Mathematical logic Books

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Book SynopsisThrough three editions, Cryptography: Theory and Practice, has been embraced by instructors and students alike. It offers a comprehensive primer for the subjectâs fundamentals while presenting the most current advances in cryptography.The authors offer comprehensive, in-depth treatment of the methods and protocols that are vital to safeguarding the seemingly infinite and increasing amount of information circulating around the world.Key Features of the Fourth Edition: New chapter on the exciting, emerging new area of post-quantum cryptography (Chapter 9). New high-level, nontechnical overview of the goals and tools of cryptography (Chapter 1). New mathematical appendix that summarizes definitions and main results on number theory and algebra (Appendix A). An expanded treatment of stream ciphers, incluTable of ContentsIntroduction to Cryptography. Classical Cryptography. Shannon's Theory, Perfect Secrecy and the One-Time Pad. Block Ciphers and Stream Ciphers. Hash Functions and Message Authentication. The RSA Cryptosystem and Factoring Integers. Public-Key Cryptography and Discrete Logarithms. Post-quantum Cryptography. Identification Schemes and Entity Authentication. Key Distribution. Key Agreement Schemes. Miscellaneous Topics. Appendix A: Number Theory and Algebraic Concepts for Cryptography, Appendix B: Pseudorandom Bit Generation for Cryptography.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisIn these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman''s modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which sets occur. In the first of three new essays written for this volume, Hellman shows how extendability can be deployed to derive the axiom of Infinity and that of Replacement, improving on earlier accounts; he also shows how extendability leads to attractive, novel resolutions of the set-theoretic paradoxes. Other essays explore advantages and limitations of restrictive systems - nominalist, predicativist, and constructivist. Also included are two essays, with Solomon Feferman, on predicative foundations of arithmetic.Table of ContentsIntroduction; Part I. Structuralism, Extendability, and Nominalism: 1. Structuralism without Structures?; 2. What Is Categorical Structuralism?; 3. On the Significance of the Burali-Forti Paradox; 4. Extending the Iterative Conception of Set: A Height-Potentialist Perspective; 5. On Nominalism; 6. Maoist Mathematics? Critical Study of John Burgess and Gideon Rosen, A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics (Oxford, 1997); Part II. Predicative Mathematics and Beyond: 7. Predicative Foundations of Arithmetic (with Solomon Feferman); 8. Challenges to Predicative Foundations of Arithmetic (with Solomon Feferman); 9. Predicativism as a Philosophical Position; 10. On the Gödel-Friedman Program; Part III. Logics of Mathematics: 11. Logical Truth by Linguistic Convention; 12. Never Say 'Never'! On the Communication Problem between Intuitionism and Classicism; 13. Constructive Mathematics and Quantum Mechanics: Unbounded Operators and the Spectral Theorem; 14. If 'If-Then' Then What?; 15. Mathematical Pluralism: The Case of Smooth Infinitesimal Analysis.

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Book SynopsisCategory theory reveals commonalities between structures of all sorts. This self-contained tour of applied category theory shows its potential in science, engineering, and beyond. Each chapter discusses a real-world application using category-theoretic tools, all of which are introduced in an accessible way with many examples and exercises.Trade Review'Category theory was always applied, but traditionally within pure mathematics. Now it is being used to clarify and synthesize a broad range of topics outside mathematics: from computer science to linguistics, from quantum theory to chemistry, and beyond. Charmingly informal yet crystal clear, Fong and Spivak's book does a wonderful job of demonstrating the power of category theory to beginners – even beginners without much background in pure mathematics.' John Baez, University of California, Riverside'The authors quite rightly describe category theory as a tool for thinking. So if your work requires thinking, this book is for you.' Bartosz Milewski, author of Category Theory for Programmers'This book provides a fantastic introduction to how category is not just abstract nonsense but can be applied to real-world engineering problems, pedagogical while still broad, and fun. A must read for all those entering the exciting emerging field of applied category theory by two key players of this community.' Bob Coecke, University of Oxford'An invitation to Applied Category Theory: Seven Sketches in Compositionality provides a grand tour of the fascinating emergent field of applied category theory that centers examples and use cases before gently introducing the accompanying abstract notions. Fong and Spivak should be congratulated for providing this accessible broad viewpoint to illustrate what category theory is all about vis-à-vis the real world.' Emily Riehl, The Johns Hopkins University'An Invitation to Applied Category Theory is clearly and entertainingly written, and provides a great entry into the world of applied category theory. It is chock full of concrete examples and illustrated with clear diagrams … Fong and Spivak will whet your appetite for learning about categories and how they - and the categorical way of thinking - can be applied in and beyond mathematics. And they will give you the means to do that in a self-contained text.' David Jaz Myers, MAA Reviews'Fong and Spivak's book is highly recommendable for anyone with even a passing interest in category theory in general. And it is mandatory reading for scholars aiming to apply category theory to real world problems.' Fernando A. Tohme, MathSciNet'The presentation is highly visual, employing graphs (nodes and edges), directed graphs, and hypergraphs. In addition, exercises intersperse each presentation, and the solutions to many of the exercises are included. Finally, the chapters include concluding summaries, with suggestions for further study. The book contains scores of references. In short, an excellent self-study resource for those interested in learning about applications of category theory to real-world problems.' J. T. Saccoman, Choice'… highly recommended.' Berthold Stoge, IUCr Journals CRYSTALLOGRAPHY JOURNALS ONLINETable of ContentsPreface; 1. Generative effects: orders and Galois connections; 2. Resource theories: monoidal preorders and enrichment; 3. Databases: categories, functors, and universal constructions; 4. Collaborative design: profunctors, categorification, and monoidal categories; 5. Signal flow graphs: props, presentations, and proofs; 6. Electric circuits: hypergraph categories and operads; 7. Logic of behavior: sheaves, toposes, and internal languages; Appendix. Exercise solutions; References; Index.

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Book SynopsisThe language of ends and (co)ends provides a natural and general way of expressing many phenomena in category theory, in the abstract and in applications. Yet although category-theoretic methods are now widely used by mathematicians, since (co)ends lie just beyond a first course in category theory, they are typically only used by category theorists, for whom they are something of a secret weapon. This book is the first systematic treatment of the theory of (co)ends. Aimed at a wide audience, it presents the (co)end calculus as a powerful tool to clarify and simplify definitions and results in category theory and export them for use in diverse areas of mathematics and computer science. It is organised as an easy-to-cite reference manual, and will be of interest to category theorists and users of category theory alike.Table of ContentsPreface; 1. Dinaturality and (co)ends; 2. Yoneda and Kan; 3. Nerves and realisations; 4. Weighted (co)limits; 5. Profunctors; 6. Operads; 7. Higher dimensional (co)ends; Appendix A. Review of category theory; Appendix B; References; Index.

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Book SynopsisThis textbook offers a detailed introduction to the methodology and applications of sequent calculi in propositional logic. Unlike other texts concerned with proof theory, emphasis is placed on illustrating how to use sequent calculi to prove a wide range of metatheoretical results. The presentation is elementary and self-contained, with all technical details both formally stated and also informally explained. Numerous proofs are worked through to demonstrate methods of proving important results, such as the cut-elimination theorem, completeness, decidability, and interpolation. Other proofs are presented with portions left as exercises for readers, allowing them to practice techniques of sequent calculus.After a brief introduction to classical propositional logic, the text explores three variants of sequent calculus and their features and applications. The remaining chapters then show how sequent calculi can be extended, modified, and applied to non-classical logics, including modal, intuitionistic, substructural, and many-valued logics.Sequents and Trees is suitable for graduate and advanced undergraduate students in logic taking courses on proof theory and its application to non-classical logics. It will also be of interest to researchers in computer science and philosophers.Trade Review“Each chapter of the book is structured in a similar way and contains the basic definitions, facts and necessary discussion regarding the key notions, accompanied with new ideas and a wide reference list, followed by the author's clear and approachable style. This book is self-contained, presenting an extensive survey of the applications and usefulness of cut elimination, and seems to be an extremely interesting source not only for logicians and philosophers, but also for researchers in computer science.” (Branislav Boričić, Mathematical Reviews, May, 2022)Table of ContentsIntroduction.- Analytic Sequent Calculus for CPL.- Gentzen's Sequent Calculus LK.- Purely Logical Sequent Calculus.- Sequent Calculi for Modal Logics.- Alternatives to CPL.- Appendix.

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Book SynopsisThis survey of computability theory offers the techniques and tools that computer scientists (as well as mathematicians and philosophers studying the mathematical foundations of computing) need to mathematically analyze computational processes and investigate the theoretical limitations of computing. Beginning with an introduction to the mathematisation of “mechanical process” using URM programs, this textbook explains basic theory such as primitive recursive functions and predicates and sequence-coding, partial recursive functions and predicates, and loop programs. Advanced chapters cover the Ackerman function, Tarski’s theorem on the non-representability of truth, Goedel’s incompleteness and Rosser’s incompleteness theorems, two short proofs of the incompleteness theorem that are based on Lob's deliverability conditions, Church’s thesis, the second recursion theorem and applications, a provably recursive universal function for the primitive recursive functions, Oracle computations and various classes of computable functionals, the Arithmetical hierarchy, Turing reducibility and Turing degrees and the priority method, a thorough exposition of various versions of the first recursive theorem, Blum’s complexity, Hierarchies of primitive recursive functions, and a machine-independent characterisation of Cobham's feasibly computable functions.Trade Review“This textbook is suited for self-study … . As a second reading however a reader interested in rigorous proofs and/or different approaches to known concepts will benefit from this wealth of material.” (Dieter Riebesehl, zbMATH 1507.03002, 2023)Table of ContentsMathematical Background; a Review.- A Theory of Computability.- Primitive Recursive Functions.- Loop Programs.-The Ackermann Function.- (Un)Computability via Church's Thesis.- Semi-Recursiveness.- Yet another number-theoretic characterisation of P.- Godel's Incompleteness Theorem via the Halting Problem.- The Recursion Theorem.- A Universal (non-PR) Function for PR.- Enumerations of Recursive and Semi-Recursive Sets.- Creative and Productive Sets Completeness.- Relativised Computability.- POSSIBILITY: Complexity of P Functions.- Complexity of PR Functions.- Turing Machines and NP-Completeness.

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Book SynopsisThis monograph offers a new foundation for information theory that is based on the notion of information-as-distinctions, being directly measured by logical entropy, and on the re-quantification as Shannon entropy, which is the fundamental concept for the theory of coding and communications.Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or “dit” of the partition will be obtained. The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy (which is not defined as a measure in the sense of measure theory) and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bits—so the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in general—and to Hilbert spaces in particular—for quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement.Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory.Table of Contents- Logical entropy.- The relationship between logical entropy and Shannon entropy.- The compound notions for logical and Shannon entropies.- Further developments of logical entropy.- Logical Quantum Information Theory.- Conclusion.- Appendix: Introduction to the logic of partitions.

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Book SynopsisThis book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2022, held in Deerfield Beach, FL, USA, in January 2022. The 23 revised full papers were carefully reviewed and selected from 35 submissions. The scope of the Symposium is broad and includes constructive mathematics and type theory; homotopy type theory; logic, automata, and automatic structures; computability and randomness; logical foundations of programming; logical aspects of computational complexity; parameterized complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple-agent system logics; logics of proof and justification; non-monotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; mathematical fuzzy logic; system design logics; other logics in computer science.Table of ContentsA Non-Hyperarithmetical Gödel Logic.- Shorten Resolution Proofs Non-Elementarily.- The Isomorphism Problem for FST Injection Structures.- Justification Logic and Type Theory as Formalizations of Intuitionistic Propositional Logic.- Hyperarithmetical Worm Battles.- Parametric Church’s Thesis: Synthetic Computability Without Choice.- Constructive and Mechanised Meta-Theory of Intuitionistic Epistemic Logic.- A Parametrized Family of Tversky Metrics Connecting the Jaccard Distance to an Analogue of the Normalized Information Distance.- A Parameterized View on the Complexity of Dependence Logic.- A Logic of Interactive Proofs.- Recursive Rules With Aggregation: A Simple Unified Semantics.- Computational Properties of Partial Non-deterministic Matrices and Their Logics.- Soundness and Completeness Results for LEA and Probability Semantics.- On Inverse Operators in Dynamic Epistemic Logic.- Computability Models Over Categories and Presheaves.- Reducts of Relation Algebras: The Aspects of Axiomatisability and Finite Representability.- Between Turing and Kleene.- Propositional Dynamic Logic With Quantification Over Regular Computation Sequences.- Finite Generation and Presentation Problems for Lambda Calculus and Combinatory Logic.- Exact and Parameterized Algorithms for Read-Once Refutations in Horn Constraint Systems.- Logical Principles.- Small Model Property Reflects in Games and Automata.

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Book SynopsisNow in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing.Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers The user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.Trade Review“In this text, Fine (mathematics, Fairfield Univ.) and Rosenberger (Univ. of Hamburg, Germany) successfully present number theory from the inception of primes to recent developments in algebraic and analytic number theory and cryptography. … Numerous exercises and open problems are provided. The breadth and depth of topics covered are impressive, making this an excellent text for those interested in the field of number theory. Summing Up: Recommended. Upper-division undergraduates and graduate students.” (J. T. Zerger, Choice, Vol. 54 (9), May, 2017)“The book is chatty and leisurely, with lots of historical notes and lots of worked examples. The exercises at the end of each chapter are good and there are a reasonable number of them. … a good text for an introductory course … .” (Allen Stenger, MAA Reviews, maa.org, November, 2016)Table of ContentsIntroduction and Historical Remarks.- Basic Number Theory.- The Infinitude of Primes.- The Density of Primes.- Primality Testing: An Overview.- Primes and Algebraic Number Theory.- The Fields Q_p of p-adic Numbers: Hensel's Lemma.- References.- Index.

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Book SynopsisThis book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis’ work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert’s tenth problem. The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis’ work. As a whole, this book shows how Davis’ scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic.Trade Review“It is welcome indeed to have the book under review on my desk and in my possession, particularly given that it’s something of a Festschrift, sporting all sorts of goodies. … To real logicians or even to folks like me … this is a wonderful book to have.” (Michael Berg, MAA Reviews, January 2018)Table of ContentsChapter 1. My Life as a Logician (Martin Davis).- Chapter 2. Martin Davis and Hilbert’s Tenth Problem (Yuri Matiyasevich).- Chapter 3. Extensions of Hilbert’s Tenth Problem: Definability and Decidability in Number Theory (Alexandra Shlapentokh).- Chapter 4. A Story of Hilbert’s Tenth Problem (Laura Elena Morales Guerrero).- Chapter 5. Hyperarithmetical Sets (Yiannis N. Moschovakis).- Chapter 6. Honest Computability and Complexity (Udi Boker and Nachum Dershowitz).- Chapter 7. Why Post Did [Not] Have Turing’s Thesis (Wilfried Sieg).- Chapter 8. On Quantum Computation, Anyons, and Categories (Andreas Blass).

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Book SynopsisParameterized Complexity in the Polynomial Hierarchy was co-recipient of the E.W. Beth Dissertation Prize 2017 for outstanding dissertations in the fields of logic, language, and information. This work extends the theory of parameterized complexity to higher levels of the Polynomial Hierarchy (PH). For problems at higher levels of the PH, a promising solving approach is to develop fixed-parameter tractable reductions to SAT, and to subsequently use a SAT solving algorithm to solve the problem. In this dissertation, a theoretical toolbox is developed that can be used to classify in which cases this is possible. The use of this toolbox is illustrated by applying it to analyze a wide range of problems from various areas of computer science and artificial intelligence.Table of ContentsComplexity Theory and Non-determinism.- Parameterized Complexity Theory.- Fpt-Reducibility to SAT.- The Need for a New Completeness Theory.- A New Completeness Theory.- Fpt-algorithms with Access to a SAT Oracle.- Problems in Knowledge Representation and Reasoning.- Model Checking for Temporal Logics.- Problems Related to Propositional Satisfiability.- Problems in Judgment Aggregation.- Planning Problems.- Graph Problems.- Relation to Other Topics in Complexity Theory.- Subexponential-Time Reductions.- Non-Uniform Parameterized Complexity.- Open Problems and Future Research Directions.- Conclusion.- Compendium of Parameterized Problems.- Generalization to Higher Levels of the Polynomial Hierarchy.

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Book Synopsis1 Historic Notes.- 2 Formal Systems.- 3 Foundations of Mathematics.- 4 Peoof Thory.- 5 Computability Theory.- 6 Algorithmic Information Theory.- Model Theory.

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Book SynopsisTopics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory.An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills.All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers.As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book.

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Book SynopsisThis text is the fifth and final in the series of educational books written by Israel Gelfand with his colleagues for high school students. These books cover the basics of mathematics in a clear and simple format - the style Gelfand was known for internationally. Gelfand prepared these materials so as to be suitable for independent studies, thus allowing students to learn and practice the material at their own pace without a class. Geometry takes a different approach to presenting basic geometry for high-school students and others new to the subject.  Rather than following the traditional axiomatic method that emphasizes formulae and logical deduction, it focuses on geometric constructions. Illustrations and problems are abundant throughout, and readers are encouraged to draw figures and move them in the plane, allowing them to develop and enhance their geometrical vision, imagination, and creativity. Chapters are structured so that only certaiTrade Review“This book is intended to engage the reader visually, tactilely, and kinesthetically. … It has a good set of material to enliven more traditional geometry instruction. … There are problems and exercises throughout. The exercises are accompanied by solutions.” (MAA Reviews, October 10, 2020) Table of ContentsPoints and Lines: A Look at Projective Geometry.- Parallel Lines: A Look at Affine Geometry.- Area: A Look at Symplectic Geometry.- Circles: A Look at Euclidean Geometry.

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Book SynopsisJourney through the world of abstract mathematics into category theory with popular science author Eugenia Cheng. Featuring humanizing examples and demystification of mathematical thought processes, this book is for fans of How to Bake Pi who want to dig deeper into mathematical concepts and build their mathematical background.Trade Review'This book is an educational tour de force that presents mathematical thinking as a right-brained activity. Most 'left brain/right brain' education-talk is at best a crude metaphor; but by putting the main focus on the process of (mathematical) abstraction, Eugenia Cheng supplies the reader (whatever their 'brain-type') with the mental tools to make that distinction precise and potentially useful. The book takes the reader along in small steps; but make no mistake, this is a major intellectual journey. Starting not with numbers, but everyday experiences, it develops what is regarded as a very advanced branch of abstract mathematics (category theory, though Cheng really uses this as a proxy for mathematical thinking generally). This is not watered-down math; it's the real thing. And it challenges the reader to think-deeply at times. We 'left-brainers' can learn plenty from it too.' Keith Devlin, Stanford University (Emeritus), author of The Joy of Sets'Eugenia Cheng loves mathematics—not the ordinary sort that most people encounter, but the most abstract sort that she calls 'the mathematics of mathematics.' And in this lovely excursion through her abstract world of Category Theory, she aims to give those who are willing to join her a glimpse of that world. The journey will change how they view mathematics. Cheng is a brilliant writer, with prose that feels like poetry. Her contagious enthusiasm makes her the perfect guide.' John Ewing, President, Math for America'Eugenia Cheng's singular contribution is in making abstract mathematics relevant to all through her great ingenuity in developing novel connections between logic and life. Her latest book, The Joy of Abstraction, provides a long awaited fully rigorous yet gentle introduction to the 'mathematics of mathematics,' allowing anyone to experience the joy of learning to think categorically.' Emily Riehl, Johns Hopkins University, author of Category Theory in Context'Archimedes is quoted as having said once: 'Mathematics reveals its secrets only to those who approach it with pure love, for its own beauty.' In this fascinating book, Eugenia Cheng approaches the abstract mathematical area of Category Theory with pure love, to reveal its beauty to anybody interested in learning something about contemporary mathematics.' Mario Livio, astrophysicist, author of The Golden Ratio and Brilliant Blunders'Eugenia Cheng's latest book will appeal to a remarkably broad and diverse audience, from non-mathematicians who would like to get a sense of what mathematics is really about, to experienced mathematicians who are not category theorists but would like a basic understanding of category theory. Speaking as one of the latter, I found it a real pleasure to be able to read the book without constantly having to stop and puzzle over the details. I have learnt a lot from it already, including what the famous Yoneda lemma is all about, and I look forward to learning more from it in the future.' Sir Timothy Gowers, Collège de France, Fields Medalist, main editor of The Princeton Companion to Mathematics'At last: a book that makes category theory as simple as it really is. Cheng explains the subject in a clear and friendly way, in detail, not relying on material that only mathematics majors learn. Category theory – indeed, mathematics as a whole – has been waiting for a book like this.' John Baez, University of California, Riverside'Many people speak derisively of category theory as the most abstract area of mathematics, but Eugenia Cheng succeeds in redeeming the word 'abstract'. This book is loquacious, conversational, and inviting. Reading this book convinced me I could teach category theory as an introductory course, and that is a real marvel, since it is a subject most people leave for experts.' Francis Su, Harvey Mudd College, author of Mathematics for Human Flourishing'Finally, a book about category theory that doesn't assume you already know category theory! In this inviting but rigorous introduction to what she calls 'the mathematics of mathematics', Eugenia Cheng brings the subject to us with insight, wit, and a point of view. Her story of finding joy-and advantage-in abstraction will inspire you to find it, too.' Patrick Honner, award-winning high school math teacher, columnist for Quanta Magazine, author of Painless Statistics'This higher category theory is the mathematics of the twenty-first century (at least my corner of it). If you'd like a taste of it, I recommend Dr. Cheng's book. The first half is an accessible and thought-provoking insight into categorical thinking. The second half climbs into the rarified air of theoretic math, but it is worth a read to get a feel for what some parts of modern mathematics look like.' Jonathan Kujawa, 3 Quarks Daily'… a successful addition to the literature that I am sure students will use in the future and I would be happy to recommend.' Constanze Roitzheim, Mathematische SemesterberichteTable of ContentsPrologue; Part I. Building Up to Categories: 1. Categories: the idea; 2. Abstraction; 3. Patterns; 4. Context; 5. Relationships; 6. Formalism; 7. Equivalence relations; 8. Categories: the definition; Interlude: A Tour of Math: 9. Examples we've already seen, secretly; 10. Ordered sets; 11. Small mathematical structures; 12. Sets and functions; 13. Large worlds of mathematical structures; Part II. Doing Category Theory: 14. Isomorphisms; 15. Monics and epics; 16. Universal properties; 17. Duality; 18. Products and coproducts; 19. Pullbacks and pushouts; 20. Functors; 21. Categories of categories; 22. Natural transformations; 23. Yoneda; 24. Higher dimensions; 25. Epilogue: thinking categorically; Appendices: A. Background on alphabets; B. Background on basic logic; C. Background on set theory; D. Background on topological spaces; Glossary; Further reading; Acknowledgements; Index.

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Book SynopsisFirst English translation of revolutionary paper (1931) that established that even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. Introduction by R. B. Braithwaite.

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Book SynopsisMichael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart.Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels.What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its PhilosophTrade Reviewa wonderful new book . . . Potter has written the best philosophical introduction to set theory on the market * Timothy Bays, Notre Dame Philosophical Reviews *Table of ContentsI. SETS ; II. NUMBERS ; III. CARDINALS AND ORDINALS ; IV. FURTHER AXIOMS

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Book SynopsisThird printing. First paperback printing. Original copyright date: 2013.Trade Review"Meshing logic problems with the stories of two extraordinary men ... Paul Nahin fashions a tale of innovation and discovery. Alongside a gripping account of how Shannon built on Boole's work, Nahin explores others key to the technological revolution, from Georg Cantor to Alan Turing."--Nature "Engaging... Nahin assumes some rudimentary knowledge but expertly explains concepts such as relay circuits, Turing machines, and quantum computing. Reasoning through the problems and diagrams will give persistent readers genuine aha moments and an understanding of the two revolutionaries who helped to lay the foundation of our digital world."--Scientific American "Part biography, part history, and part a review of basic information theory, this book does an excellent job of fitting these interlocking elements together."--Library Journal "The reader is taken on a journey from the development of some abstract mathematical ideas through a nearly ubiquitous application of those ideas within the modern world with so many embedded digital computers... I enjoyed the discussion of Claude Shannon. In the history of the computer and development of the internet and World Wide Web, his ideas and contributions are too often overlooked. He is one of my heroes and I believe that everyone that reads this book will come to the same conclusion."--Charles Ashbacher, MAA Reviews "Paul J. Nahin really knows how to tell a good story... The Logician and the Engineer is truly a gem."--New York Journal of Books "A short but fairly detailed exploration of the genesis of Boolean logic and Shannon's information theory... [G]ood background reading for anyone studying electronics or computer science."--Christine Evans-Pughe, Engineering & Technology "Although the book is technical, it is always easily understandable for anyone (for those who need it, some basic rules for electrical circuits are collected in a short appendix). It is not only understandable but also pleasantly bantering and at occasions even facetious."--A. Bultheel, European Mathematical Society "Most valuable to this reviewer, and likely to many potential readers, is the closing chapter, aptly titled Beyond Boole and Shannon. Here is provided an introduction to quantum computing and its logic, possibly portending the future of computers, yet unmistakably bearing the footprints of the two early pioneers. It is an unexpected yet fitting conclusion to this thoroughly enjoyable read."--Ronald E. Prather, Mathematical Reviews Clippings "Nahin has had the very good idea of connecting the very different worlds and times of Boole, Shannon, and others to demonstrate that a little Victorian algebra can turn out to be very useful."--SIAM Review "The exposition is clear and does not assume any prior knowledge except elementary mathematics and a few basic facts from physics. I recommend this well-written book to all readers interested in the history of computer science, as well as those who are curious about the fundamental principles of digital computing."--Antonin Slavik, Zentralblatt MATH "[T]his is a useful and often interesting introduction to the life and work of two intellectual giants who are largely unknown to the general public."--Gareth and Mary Jones, London Mathematical Society Newsletter "The problems are varied and indeed intriguing, and the solutions are delightful."--Mathematics Magazine "This book is not light reading. It would be excellent for advanced high school juniors or seniors with a strong interest in computer science as well as mathematics."--Tom Ottinger, Mathematics Teacher "Nahin leavens the math and engineering with humor and an infectious intellectual curiosity, and the parallels between Boole and Shannon are convincingly drawn... [The Logician and the Engineer] will give your brain a workout, but an enjoyable one."--San Francisco Book ReviewTable of ContentsPreface xi 1 What You Need to Know to Read This Book 1 Notes and References 5 2 Introduction 6 Notes and References 14 3 George Boole and Claude Shannon: Two Mini-Biographies 17 *3.1 The Mathematician 17 *3.2 The Electrical Engineer 28 * Notes and References 39 4 Boolean Algebra 43 *4.1 Boole's Early Interest in Symbolic Analysis 43 *4.2 Visualizing Sets 44 *4.3 Boole's Algebra of Sets 45 *4.4 Propositional Calculus 48 *4.5 Some Examples of Boolean Analysis 52 *4.6 Visualizing Boolean Functions 59 * Notes and References 65 5 Logical Switching Circuits 67 *5.1 Digital Technology: Relays versus Electronics 67 *5.2 Switches and the Logical Connectives 68 *5.3 A Classic Switching Design Problem 71 *5.4 The Electromagnetic Relay and the Logical NOT 73 *5.5 The Ideal Diode and the Relay Logical AND and OR 76 *5.6 The Bi-Stable Relay Latch 81 * Notes and References 84 6 Boole, Shannon, and Probability 88 *6.1 A Common Mathematical Interest 88 *6.2 Some Fundamental Probability Concepts 89 *6.3 Boole and Conditional Probability 96 *6.4 Shannon, Conditional Probability, and Relay Reliability 99 *6.5 Majority Logic 106 * Notes and References 110 7 Some Combinatorial Logic Examples 114 *7.1 Channel Capacity, Shannon's Theorem, and Error-Detection Theory 114 *7.2 The Exclusive-OR Gate (XOR) 122 *7.3 Error-Detection Logic 127 *7.4 Error-Correction Theory 128 *7.5 Error-Correction Logic 132 * Notes and References 137 8 Sequential-State Digital Circuits 139 *8.1 Two Sequential-State Problems 139 *8.2 The NOR Latch 142 *8.3 The Clocked RS Flip-Flop 146 *8.4 More Flip-Flops 154 *8.5 A Synchronous, Sequential-State Digital Machine Design Example 158 * Notes and References 160 9 Turing Machines 161 *9.1 The First Modern Computer 162 *9.2 Two Turing Machines 164 *9.3 Numbers We Can't Compute 168 * Notes and References 173 10 Beyond Boole and Shannon 176 *10.1 Computation and Fundamental Physics 176 *10.2 Energy and Information 178 *10.3 Logically Reversible Gates 180 *10.4 Thermodynamics of Logic 184 *10.5 A Peek into the Twilight Zone: Quantum Computers 188 *10.6 Quantum Logic--and Time Travel, Too! 197 Notes and References 205 Epilogue For the Future: The Anti-Amphibological Machine 210 Appendix Fundamental Electric Circuit Concepts 219 Acknowledgments 223 Index 225

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Book SynopsisAttempts to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. This book is useful for logicians, type theorists, category theorists and (theoretical) computer scientists.Trade Review"The author's achievement in collecting and organizing a very large body of material in coherent form,... this is first and foremost an encyclopaedic work, into which specialists will delve with much pleasure and profit... One very welcome feature of the book is a comprehensive bibliography of nearly 350 items..." --Zentralblatt für Mathematik, vol.905R.A.G. Seely"This book will be the standard reference in its field for some time to come." --The Bulletin of Symbolic Logic, Vol. 6Table of ContentsChapter Headings only. Preface. Contents. Preliminaries. Prospectus. Introduction to fibred category theory. Simple type theory. Equational logic. First order predicate logic. Higher order predicate logic. The effective topos. Internal category theory. Polymorphic type theory. Advanced fibred category theory. First order dependent type theory. Higher order dependent type theory. References. Notation index. Subject index.

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Book SynopsisThis concise introduction takes the reader from standard notions to more advanced topics. It introduces the classic results, as well as more recent developments in this vibrant area of mathematical logic. Many worked examples and exercises make the book a useful resource for graduate students as well as researchers.Trade Review'The book is very well written and a pleasure to read.' Tim Netzer, Zentralblatt MATHTable of ContentsPreface; 1. The basics; 2. Elementary extensions and compactness; 3. Quantifier elimination; 4. Countable models; 5. Aleph-1-categorical theories; 6. Morley rank; 7. Simple theories; 8. Stable theories; 9. Prime extensions; 10. The fine structure of 1-categorical theories; A. Set theory; B. Fields; C. Combinatorics; D. Solutions of exercises; Bibliography; Index.

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Book SynopsisThis graphic novel is both a historical novel as well as an entertaining way of using mathematics to solve a crime. The plot, the possible motive of every suspect, and the elements of his or her character are based on actual historical figures.The 2nd International Congress of Mathematicians is being held in Paris in 1900. The main speaker, the renowned Professor X, is found dead in the hotel dining room. Foul play is suspected. The greatest mathematicians of all time (who are attending the Congress) are called in for questioning. Their statements to the police, however, take the form of mathematical problems. The Chief Inspector enlists the aid of a young mathematician to help solve the crime. Do numbers always tell the truth? Or don’t they?Trade Review“It is a detective story in which several of the greatest historic mathematicians become all suspects for a murder on a colleague. … This is a wonderful booklet of fiction, but based on historical incidents. … It is a fantastic present that you can give to anybody between 9 and 99.” (Adhemar Bultheel, euro-math-soc.eu, June, 2015)Table of ContentsThe Crime.- The Suspects: Mathematicians.- Credits.- Examination of the Statements.

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Book SynopsisIn 1931 Kurt Godel published his fundamental paper, On Formally Undecidable Propositions of Principia Mathematica and Related Systems. This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. The authors provide an explanation of the main ideas and broad implications of Godel's discovery.Trade ReviewA little masterpiece of exegesis. * Nature *An excellent nontechnical account of the substance of Gödel's celebrated paper. -- American Mathematical SocietyTable of ContentsContents Foreword to the New Edition by Douglas R. Hofstadter ix Acknowledgments xxiii i Introduction 1 ii The Problem of Consistency 7 iii Absolute Proofs of Consistency 25 iv The Systematic Codification of Formal Logic 37 v An Example of a Successful Absolute Proof of Consistency 45 vi The Idea of Mapping and Its Use in Mathematics 57 vii Godel's Proofs 68 a Godel numbering 68 b The arithmetization of meta-mathematics 80 c The heart of Godel's argument 92 viii Concluding Reflections 109 Appendix: Notes 114 Brief Bibliography 125 Index 127

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Book SynopsisThis book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed. Originally published in 1986. The Princeton LegacyTable of Contents*FrontMatter, pg. i*Acknowledgments, pg. v*Table of Contents, pg. vii*Chapter 1. The impredicativity of induction, pg. 1*Chapter 2. Logical terminology, pg. 3*Chapter 3. The axioms of arithmetic, pg. 8*Chapter 4. Order, pg. 10*Chapter 5. Induction by relativization, pg. 12*Chapter 6. Interpretability in Robinson's theory, pg. 16*Chapter 7. Bounded induction, pg. 23*Chapter 8. The bounded least number principle, pg. 29*Chapter 9. The euclidean algorithm, pg. 32*Chapter 10. Encoding, pg. 36*Chapter 11. Bounded separation and minimum, pg. 43*Chapter 12. Sets and functions, pg. 46*Chapter 13. Exponential functions, pg. 51*Chapter 14. Exponentiation, pg. 54*Chapter 15. A stronger relativization scheme, pg. 60*Chapter 16. Bounds on exponential functions, pg. 64*Chapter 17. Bounded replacement, pg. 70*Chapter 18. An impassable barrier, pg. 73*Chapter 19. Sequences, pg. 82*Chapter 20. Cardinality, pg. 90*Chapter 21. Existence of sets, pg. 95*Chapter 22. Semibounded Replacement, pg. 98*Chapter 23. Formulas, pg. 101*Chapter 24. Proofs, pg. 111*Chapter 25. Derived rules of inference, pg. 115*Chapter 26. Special constants, pg. 134*Chapter 27. Extensions by definition, pg. 136*Chapter 28. Interpretations, pg. 152*Chapter 29. The arithmetization of arithmetic, pg. 157*Chapter 30. The consistency theorem, pg. 162*Chapter 31. Is exponentiation total?, pg. 173*Chapter 32. A modified Hilbert program, pg. 178*Bibliography, pg. 181*General index, pg. 183*Index of defining axioms, pg. 186

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Book SynopsisThis text organizes a range of results in chromatic homotopy theory, running a single thread through theorems in bordism and a detailed understanding of the moduli of formal groups. It emphasizes the naturally occurring algebro-geometric models that presage the topological results, taking the reader through a pedagogical development of the field. In addition to forming the backbone of the stable homotopy category, these ideas have found application in other fields: the daughter subject ''elliptic cohomology'' abuts mathematical physics, manifold geometry, topological analysis, and the representation theory of loop groups. The common language employed when discussing these subjects showcases their unity and guides the reader breezily from one domain to the next, ultimately culminating in the construction of Witten''s genus for String manifolds. This text is an expansion of a set of lecture notes for a topics course delivered at Harvard University during the spring term of 2016.Trade Review'It has a down-to-earth and inviting style (no small achievement in a book about functorial algebraic geometry). It is elegant, precise, and incisive, and it is strong on both theory and calculation.' Michael Berg, MAA Reviews'This book is likely to be quite useful to graduate students in algebraic topology. For years it has been an informal tradition for students of algebraic topology to teach themselves enough of the foundations of algebraic geometry to be able to translate between theorems about Hopf algebroids and theorems about algebraic stacks, and then to proceed to translate, as much as possible, calculations and theorems in algebraic topology into equivalent formulations in terms of moduli stacks of formal groups and related objects. This book does a great service to such students (and their advisors!), as it gives good answers to many of the questions such students inevitably ask.' Andrew Salch, MatSciNet'The presentation is lucid, pedagogical, and also offers a fresh point of view on classical topics. It draws from several mostly unpublished sources, for instance Strickland's manuscripts or various sets of notes by Goerss, Hopkins, and Lurie, and combines them in a single uniform treatment. Moreover, it contains a wealth of references to the published and unpublished literature that guides the interested reader to further topics that are only discussed in passing.' Tobias Barthel, zbMATH OpenTable of ContentsForeword Matthew Ando; Preface; Introduction; 1. Unoriented bordism; 2. Complex bordism; 3. Finite spectra; 4. Unstable cooperations; 5. The σ-orientation; Appendix A. Power operations; Appendix B. Loose ends; References; Index.

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Book SynopsisFinite-state methods are the most efficient mechanisms for analysing textual and symbolic data, providing elegant solutions for an immense number of practical problems in computational linguistics and computer science. This book for graduate students and researchers gives a complete coverage of the field, starting from a conceptual introduction and building to advanced topics and applications. The central finite-state technologies are introduced with mathematical rigour, ranging from simple finite-state automata to transducers and bimachines as ''input-output'' devices. Special attention is given to the rich possibilities of simplifying, transforming and combining finite-state devices. All algorithms presented are accompanied by full correctness proofs and executable source code in a new programming language, C(M), which focuses on transparency of steps and simplicity of code. Thus, by enabling readers to obtain a deep formal understanding of the subject and to put finite-state methodsTrade Review'… this volume is well written and very detailed. It is thus a nice reference for those results for the interested graduate or researcher …' Andreas Maletti, ZB Math ReviewsTable of ContentsPart I. Formal Background: 1. Formal preliminaries; 2. Monoidal finite-state automata; 3. Classical finite-state automata and regular languages; 4. Monoidal multi-tape automata and finite-state transducers; 5. Deterministic transducers; 6. Bimachines; Part II. From Theory to Practice: 7. The C(M) language; 8. C(M) implementation of finite-state devices; 9. The Aho–Corasick algorithm; 10. The minimal deterministic finite-state automaton for a finite language; 11. Constructing finite-state devices for text rewriting; Bibliography; Index.

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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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