Three Views of Logic

Author: Donald W. Loveland
Publisher: Princeton University Press
ISBN: 9781400848751
Release Date: 2014-01-26
Genre: Mathematics

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

Philosophy and Computer Science

Author: Timothy R. Colburn
Publisher: M.E. Sharpe
ISBN: 1563249901
Release Date: 2000
Genre: Computers

Colburn (computer science, U. of Minnesota-Duluth) has a doctorate in philosophy and an advanced degree in computer science; he's worked as a philosophy professor, a computer programmer, and a research scientist in artificial intelligence. Here he discusses the philosophical foundations of artificial intelligence; the new encounter of science and philosophy (logic, models of the mind and of reasoning, epistemology); and the philosophy of computer science (touching on math, abstraction, software, and ontology).

Philosophical Perceptions on Logic and Order

Author: Horne, Jeremy
Publisher: IGI Global
ISBN: 9781522524441
Release Date: 2017-05-19
Genre: Philosophy

Strong reasoning skills are an important aspect to cultivate in life, as they directly impact decision making on a daily basis. By examining the different ways the world views logic and order, new methods and techniques can be employed to help expand on this skill further in the future. Philosophical Perceptions on Logic and Order is a pivotal scholarly resource that discusses the evolution of logical reasoning and future applications for these types of processes. Highlighting relevant topics including logic patterns, deductive logic, and inductive logic, this publication is an ideal reference source for academicians, students, and researchers that would like to expand their understanding of how society currently employs the use of logical reasoning techniques.

Modern Logic 1850 1950 East and West

Author: Francine F. Abeles
Publisher: Birkhäuser
ISBN: 9783319247564
Release Date: 2016-05-26
Genre: Mathematics

This book presents diverse topics in mathematical logic such as proof theory, meta-mathematics, and applications of logic to mathematical structures. The collection spans the first 100 years of modern logic and is dedicated to the memory of Irving Anellis, founder of the journal 'Modern Logic', whose academic work was essential in promoting the algebraic tradition of logic, as represented by Charles Sanders Peirce. Anellis’s association with the Russian logic community introduced their school of logic to a wider audience in the USA, Canada and Western Europe. In addition, the collection takes a historical perspective on proof theory and the development of logic and mathematics in Eastern Logic, the Soviet Union and Russia. The book will be of interest to historians and philosophers in logic and mathematics, and the more specialized papers will also appeal to mathematicians and logicians.

Eliminating The Universe Logical Properties Of Natural Language

Author: Keenan Edward L
Publisher: World Scientific
ISBN: 9789814719858
Release Date: 2018-07-06
Genre: Mathematics

This book synthesizes the author's work (1980s-2015) on the logical expressive power of natural language. It extends the tools and concepts of model theory as used in (higher order) predicate logic to the study of natural language semantics. It focuses on boolean structure, generalized quantification (separated from variable binding), covering some cases of anaphora. Different categories — predicates, adjective, quantifiers — are modeled by non-isomorphic boolean lattices.Of empirical linguistic interest is the expressibility of many natural classes of quantifiers defined in terms of their logical (automorphism invariant) properties. Some of these correlate with classes used syntactically in generative grammar. In other cases we find general (possibly universal) constraints on possible quantifier denotations in natural language.Also of novel logical interest are entailment paradigms that depend on relations between pairs or triples of generalized quantifier denoting expressions, ones that are in some cases inherently vague. In addition we note novel binary quantifiers that lie beyond the 'Frege boundary' in that they are provably not identical to any iterated application of unary quantifiers.Of philosophical interest is the existence of models which make the same sentences true as standard models but which lack a universe and hence, seemingly, a notion of 'reference'. Moreover, these models generalize to ones in which we can represent (some) intensional expressions without the use of novel ontological objects, such as 'possible worlds' or 'propositions'.

Advances in Contemporary Logic and Computer Science

Author: Walter Alexandre Carnielli
Publisher: American Mathematical Soc.
ISBN: 9780821813645
Release Date: 1999
Genre: Computers

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society (co-sponsored by the Centre for Logic, Epistemology and the History of Science, State University of Campinas, Sao Paolo) 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.

Principia Mathematica

Author: Alfred North Whitehead
ISBN: STANFORD:36105039675058
Release Date: 1984-01
Genre: Logic, Symbolic and mathematical

Proof and Other Dilemmas

Author: Roger Simons
Publisher: MAA
ISBN: 0883855674
Release Date: 2008
Genre: Mathematics

For the majority of the twentieth century, philosophers of mathematics focused their attention on foundational questions. However, in the last quarter of the century they began to return to basics, and two new schools of thought were created: social constructivism and structuralism. The advent of the computer also led to proofs and development of mathematics assisted by computer, and to questions concerning the role of the computer in mathematics. This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers have not yet discussed. The other half, written by philosophers of mathematics, summarise the discussion in that community during the last 35 years. A connection is made in each case to issues relevant to the teaching of mathematics.

Thinking about Godel and Turing

Author: Gregory J. Chaitin
Publisher: World Scientific
ISBN: 9789812708977
Release Date: 2007
Genre: Computational complexity

Dr Gregory Chaitin, one of the world's leading mathematicians, is best known for his discovery of the remarkable O number, a concrete example of irreducible complexity in pure mathematics which shows that mathematics is infinitely complex. In this volume, Chaitin discusses the evolution of these ideas, tracing them back to Leibniz and Borel as well as GAdel and Turing.This book contains 23 non-technical papers by Chaitin, his favorite tutorial and survey papers, including Chaitin's three Scientific American articles. These essays summarize a lifetime effort to use the notion of program-size complexity or algorithmic information content in order to shed further light on the fundamental work of GAdel and Turing on the limits of mathematical methods, both in logic and in computation. Chaitin argues here that his information-theoretic approach to metamathematics suggests a quasi-empirical view of mathematics that emphasizes the similarities rather than the differences between mathematics and physics. He also develops his own brand of digital philosophy, which views the entire universe as a giant computation, and speculates that perhaps everything is discrete software, everything is 0's and 1's.Chaitin's fundamental mathematical work will be of interest to philosophers concerned with the limits of knowledge and to physicists interested in the nature of complexity."

Der Turing Omnibus

Author: A.K. Dewdney
Publisher: Springer-Verlag
ISBN: 9783642788727
Release Date: 2013-03-12
Genre: Computers

Der Turing Omnibus macht in 66 exzellent geschriebenen Beiträgen Station bei den interessantesten Themen aus der Informatik, der Computertechnologie und ihren Anwendungen.

Mathematics Computer Science and Logic A Never Ending Story

Author: Peter Paule
Publisher: Springer Science & Business Media
ISBN: 9783319009667
Release Date: 2013-09-17
Genre: Computers

This book presents four mathematical essays which explore the foundations of mathematics and related topics ranging from philosophy and logic to modern computer mathematics. While connected to the historical evolution of these concepts, the essays place strong emphasis on developments still to come. The book originated in a 2002 symposium celebrating the work of Bruno Buchberger, Professor of Computer Mathematics at Johannes Kepler University, Linz, Austria, on the occasion of his 60th birthday. Among many other accomplishments, Professor Buchberger in 1985 was the founding editor of the Journal of Symbolic Computation; the founder of the Research Institute for Symbolic Computation (RISC) and its chairman from 1987-2000; the founder in 1990 of the Softwarepark Hagenberg, Austria, and since then its director. More than a decade in the making, Mathematics, Computer Science and Logic - A Never Ending Story includes essays by leading authorities, on such topics as mathematical foundations from the perspective of computer verification; a symbolic-computational philosophy and methodology for mathematics; the role of logic and algebra in software engineering; and new directions in the foundations of mathematics. These inspiring essays invite general, mathematically interested readers to share state-of-the-art ideas which advance the never ending story of mathematics, computer science and logic. Mathematics, Computer Science and Logic - A Never Ending Story is edited by Professor Peter Paule, Bruno Buchberger’s successor as director of the Research Institute for Symbolic Computation.

Logik f r Informatiker

Author: Uwe Schöning
Publisher: Spektrum Akademischer Verlag
ISBN: 3827410053
Release Date: 2000-01-20
Genre: Computers

Das Buch macht den Leser mit den wesentlichen Teilgebieten der formalen Logik vertraut, die Bestandteil der Ausbildung in Theoretischer Informatik sind. Die Darstellung orientiert sich an den Bedürfnissen von Informatikstudierenden. Insbesondere werden viele mehr auf das Prinzipielle ausgerichtete Resultate der formalen Logik unter einem algorithmischen Gesichtspunkt behandelt. Diese Vorgehensweise erleichtert entscheidend den Zugang zu dem abstrakten Themengebiet. Prof. Schöning gelingt eine kompakte und verständliche Darstellung der Aussagen- und Prädikatenlogik, bei der die benötigten Begriffe präzise eingeführt und durch Beispiele veranschaulicht werden. Darauf beruhend werden Anwendungen der Logik in der Informatik, wie z. B. Resolution, Automatisches Beweisen und Logik-Programmierung behandelt. Zahlreiche Übungsaufgaben mit ausführlichen Lösungshinweisen erleichtern die Vertiefung des Lernstoffes.

Der Tod und das Leben danach

Author: Samuel Scheffler
Publisher: Suhrkamp Verlag
ISBN: 9783518741085
Release Date: 2015-03-07
Genre: Philosophy

Wie würden Sie reagieren, wenn Sie wüssten, dass 30 Tage nach Ihrem Tod die Erde und damit alles Leben auf ihr unwiederbringlich zerstört würden? Würde dieses Wissen die Art und Weise, wie Sie Ihr Leben führen, beeinflussen? Das ist das Gedankenexperiment, zu dem uns der amerikanische Philosoph Samuel Scheffler in seinem faszinierenden Buch einlädt. Er zeigt, dass ein solches Wissen weitreichende Folgen für unser Leben hätte – nichts wäre mehr wie zuvor! In ebenso luziden wie psychologisch verblüffenden Analysen, die immer wieder auf geniale Weise Beispiele aus der Populärkultur heranziehen, zeigt Scheffler, dass ein solches Wissen um den Untergang der Menschheit den Wert zahlreicher unserer Tätigkeiten in Frage stellen würde: Die langfristige medizinische Forschung nach einer Krebstherapie verlöre ihren Sinn, aber auch der Kampf gegen den Klimawandel oder der Einsatz für internationale Gerechtigkeit. Und würden wir noch Kunstwerke schaffen, Traditionen und Bräuche pflegen, uns verlieben, Kinder kriegen? Wohl kaum. Vielmehr steht zu befürchten, dass gesellschaftliche Regeln und Konventionen nicht mehr beachtet würden und anarchische Zustände drohten, wie Scheffler anhand des Romans Children of Men von P. D. James und seiner Verfilmung vorführt. Könnte es daher sein, dass uns das Überleben der Menschheit wichtiger ist als unser eigenes? Und was folgt daraus für unser Denken und Handeln in der Welt von heute? Ein kleines philosophisches Meisterwerk, das unser eigenes Leben in einem ganz anderen Licht erscheinen lässt.

Was ist Mathematik

Author: Richard Courant
Publisher: Springer-Verlag
ISBN: 9783662000533
Release Date: 2013-03-09
Genre: Mathematics

47 brauchen nur den Nenner n so groß zu wählen, daß das Intervall [0, IJn] kleiner wird als das fragliche Intervall [A, B], dann muß mindestens einer der Brüche m/n innerhalb des Intervalls liegen. Also kann es kein noch so kleines Intervall auf der Achse geben, das von rationalen Punkten frei wäre. Es folgt weiterhin, daß es in jedem Intervall unendlich viele rationale Punkte geben muß; denn wenn es nur eine endliche Anzahl gäbe, so könnte das Intervall zwischen zwei beliebigen benachbarten Punkten keine rationalen Punkte enthalten, was, wie wir eben sahen, unmöglich ist. § 2. Inkommensurable Strecken, irrationale Zahlen und der Grenzwertbegriff 1. Einleitung Vergleicht man zwei Strecken a und b hinsichtlich ihrer Größe, so kann es vor kommen, daß a in b genau r-mal enthalten ist, wobei r eine ganze Zahl darstellt. In diesem Fall können wir das Maß der Strecke b durch das von a ausdrücken, indem wir sagen, daß die Länge von b das r-fache der Länge von a ist.