 sections in this congress: 1. proof theory and foundations of mathematics 2. model theory and its applica ti on 3. recursion theory and theory of computation 4. axiomatic set theory 5. philosophical logic 6. general methodology of science 7. (...) class='Hi'>foundations of probability and induction 8. foundations and philosophy of the physical sciences 9. foundati ons and phi 1 osophy of biology 10. foundations and philosophy of psychology foundations and philosophy 11. of the social sciences 12. foundati ons and philosophy of linguistics 13. history of logic, methodology and philosophy of science 14. fundamental principles of the ethics of science In each section, three or four invited addresses were given, which will be published in the Congress Proceedings (Ruth Barcan Marcus, Georg J. W. Dorn and Paul Weingartner, eds. : Logic, Metho dology and Philosophy of Science VII. Proceedings of the Seventh International Congress of Logic, Methodology and Philosophy of v PREFACE Science, Salzburg, 1983. - Amsterdam, New York, Oxford: North-Holland Publishing 'Company, 1985. ) Every section with the exception of section 14 also contained contributed papers. (shrink)

This is a concise book that introduces students to the basics of logical thinking and important mathematical structures that are critical for a solid understanding of logical formalisms themselves as well as for building the necessary background to tackle other fields that are based on these logical principles. Despite its compact and small size, it includes many solved problems and quite a few end-of-section exercises that will help readers consolidate their understanding of the material. This textbook is essential reading for (...) anyone interested in the logical foundations of Informatics, Computer Science, Data Science, Artificial Intelligence, and other related areas. Written with undergraduate students in these disciplines in mind, this book can very well serve the needs of interested and curious readers who wish to get a grasp of the logical principles upon which these fields are built. This book does not require readers to possess math skills beyond those learned in high school. (shrink)

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

With material on his early philosophical views, his contributions to set theory and his work on nominalism and higher-order quantification, this book offers a uniquely expansive critical commentary on one of analytical philosophy’s great ...

The aim of this paper is to present and discuss main philosophical ideas concerning logic and mathematics of a significant but forgotten Polish philosopher Benedykt Bornstein. He received his doctoral degree with Kazimierz Twardowski but is not included into the Lvov–Warsaw School of Philosophy founded by the latter. His philosophical views were unique and quite different from the views of main representatives of Lvov–Warsaw School. We shall discuss Bornstein’s considerations on the philosophy of geometry, on the infinity, on (...) the foundations of set theory and his polemics with Stanisław Leśniewski as well as his conception of a geometrization of logic, of the categorial logic and of the mathematics of quality. (shrink)

"Foundations of the Formal Sciences" is a series of interdisciplinary conferences in mathematics, philosophy, computer science and linguistics. The main goal is to reestablish the traditionally strong links between these areas of research that have been lost in the past decades. The second conference in the series had the subtitle "Applications of Mathematical Logic in Philosophy and Linguistics" and brought speakers from all parts of the Formal Sciences together to give a holistic view of how mathematical methods (...) can improve our philosophical and technical understanding of language and scientific discourse, ranging from the theoretical level up to applications in language recognition software. Audience: This volume is of interest to all formal philosophers and theoretical linguists. In addition to that, logicians interested in the applications of their field and logic students in mathematics, computer science, philosophy and linguistics can use the volume to broaden their knowledge of applications of logic. (shrink)

The book is a collection of the author’s selected works in the philosophy and history of logic and mathematics. Papers in Part I include both general surveys of contemporary philosophy of mathematics as well as studies devoted to specialized topics, like Cantor's philosophy of set theory, the Church thesis and its epistemological status, the history of the philosophical background of the concept of number, the structuralist epistemology of mathematics and the phenomenological philosophy of mathematics. Part (...) II contains essays in the history of logic and mathematics. They address such issues as the philosophical background of the development of symbolism in mathematical logic, Giuseppe Peano and his role in the creation of contemporary logical symbolism, Emil L. Post's works in mathematical logic and recursion theory, the formalist school in the foundations of mathematics and the algebra of logic in England in the 19th century. The history of mathematics and logic in Poland is also considered.This volume is of interest to historians and philosophers of science and mathematics as well as to logicians and mathematicians interested in the philosophy and history of their fields. (shrink)

The IOth International Congress of Logic, Methodology and Philosophy of Science, which took place in Florence in August 1995, offered a vivid and comprehensive picture of the present state of research in all directions of Logic and Philosophy of Science. The final program counted 51 invited lectures and around 700 contributed papers, distributed in 15 sections. Following the tradition of previous LMPS-meetings, some authors, whose papers aroused particular interest, were invited to submit their works for publication in a (...) collection of selected contributed papers. Due to the large number of interesting contributions, it was decided to split the collection into two distinct volumes: one covering the areas of Logic, Foundations of Mathematics and Computer Science, the other focusing on the general Philosophy of Science and the Foundations of Physics. As a leading choice criterion for the present volume, we tried to combine papers containing relevant technical results in pure and applied logic with papers devoted to conceptual analyses, deeply rooted in advanced present-day research. After all, we believe this is part of the genuine spirit underlying the whole enterprise of LMPS studies. (shrink)

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.

First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

LN , so f lies in the elementary submodel M'. Clearly co 9 M' . It follows that 6 = {f(n): n em} is included in M'. Hence the ordinals of M' form an initial ...

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated (...) in this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims atproviding an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic hasfewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic PPC3, with two kinds of falsity and two negations. It introduces the notion of -space for a sentence A as the basis of logical model theory, and the notion of P A /, functioning as a `private' subuniverse for A/A. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. PPC3 and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of PPC3 as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: 1a for the conjoined presuppositions of a, ã for the complement of a within 1a, and â for the complement of 1a within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights (...) elaborated in this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims at providing an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic has fewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic ${\rm PPC}_{3}$ , with two kinds of falsity and two negations. It introduces the notion of Σ-space for a sentence A (or / A /, the set of situations in which A is true) as the basis of logical model theory, and the notion of / ${\bf P}^{A}$ / (the Σ-space of the presuppositions of A), functioning as a 'private' subuniverse for / A /. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. ${\rm PPC}_{3}$ and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of ${\rm PPC}_{3}$ as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: $1_{a}$ for the conjoined presuppositions of a, ã for the complement of a within $1_{a}$ , and â for the complement of $1_{a}$ within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

Graduate-level historical study is ideal for students intending to specialize in the topic, as well as those who only need a general treatment. Part I discusses traditional and symbolic logic. Part II explores the foundations of mathematics, emphasizing Hilbert’s metamathematics. Part III focuses on the philosophy of mathematics. Each chapter has extensive supplementary notes; a detailed appendix charts modern developments.

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)

Frege attempted to provide arithmetic with a foundation in logic. But his attempt to do so was confounded by Russell's discovery of paradox at the heart of Frege's system. The papers collected in this special issue contribute to the on-going investigation into the foundations of mathematics and logic. After sketching the historical background, this introduction provides an overview of the papers collected here, tracing some of the themes that connect them.

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and (...) the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics. (shrink)

This volume comprises seven of the eight addresses presented before the International Colloquium on Mathematical Logic and Foundations of Set theory held at the Acadmey Building in Jerusalem, Israel, On November 11-14, 1968.

The four volumes of this collection bring together some of the major contributions to the literature on Gottlob Frege (1848-1925), one of the most formative influences on the course of philosophy during the last hundred years. The first volume provided general assessments of Frege's work and examined its historical context. The present volume deals with Frege's contributions to logic and the foundations of mathematics. The essays are arranged in order of their first publication, providing insight into the (...) historical evolution of the Frege literature. Annotation copyright by Book News, Inc., Portland, OR. (shrink)