Mathematical logic Books

Book SynopsisAn Introduction to Fuzzy Sets provides a comparison of the quality of life in urban, intermediate and rural NUTS III regions in Portugal, with the main goal of identifying and analysing the necessary and conditions for a high quality of life in those different regions. The authors assess the necessary and sufficient conditions for higher Human Development Index levels, aiming to determine whether the same pattern could be used to explain the happiness index. In order to represent the applications of fuzzy set theory as well as neuro-fuzzy in industry, a literature review of these topics is carried out. As some researchers have efficiently utilized fuzzy logic and neuro-fuzzy, in-depth discussions are provided for stimulating future investigations. Following this, using the L. Zadeh theory of fuzzy sets, the authors consider all types of uncertainties in oil fields and oil production to make a decision as to what model is best in such a fuzzy environment. Additionally, several challenges are explored, such as: the fuzzy random finite difference numerical method, possibilistic uncertainty modelling, and the development of a fuzzy Wilks' theorem to model the hybrid structure of randomness and fuzziness modelling. In closing, a standard fuzzy arithmetic method is used for solving fuzzy equations, as well as for the optimization of fuzzy objectives. The fuzzy variable of the equation is fuzzified using a fuzzy set.Table of ContentsPreface; Quality of Life: Urban versus Rural Analysis Based on Fuzzy Sets Approach; Economically Speaking, Are Happiness and HDI the Same? The Fuzzy-Set Approach; Implementation of Fuzzy Logic and Neuro-Fuzzy in Industry; Lotfi Zadehs Theory of Fuzzy Sets in Decision-Making Process for Oil and Gas Production; Fuzziness-Randomness Modeling of Plasma Disruption in First Wall of Fusion Reactor Using Type I Fuzzy Random Set; Application of a Standard Fuzzy Arithmetic Method; Index.

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Book SynopsisFormal Logic is an undergraduate text suitable for introductory, intermediate, and advanced courses in symbolic logic. The book’s nine chapters offer thorough coverage of truth-functional and quantificational logic, as well as the basics of more advanced topics such as set theory and modal logic. Complex ideas are explained in plain language that doesn’t presuppose any background in logic or mathematics, and derivation strategies are illustrated with numerous examples. Translations, tables, trees, natural deduction, and simple meta-proofs are taught through over 400 exercises. A companion website (complimentary for anyone who buys the book) offers supplemental practice software and tutorial videos.Trade Review“Formal Logic is clear, accessible, and intuitive, but it is also precise, explicit, and thorough. Complex and often confusing concepts are rolled out in a no-nonsense and direct manner with funny and demystifying terminology and helpful analogies. It's a pedagogical gem.” — Mary Kate McGowan, Wellesley College“This is an excellent introductory text in symbolic logic. It is accessible, with clear and concise explanations of key concepts, along with many helpful examples and practice problems, but also rigorous enough to prepare students for a second course in logic; indeed, I do not know of any book that better combines these virtues. I am looking forward to using Formal Logic in my courses.” — Kevin Morris, Tulane University“This book makes the ideas of sentential logic, predicate logic, and formal proof easily accessible by getting directly to the point of each in natural, non-technical language. It is concise while never hurried. It gets the details right, not by focusing on them as details, but through clear insight into why they are as they are.” — Colin McLarty, Case Western Reserve University“Paul Gregory’s Formal Logic is worth careful consideration for anyone adopting a new logic text. The inclusion of chapters on set theory and modal logic makes it a valuable resource for students looking to go beyond the standard introduction to logic.” — Michael Hicks, Miami UniversityTable of ContentsI: Informal Notions1: Informal Introduction1.1 Logic: What, Why, How?1.2 Arguments, Forms, and Truth Values1.3 Deductive Criteria1.3.1Quirky Cases of Deductive Validity1.4 Inductive Criteria1.5 Other Deductive Properties1.6 Exercises1.7 Chapter GlossaryII: Truth-Functional Logic2: The Language S2.1 Introducing S2.1.1 Compound Sentences and Truth-Functional Logic 2.1.2 Negation—It is not the case that…2.1.3 Conjunction—Both…and---2.1.4 Disjunction—Either…or---2.1.5 Material Conditional—If …, then---2.1.6 Material Biconditional—…if and only if---2.1.7 Conditionals and Non-Truth-Functionality2.2 Some Technical Bits2.2.1 Object Language and Metalanguage2.2.2 Use and Mention2.2.3 Metavariables2.2.4 Syntax and Semantics 2.3 The Syntax of S2.3.1 Defining the Language2.3.2 Syntactic Concepts and Conventions2.3.3 Exercises2.4 Alternate Symbols and Other Choices2.5 Chapter Glossary3: Formal Semantics for S3.1 Truth Value Assignments and Truth Tables3.2 Semantic Properties of Individual Wffs3.2.1 Exercises3.3 Semantic Properties of Sets of Wffs3.3.1 Exercises3.4 Semantic Properties, Their Interrelations, and Simple Metalogic3.4.1 Exercises3.5 Truth Trees3.5.1 Tests with Truth Trees3.5.2 Exercises3.6 Chapter Glossary4: SD: Natural Deduction in S4.1 The Basic Idea4.1.1 Reiteration4.1.2 Wedge Rules4.1.3 Arrow Rules4.1.4 Hook Rules4.1.5 Vee Rules4.1.6 Double Arrow Rules4.1.7 Exercises4.2 Derivations: Strategies and Notes4.3 Proof Theory in SD4.3.1 Exercises4.4 SDE, an Extension to SD4.4.1 The Inference Rules of SDE4.4.2 Exercises4.4.3 The Replacement Rules of SDE4.4.4 Exercises4.5 Chapter GlossaryIII: Quantificational Logic5: The Language P5.1 Introducing P5.1.1 Quantificational Logic5.1.2 Predicates and Singular Terms5.1.3 Predicate Letters and Individual Constants in P5.1.4 Pronouns and Quantifiers5.1.5 Variables and Quantifiers in P 5.2 The Syntax of P5.2.1 Defining the Language5.2.2 Syntactic Concepts and Conventions5.2.3 Exercises5.3 Simple Symbolizations5.3.1 Non-categorical Claims5.3.2 Exercises5.3.3 Categorical Claims5.3.4 Exercises5.4 Complex Symbolizations5.4.1 Basics of Overlapping Quantifiers5.4.2 Exercises5.4.3 Identity, Numerical Quantification, and Definite Descriptions5.4.4 Exercises5.5 Chapter Glossary6: Formal Semantics for P6.1 Semantics and Interpretations6.1.1 Basics of Interpretations6.1.2 Interlude: A Little Bit of Set Theory6.1.3 Formal Interpretation of P6.1.4 Constructing Interpretations6.2 Semantic Properties of Individual Wffs6.2.1 Exercises6.3 Semantic Properties of Sets of Wffs6.3.1 Exercises6.4 Quantifier Scope and Distribution6.4.1 Exercises6.5 Properties of Relations6.5.1 Exercises6.6 Chapter Glossary7: PD: Natural Deduction in P7.1 Derivation Rules for the Quantifiers7.1.1 Universal Elimination7.1.2 Existential Introduction7.1.3 Universal Introduction7.1.4 Existential Elimination7.1.5 Exercises7.2 Derivations: Strategies and Notes7.3 Proof Theory in PD7.3.1 Exercises7.4 PDE, an Extension to PD7.4.1 Quantifier Negation7.4.2 Exercises7.4 Chapter GlossaryIV: Advanced Topics8: Basic Set Theory, Paradox, and Infinity8.1 Basics of Sets8.2 Russell’s Paradox8.3 The Axiom Schema of Separation8.4 Subset, Intersection, Union, Difference8.4.1 Exercises8.5 Pairs, Ordered Pairs, Power Sets, Relations, and Functions8.6 Infinite Sets and Cantor’s Proof8.6.1 Exercises8.7 Chapter Glossary9: Modal Logic9.1 Necessity, Possibility, and Impossibility9.1.1 Modalities9.1.2 Logical, Metaphysical, Physical9.1.3 Possible Worlds9.2 The Language S9.2.1 The Syntax of S9.2.2 Exercises9.3 Basic Possible Worlds Semantics for S9.3.1 Semantic Properties of Wffs and Sets of Wffs9.3.2 Exercises9.3.3 Possible Worlds and Trees9.3.4 Exercises9.4 Natural Deduction in S9.4.1 System K9.4.2 System D9.4.3 System T9.4.4 System B9.4.5 System S49.4.6 System S59.4.7 Relations Between Modal Systems9.4.8 Exercises9.5 Chapter GlossaryV: AppendicesA: Answers to ExercisesB: GlossaryC: Truth Tables, Tree Rules, and Derivation RulesC.1 Characteristic Truth TablesC.2 Truth Tree Rules for SC.3 The Derivation System SDC.4 The Derivation System SDEC.5 The Derivation System PD

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Book SynopsisFROM THE PRESENTER OF THE TEDx TALK 'You weren't bad at maths - you just weren't looking at it the right way' 'Compelling and wonderfully readable' - Ian Stewart, bestselling author of Seventeen Equations that Changed the World 'AI is powerful, but human thinking is differently powerful, and Junaid Mubeen deftly shows us how' - Eugenia Cheng, author of How to Bake Pi There's so much talk about the threat posed by intelligent machines that it sometimes seems as though we should surrender to our robot overlords now. But Junaid Mubeen isn't ready to throw in the towel just yet. As far as he is concerned, we have the edge over machines because of a remarkable system of thought developed over the millennia. It's familiar to us all, but often badly taught and misrepresented in popular discourse - maths. Computers are brilliant at totting up sums, pattern-seeking and performing, well, computation. For all things calculation, machines reign supreme. But Junaid identifies seven areas of intelligence where humans can retain a crucial edge. And in exploring these areas, he opens up a fascinating world where we can develop our uniquely human mathematical superpowers.Trade Review[An] intelligent analysis * Nature *Insightful * Popular Science *A compelling and wonderfully readable analysis of why computers won't replace mathematicians, but why the two together are superior to either on its own. A rallying-cry for real intelligence in the age of algorithms and artificial intelligence. -- Ian Stewart, author * What's the Use? *Maths needs more demystifiers, and Junaid Mubeen is here to lift back the veil to show the inner workings of maths and mathematicians. This book importantly shows that computers and AI do not make mathematicians redundant - in fact, Mubeen uses the advances and stumbling blocks in AI to illuminate the crucial contribution that human mathematicians continue to make. I recommend this to anyone who thinks - or knows someone who thinks - that AI will make the study of maths redundant. AI is powerful, but human thinking is differently powerful, and Junaid Mubeen deftly shows us how. -- Eugenia Cheng, author * X + Y: A Mathematician's Manifesto for Rethinking Gender *

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Book SynopsisThe fourth edition of a classic book on logic has been thoroughly revised by the author's son. It is a fundamental guide to modern mathematical logic and to the construction of mathematical theories. The first half covers the elements of logic, and the second half covers the applications of logic in theory building. A short biographical sketch of Alfred Tarski is a newly-added section.Trade Review"For Tarski logic was not only an essential tool of mathematics but the very foundation of it. What is more, he credited logic with having even more general meaning and significance. This new edition of Tarski's classic book will certainly help a new generation of readers in this respect." -- Roman Murawski, Modern Logic, Vol 8, No 1/2 (January 1998 - April 2000) "For Tarski logic was not only an essential tool of mathematics but the very foundation of it. What is more, he credited logic with having even more general meaning and significance. This new edition of Tarski's classic book will certainly help a new generation of readers in this respect." -- Roman Murawski, Modern Logic, Vol 8, No 1/2 (January 1998 - April 2000)Table of ContentsFIRST PART: Elements of Logic. Deductive Method 1: On the Use of Variables 2: On the Sentential Calculus 3: On the Theory of Identity 4: On the Theory of Classes 5: On the Theory of Relations 6: On the Deductive Method SECOND PART: Applications of Logic and Methodology in Constructing Mathematical Theories 7: Construction of a Mathematical Theory: Laws of Order for Numbers 8: Construction of a Mathematical Theory: Laws of Addition and Subtraction 9: Methodological Considerations of the Constructed Theory 10: Extension of the Constructed Theory: Foundations of Arithmetic of Real Numbers

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Book SynopsisKurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein''s equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel''s publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel''s Nachlass. These long-awaited final two volumes contain Gödel''s corTrade ReviewThe whole enterprise is superbly coordinated and assembled under the direction of Solomon Feferman ... The book is a tour de force and a labour of love. Superbly crafted and presented, what a bargain, given the many gems it contains! * The Mathematical Gazette *The books are carefully and beautifully produced and offer rich material, illuminating not only the outstanding work of Gödel, but also the whole mathematical logic of the twentieth century, including some philosophical and historical aspects. * EMS *Table of ContentsGödel's life and workSolomon Feferman: A Gödel chronologyJohn W. Dawson, Jr.: Gödel 1929: Introductory note to 1929, 1930 and 1930aBurton Dreben and Jean van Heijenoort: Über die Vollständigkeit des Logikkalküls On the completeness of the calculus of logic Gödel 1930: (See introductory note under Gödel 1929.) Die Vollständigkeit der Axiome des logischen Funktionenkalküls The completeness of the axioms of the functional calculus of logic Gödel 1930a: (See introductory note under Gödel 1929.) Über die Vollständigkeit des Logikkalküls On the completeness of the calculus of logic Gödel 1930b: Introductory note to 1930b, 1931 and 1932bStephen C. Kleene: Einige metamathematische Resultate über Entscheidungs-definitheit und Widerspruchsfreiheit Some metamathematical results on completeness and consistency Gödel 1931: (See introductory note under Gödel 1930b.) Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I On formally undecidable propositions of Principia mathematica and related systems I Gödel 1931a: Introductory note to 1931a, 1932e, f and gJohn W. Dawson, Jr.: Diskussion zur Grundlegung der Mathematik Discussion on providing a foundation for mathematics Gödel 1931b: Review of Neder 1931 Gödel 1931c: Introductory note to 1931cSolomon Feferman: Review of Hilbert 1931 Gödel 1931d: Review of Betsch 1926 Gödel 1931e: Review of Becker 1930 Gödel 1931f: Review of Hasse and Scholz 1928 Gödel 1931g: Review of von Juhos 1930 Gödel 1932: Introductory note to 1932A. S. Troelstra: Zum intuitionistischen aussagenkalkül On the intuitionistic propositional calculus Gödel 1932a: Introductory note to 1932a, 1933i and lWarren D. Goldfarb: Ein Spezialfall des Enscheidungsproblems der theoretischen Logik A special case of the decision problem for theoretical logic Gödel 1932b: (See introductory note under Gödel 1930b.) Über Vollständigkeit und Widerspruchsfreiheit On completeness and consistency Gödel 1932c: Introductory note to 1932cW. V. Quine: Eine Eigenschaft der Realisierungen des Aussagenkalküls A property of the realizations of the propositional calculus Gödel 1932d: Review of Skolem 1931 Gödel 1932e: (See introductory note under Gödel 1931a.) Review of Carnap 1931 Gödel 1932f: (See introductory note under Gödel 1931a.) Review of Heyting 1931 Gödel 1932g: (See introductory note under Gödel 1931a.) Review of von Neumann 1931 Gödel 1932h: Review of Klein 1931 Gödel 1932i: Review of Hoensbroech 1931 Gödel 1932j: Review of Klein 1932 Gödel 1932k: Introductory note to 1932k, 1934e and 1936bStephen C. Kleene: Review of Church 1932 Gödel 1932l: Review of Kalmár 1932 Gödel 1932m: Review of Huntington 1932 Gödel 1932n: Review of Skolem 1932 Gödel 1932o: Review of Dingler 1931 Gödel 1933: Introductory note to 1933W. V. Quine: [[Über die Parryschen Axiome]] [[On Parry's axioms]] Gödel 1933a: Introductory note to 1933aW. V. Quine: Über Unabhängigkeitsbeweise im Aussagenkalkül On independence proofs in the propositional calculus Gödel 1933b: Introductory note to 1933b, c, d, g and hJudson Webb: Über die metrische Einbettbarkeit der Quadrupel des R[3 in Kugelflächen On the isometric embeddability of quadruples of points of R[3 in the surface of a sphere Gödel 1933c: (See introductory note under Gödel 1933b.) Über die Waldsche Axiomatik des Zwichenbegriffes On Wald's axiomization of the notion of betweenness Gödel 1933d: (See introductory note under Gödel 1933b.) Zur Axiomatik der elementargeometrischen Verknüpfungs-relationen On the axiomatization of the relations of connection in elementary geometry Gödel 1933e: Introductory note to 1933eA. S. Troelstra: Zur institutionistischen Arithmetik und Zahlentheorie On intuitionistic arithmetic and number theory Gödel 1933f: Introductory note to 1933fA. S. Troelstra: Eine Interpretation des institutionistischen Aussagenkalküls An interpretation of the intuitionistic propositional calculus Gödel 1933g: (See introductory note under Gödel 1933b.) Bemerkung über projektive Abbildungen Remark concerning projective mappings Gödel 1933h: (See introductory note under Gödel 1933b.) Diskussion über koordinatenlose Differentialgeometrie Discussion concerning coordinate-free differential geometry Gödel 1933i: (See introductory note under Gödel 1932a.) Zum Enscheidungsproblem des logischen Funktionenkalküls On the decision probelm for the functional calculus of logic Gödel 1933j: Review of Kaczmarz 1932 Gödel 1933k: Review of Lewis 1932 Gödel 1933l: (See introductory note under Gödel 1932a.) Review of Kalmár 1933 Gödel 1933m: Review of Hahn 1932 Gödel 1934: Introductory note to 1934Stephen C. Kleene: On undecidable propositions of formal mathematical systems Gödel 1934a: Review of Skolem 1933 Gödel 1934b: Introductory note to 1934bW. V. Quine: Review of Quine 1933 Gödel 1934c: Introductory note to 1934c and 1935Robert L. Vaught: Review of Skolem 1933a Gödel 1934d: Review of Chen 1933 Gödel 1934e: (See introductory note under Gödel 1932k.) Review of Church 1933 Gödel 1934f: Review of Notcutt 1934 Gödel 1935: (See introductory note under Gödel 1934c.) Review of Skolem 1934 Gödel 1935a: Introductory note to 1935aW. V. Quine: Review of Huntington 1934 Gödel 1935b: Review of Carnap 1934 Gödel 1935c: Review of Kalmár 1934 Gödel 1936: Introductory note to 1936John W. Dawson, Jr.: Diskussionsbemerkung Discussion remark Gödel 1936a: Introductory note to 1936aRohit Parikh: Über die Länge von Beweisen On the length of proofs Gödel 1936b: (See introductory note under Gödel 1932k.) Review of Church 1935 Textual notes References Index

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Book SynopsisKurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein''s equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel''s publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel''s Nachlass. These long-awaited final two volumes contain Gödel''s corTrade Review"The book....will certainly enlarge our appreciation of Gödel's scientific and philosophical thought as well as our understanding of his motivations. With great impatience we await now the succeeding volume...." --Mathematical Reviews"As a whole this volume is as indispensable as the two former ones for any serious student of Godel's ideas and achievements, but in this case it is also indispensable for philosophers interested in logic and mathematics. The fourth (and last?) volume of this formidable series will be devoted to Godel's correspondance, so we should look forward to having it to study."--Modern Logic"On the whole....the editors are to be wholeheartedly congratulated on bringing to the public work whi deserves careful study and which ought to do something to revitalise the philosophy of mathematics by presenting a point of view that, unusualy, combines intellectual rogour with a willingness to make bold and sweeping metaphysical claims." --Times Higher Education Supplement"This is the third volume of a comprehensive and critical edition of the works of Kurt Gödel. . .All these essays and lectures are most carefully written and remarkably rich. They give considerable insight into Gödel's own achievements in logic, set theory and physics and also into his philosophical views. . . .This volume was a desideratum for a long time. We also hope very strongly that volume 3 is not the last volume." --Vienna Circle Institute Yearbook 1997 contains unpublished materialTable of Contents1. The Nachlass of Kurt Godel: an overview ; 2. Godel's Gabelsberger shorthand ; 3. Godel *1930c: Introductory note to *1930c ; 4. Lecture on completeness of the functional calculus ; 5. Godel *1931?: Introductory note to *1931? ; 6. On undecidable sentences ; 7. Godel *1933c: Introductory note to *1933c ; 8. The present situation in the foundations of mathematics ; 9. Godel *1933?: Introductory note to *1933? ; 10. Simplified proof of a theorem of Steinitz ; 11. Godel *1938a: Introductory note to *1938a ; 12. Lecture at Zilsel's ; 13. Godel *1939b: Introductory note to *1939b and *1940a ; 14. Lecture at Gottingen ; 15. Godel *193?: Introductory note to *193? ; 16. Undecidable diophantine propositions ; 17. Godel *1940a ; 18. Lecture on the consistency of the continuum hypothesis ; 19. Godel *1941: Introductory note to *1941 ; 20. In what sense is intuitionistic logic constructive? ; 21. Godel *1946/9: Introductory note to *1946/9 ; 22. Some observations about the relationship between theory of relativity and Kantian philosophy ; 23. Godel *1949b: Introductory note to *1949b ; 24. Lecture on rotating universes ; 25. Godel *1951: Introductory note to *1951 ; 26. Some basic theorems on the foundations of mathematics and their implications ; 27. Godel *1953/9: Introductory note to *1953/9 ; 28. Is mathematics syntax of language? Version III ; 29. Is mathematics syntax of language? Version V ; 30. Godel *1961/?: Introductory note to *1961/? ; 31. The modern development of the foundations of mathematics in the light of philosophy ; 32. Godel *1970: Introductory note to *1970 ; 32. Ontological proof ; 33. Godel *1970a: Introductory note to *1970a, *1970b and *1970c ; 34. Some considerations leading to the probable conclusion that the true power of the continuum is N[2 ; 35. Godel *1970b ; 36. A proof of Cantor's continuum hypothesis from a highly plausible axiom about orders of growth ; 37. Godel *1970c ; 38. Unsent letter to Alfred Tarski ; Appendix A: Excerpt from *1946/9-A ; Appendix B: Texts relating to the ontological proof

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Book SynopsisThe Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with many influential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.Trade ReviewOverall, the book presents a clear picture of the Quinean world view. * Mathematical Reviews *

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Book SynopsisMathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positionsTrade Review"The Oxford Handbook of the Philosophy of Mathematics and Logic is most certainly here to stay for a very long time. The quality of each of the contributions is reflected in the authors' stimulating writing. The handbook can add substantially to the emerging thoughts and studies on the subject."--Current Engineering Practice"The Oxford Handbook of the Philosophy of Mathematics and Logic is a very accessible, wide ranging work that serves not only to indicate the 'state of the art' in the given area, but, remarkably, also serves as a very fine introduction to the field. I recommend it highly, both to workers in the given field and, equally, to the 'general philosopher,' regardless of one's main area." --Notre Dame Philosophical Reviews

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Book SynopsisStewart Shapiro presents a distinctive and persuasive view of the foundations of mathematics, arguing controversially that second-order logic has a central role to play in laying these foundations. To support this contention, he first gives a detailed development of second-order and higher-order logic, in a way that will be accessible to graduate students. He then demonstrates that second-order notions are prevalent in mathematics as practised, and that higher-order logic is needed to codify many contemporary mathematical concepts. Throughout, he emphasizes philosophical and historical issues that the subject raises. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic. ''In this excellent treatise Shapiro defends the use of second-order languages and logic as framework for mathematics. His coverage of the wide range of logical and philosophical topics required for understanding the controversy over second-order logic is Trade ReviewContains more on second-order logic than is readily available in any other textbook or survey. Philosophically, the book also contains many words of wisdom. * Journal of Symbolic Logic *Table of ContentsPART I: ORIENTATION; 1. TERMS AND QUESTIONS; 2. FOUNDATIONALISM AND FOUNDATIONS OF MATHEMATICS; PART II: LOGIC AND MATHEMATICS; 3. THEORY; 4. METATHEORY; 5. SECOND-ORDER LOGIC AND MATHEMATICS; 6. ADVANCED METATHEORY; PART III: HISTORY AND PHILOSOPHY; 7. THE HISTORICAL 'TRIUMPH' OF FIRST-ORDER LANGUAGES; 8. SECOND-ORDER LOGIC AND RULE-FOLLOWING; 9. THE COMPETITION; REFERENCES; INDEX

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Book SynopsisSet theory is concerned with the foundation of mathematics. In the original formulations of set theory, there were paradoxes contained in the idea of the set of all sets. Current standard theory (Zermelo-Fraenkel) avoids these paradoxes by restricting the way sets may be formed by other sets, specifically to disallow the possibility of forming the set of all sets. In the 1930s, Quine proposed a different form of set theory in which the set of all sets - the universal set - is allowed, but other restrictions are placed on these axioms. Since then, the steady interest expressed in these non-standard set theories has been boosted by their relevance to computer science.The second edition still concentrates largely on Quine''s New Foundations, reflecting the author''s belief that this provides the richest and most mysterious of the various systems dealing with set theories with a universal set. Also included is an expanded and completely revised account of the set theories of Church-Oswald and Mitchell, with descriptions of permutation models and extensions that preserve power sets. Dr Foster here presents the reader with a useful and readable introduction for those interested in this topic, and a reference work for those already involved in this area.Trade Review...a lively introductin to the current research on NF' * Maruice Boffa, Modern Logic *Table of Contents1. Introduction ; 2. NF and related systems ; 3. Permutation models ; 4. Church-Oswald models ; 5. Open problems ; 6. Bibliography

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Book SynopsisThe ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author''s teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course.Trade Review'a clear and unifying treatment of fundamental concepts underlying Computer Sciences and Foundations of Mathematics' Professor Boris Zilber (Professor of Mathematical Logic, University of Oxford)'an excellent book' Professor Dov Gabbay (King's College, London)Table of ContentsPreliminaries ; 1. Propositional Logic ; 2. Structures and First-Order Logic ; 3. Proof Theory ; 4. Properties of First-Order Logic ; 5. First-Order Theories ; 6. Models of Countable Theories ; 7. Computability and Complexity ; 8. The Incompleteness Theorems ; 9. Beyond First-Order Logic ; 10. Finite Model Theory ; Bibliography ; Index

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Book SynopsisIntermediate Logic is an ideal text for anyone who has taken a first course in logic and is progressing to further study. It examines logical theory, rather than the applications of logic, and does not assume any specific technical grounding. The author introduces and explains each concept and term, ensuring that readers have a firm foundation for study. He provides a broad, deep understanding of logic by adopting and comparing a variety of different methods and approaches.In the first section, Bostock covers such fundamental notions as truth, validity, entailment, qualification, and decision procedures. Part Two lays out a definitive introduction to four key logical tools or procedures: semantic tableaux, axiomatic proofs, natural deduction, and sequent calculi. The final section opens up new areas of existence and identity, concluding by moveing from orthodox logic to an examination of `free logic''.Intermediate Logic provides an ideal secondary course in logic for university studentTrade ReviewThis textbook covers the fundamental proof-theoretical and model-theoretical aspects of classical propositional and first-order logic. . . .The book is clearly written and ideally suited for an intermediate course on the subject, requiring just some elementary knowledge of proof theory and model theory. * Mathematical Reviews *

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Book SynopsisHow strongly should you believe the various propositions that you can express?That is the key question facing Bayesian epistemology. Subjective Bayesians hold that it is largely (though not entirely) up to the agent as to which degrees of belief to adopt. Objective Bayesians, on the other hand, maintain that appropriate degrees of belief are largely (though not entirely) determined by the agent''s evidence. This book states and defends a version of objective Bayesian epistemology. According to this version, objective Bayesianism is characterized by three norms: Probability - degrees of belief should be probabilities Calibration - they should be calibrated with evidence Equivocation - they should otherwise equivocate between basic outcomesObjective Bayesianism has been challenged on a number of different fronts. For example, some claim it is poorly motivated, or fails to handle qualitative evidence, or yields counter-intuitive degrees of belief after updating, or suffers from a failureTable of ContentsPreface ; 1. Introduction ; 2. Objective Bayesianism ; 3. Motivation ; 4. Updating ; 5. Predicate Languages ; 6. Objective Bayesian Nets ; 7. Probabilistic Logic ; 8. Judgement Aggregation ; 9. Languages and Relativity ; 10. Objective Bayesianism in Perspective ; References ; Index

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Book SynopsisPacked with more than a hundred color illustrations and a wide variety of puzzles and brainteasers, Taking Sudoku 2eriously uses this popular craze as the starting point for a fun-filled introduction to higher mathematics. How many Sudoku solution squares are there? What shapes other than three-by-three blocks can serve as acceptable Sudoku regions? What is the fewest number of starting clues a sound Sudoku puzzle can have? Does solving Sudoku require mathematics? Jason Rosenhouse and Laura Taalman show that answering these questions opens the door to a wealth of interesting mathematics. Indeed, they show that Sudoku puzzles and their variants are a gateway into mathematical thinking generally. Among many topics, the authors look at the notion of a Latin square--an object of long-standing interest to mathematicians--of which Sudoku squares are a special case; discuss how one finds interesting Sudoku puzzles; explore the connections between Sudoku, graph theory, and polynomials; and cTrade ReviewThis well-written book would be of interest to anyone, mathematician or not, who likes solving Sudoku puzzles. * Donald Keedwell, Mathematical Gazette *This is an interesting book. The style is conversational and east to read ... * John Sykes, Mathematics in School *I thoroughly enjoyed this book and do not have any criticisms to make. The authors have produced a lovely addition to any budding or practiced mathematicians bookcase. Well-presented and readable for both the novice and the maths expert, which is an admirable feat, this book is for anyone with an interest, no matter how vague or intense, in Sudoku. * Angie Wade, Significance *A beautiful book. * Paul Levrie, Karel de Grote University College *Table of Contents1. Playing the Game ; Mathematics as Applied Puzzle-Solving ; 2. Latin Squares ; What Do Mathematicians Do? ; 3. Greco-Latin Squares ; The Problem of the Thirty-Six Officers ; 4. Counting ; It's Harder Than it Looks ; 5. Equivalence Classes ; The Importance of Being Essentially Identical ; 6. Searching ; The Art of Finding Needles in Haystacks ; 7. Graphs ; Dots, Lines and Sudoku ; 8. Polynomials ; We Finally Found a Use For Algebra ; 9. Extremes ; Sudoku Pushed to its Limits ; 10. Epilogue ; You Can Never Have Too Many Puzzles ; Solutions to Puzzles

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Book SynopsisThis is the fascinating story of two women who lives were guided by a passion for mathematics and an insatiable curiosity to know and understand the world around them -- the beautiful, outrageous Émilie du Châtelet and the charmingly subversive Mary Somerville. Against great odds, Émilie and Mary taught themselves mathematics, and did it so well that they each became a world authority on Newtonian mathematical physics.Seduced by Logic begins with Émilie du Châtelet, an 18th-century French aristocrat, intellectual, and Voltaire''s lover, whose true ambition was to be a mathematician. She strove not only to further Newton''s ideas in France, but to prove that they had French connections, including to the work of Descartes, whom Newton had read. She translated the great Principia Mathematica into French, in what became the accepted French version of Newton''s work, and was instrumental in bringing Newton''s revolutionary opus to a Continental audience. A century later, in Scotland, Mary STrade Review...timely reminder of how little things have changed since the 19th century and how much women of science can accomplish. * Wall Street Journal *Table of ContentsIntroduction ; 1 Madame Newton du Chatelet ; 2 Creating the theory of gravity: the Newtonian controversy ; 3 Learning mathematics and fighting for freedom ; 4 Emilie and Voltaire's Academy of Free Thought ; 5 Testing Newton: the'New Argonauts' ; 6 The danger in Newton: life, love and politics ; 7 The nature of light: Emilie takes on Newton ; 8 Searching for 'energy': Emilie discovers Leibniz ; 9 Mathematics and free will ; 10 The re-emergence of Madame Newton du Chatelet ; 11 Love letters to Saint-Lambert ; 12 Mourning Emilie ; 13 Mary Fairfax Somerville ; 14 The long road to fame ; 15 Mechanism of the Heavens ; 16 Mary's second book: popular science in the nineteenth century ; 17 Finding light waves: the 'Newtonian Revolution' comes of age ; 18 Mary Somerville: a fortunate life ; Epilogue: Declaring a point of view

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1 The Axiom of Extension.- 2 The Axiom of Specification.- 3 Unordered Pairs.- 4 Unions and Intersections.- 5 Complements and Powers.- 6 Ordered Pairs.- 7 Relations.- 8 Functions.- 9 Families.- 10 Inverses and Composites.- 11 Numbers.- 12 The Peano Axioms.- 13 Arithmetic.- 14 Order.- 15 The Axiom of Choice.- 16 Zorn's Lemma.- 17 Well Ordering.- 18 Transfinite Recursion.- 19 Ordinal Numbers.- 20 Sets of Ordinal Numbers.- 21 Ordinal Arithmetic.- 22 The Schröder-Bernstein Theorem.- 23 Countable Sets.- 24 Cardinal Arithmetic.- 25 Cardinal Numbers.

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Book Synopsis1 Naive Set Theory.- 1.1 What is a Set?.- 1.2 Operations on Sets.- 1.3 Notation for Sets.- 1.4 Sets of Sets.- 1.5 Relations.- 1.6 Functions.- 1.7 Well-Or der ings and Ordinals.- 1.8 Problems.- 2 The ZermeloFraenkel Axioms.- 2.1 The Language of Set Theory.- 2.2 The Cumulative Hierarchy of Sets.- 2.3 The ZermeloFraenkel Axioms.- 2.4 Classes.- 2.5 Set Theory as an Axiomatic Theory.- 2.6 The Recursion Principle.- 2.7 The Axiom of Choice.- 2.8 Problems.- 3 Ordinal and Cardinal Numbers.- 3.1 Ordinal Numbers.- 3.2 Addition of Ordinals.- 3.3 Multiplication of Ordinals.- 3.4 Sequences of Ordinals.- 3.5 Ordinal Exponentiation.- 3.6 Cardinality, Cardinal Numbers.- 3.7 Arithmetic of Cardinal Numbers.- 3.8 Regular and Singular Cardinals.- 3.9 Cardinal Exponentiation.- 3.10 Inaccessible Cardinals.- 3.11 Problems.- 4 Topics in Pure Set Theory.- 4.1 The Borel Hierarchy.- 4.2 Closed Unbounded Sets.- 4.3 Stationary Sets and Regressive Functions.- 4.4 Trees.- 4.5 Extensions of Lebesgue Measure.- 4.6 A ReTable of ContentsPreface; 1. Naïve Set Theory; 2. The Zermelo-Fraenkel Axioms; 3. Ordinal and Cardinal Numbers; 4. Topics in Pure Set Theory; 5. The Axiom of Constructibility; 6. Independence Proofs in Set Theory; 7. Non-Well-Founded Set Theory; Bibliography; Glossary of Symbols; Index

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Book SynopsisA.- I Introduction.- II Syntax of First-Order Languages.- III Semantics of First-Order Languages.- IV A Sequent Calculus.- V The Completeness Theorem.- VI The Löwenheim-Skolem and the Compactness Theorem.- VII The Scope of First-Order Logic.- VIII Syntactic Interpretations and Normal Forms.- B.- IX Extensions of First-Order Logic.- X Limitations of the Formal Method.- XI Free Models and Logic Programming.- XII An Algebraic Characterization of Elementary Equivalence.- XIII Lindström's Theorems.- References.- Symbol Index.Trade Review“…the book remains my text of choice for this type of material, and I highly recommend it to anyone teaching a first logic course at this level.” – Journal of Symbolic LogicTable of ContentsPreface; Part A: 1. Introduction; 2. Syntax of First-Order Languages; 3. Semantics of first-Order Languages; 4. A Sequent Calculus; 5. The Completeness Theorem; 6. The Lowenheim-Skolem and the Compactness Theorem; 7. The Scope of First-Order Logic; 8. Syntactic Interpretations and Normal Forms; Part B: 9. Extensions of First-Order Logic; 10. Limitations of the Formal Method; 11. Free Models and Logic Programming; 12. An Algebraic Characterization of Elementary Equivalence; 13. Lindstroem's Theorems; References; Symbol Index; Subject Index

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Book SynopsisThe later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.Table of Contents1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem “P ? NP?”.- 8 Newton’s Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bézout’s Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

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Book SynopsisAn introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses.Trade Review'Proofs and Confirmations is one of the most brilliant examples of mathematical exposition that I have encountered, in many years of reading the same. This is not for the faint-hearted, nor is Proofs and Confirmations a book that can be read in an easy chair, like a novel; it demands active participation by the reader. But Bressoud rewards such readers with a panorama of combinatorics today and with renewed awe at the human ability to penetrate the deeply hidden mysteries of pure mathematics.' Herbert S. Wilf, Science'The unexpected twists and turns will hardly be matched in any novel - this book allows us all to share in the excitement … a brilliant book.' Alun O. Morris'I strongly recommend the book as an account of a remarkable mathematical development.' P. J. Cameron, Proceedings of the Edinburgh Mathematical Society'This is an excellent book which can be recommended without hesitation, not only to specialists in the field, but to any mathematician with time to read something interesting and nicely written.' EMSTable of Contents1. The conjecture; 2. Fundamental structures; 3. Lattice paths and plane partitions; 4. Symmetric functions; 5. Hypergeometric series; 6. Explorations; 7. Square ice.

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