Mathematical logic Books

Book SynopsisAdvances in Modal Logic is a unique forum for presenting the latest results and new directions of research in modal logic. The topics dealt with are of interdisciplinary interest and range from mathematical, computational, and philosophical problems to applications in knowledge representation and formal linguistics.Volume 3 presents substantial advances in the relational model theory and the algorithmic treatment of modal logics. It contains invited and contributed papers from the third conference on “Advances in Modal Logic”, held at the University of Leipzig (Germany) in October 2000. It includes papers on dynamic logic, description logic, hybrid logic, epistemic logic, combinations of modal logics, tense logic, action logic, provability logic, and modal predicate logic.Table of ContentsFrom description to hybrid logics, and back, C. Areces and M. de Rijke; homophonic theory of truth for tense logic, Torben Brauner; weak necessity on weak Kleene matrices, F. Correia; bimodal logics for reasoning about continuous dynamics, J.M. Davoren and R.P. Gore; from bisimulation quantifiers to classifying toposes, S. Ghilardi and M. Zawadowski; normal products of modal logics, Y. Hasimoto; a tableau algorithm for the clique guarded fragment, C. Hirsch and S. Tobies; the complexity of reasoning with Boolean modal logics, C. Lutz and U. Sattler; outline of a logic of action, K. Segerberg; belief, names, and modes of presentation, R. Ye and M. Fitting. (Part contents)

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Book SynopsisCellular automata are a class of spatially and temporally discrete mathematical systems characterized by local interaction and synchronous dynamical evolution. Introduced by the mathematician John von Neumann in the 1950s as simple models of biological self-reproduction, they are prototypical models for complex systems and processes consisting of a large number of simple, homogeneous, locally interacting components. Cellular automata have been the focus of great attention over the years because of their ability to generate a rich spectrum of very complex patterns of behavior out of sets of relatively simple underlying rules. Moreover, they appear to capture many essential features of complex self-organizing cooperative behavior observed in real systems.This book provides a summary of the basic properties of cellular automata, and explores in depth many important cellular-automata-related research areas, including artificial life, chaos, emergence, fractals, nonlinear dynamics, and self-organization. It also presents a broad review of the speculative proposition that cellular automata may eventually prove to be theoretical harbingers of a fundamentally new information-based, discrete physics. Designed to be accessible at the junior/senior undergraduate level and above, the book will be of interest to all students, researchers, and professionals wanting to learn about order, chaos, and the emergence of complexity. It contains an extensive bibliography and provides a listing of cellular automata resources available on the World Wide Web.Table of Contents* Introduction: Preliminary Musings * Formalism * Phenomenological Studies of Generic CA * Dynamical Systems Theory Approach * Analytic Approach * Cellular Automata and Language Theory * Probabilistic CA * Generalized Models * CA Models of Fluid Dynamics * Neural Networks * Artificial-Life * Is Nature, Underneath It All, a CA?

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Book SynopsisThis volume is devoted to the main areas of mathematical logic and applications to computer science. There are articles on weakly o-minimal theories, algorithmic complexity of relations, models within the computable model theory, hierarchies of randomness tests, computable numberings, and complexity problems of minimal unsatisfiable formulas. The problems of characterization of the deduction-detachment theorem, Δ1-induction, completeness of Leśniewski's systems, and reduction calculus for the satisfiability problem are also discussed.The coverage includes the answer to Kanovei's question about the upper bound for the complexity of equivalence relations by convergence at infinity for continuous functions. The volume also gives some applications to computer science such as solving the problems of inductive interference of languages from the full collection of positive examples and some negative data, the effects of random negative data, methods of formal specification and verification on the basis of model theory and multiple-valued logics, interval fuzzy algebraic systems, the problems of information exchange among agents on the base topological structures, and the predictions provided by inductive theories.Table of ContentsAnother Characterization of the Deduction-Detachment Theorem (S V Babyonyshev); On Behavior of 2-Formulas in Weakly o-Minimal Theories (B S Baizhanov & B Sh Kulpeshov); Arithmetic Turing Degrees and Categorical Theories of Computable Models (E Fokina); Negative Data in Learning Languages (S Jain & E Kinber); Effective Cardinals in the Nonstandard Universe (V Kanovei & M Reeken); Model-Theoretic Methods of Analysis of Computer Arithmetic (S P Kovalyov); The Functional Completeness of Le niewski's Systems (F Lepage); Hierarchies of Randomness Tests (J Reimann & F Stephan); Intransitive Linear Temporal Logic Based on Integer Numbers, Decidability, Admissible Logical Consecutions (V V Rybakov); The Logic of Prediction (E Vityaev); Conceptual Semantic Systems Theory and Applications (K E Wolff); Complexity Results on Minimal Unsatisfiable Formulas (X Zhao); and other papers.

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Book SynopsisThis volume is a collection of written versions of the talks given at the Workshop on Computational Prospects of Infinity, held at the Institute for Mathematical Sciences from 18 June to 15 August 2005. It consists of contributions from many of the leading experts in recursion theory (computability theory) and set theory. Topics covered include the structure theory of various notions of degrees of unsolvability, algorithmic randomness, reverse mathematics, forcing, large cardinals and inner model theory, and many others.Table of ContentsPrompt Simplicity, Array Computability and Cupping (R Downey et al.); A Simpler Short Extenders Forcing-Gap 3 (M Gitik); The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs (D R Hirschfeldt et al.); Absoluteness for Universally Baire Sets and the Uncountable II (I Farah et al.); Modaic Definability of Ordinals (I Neeman); Eliminating Concepts (A Nies); Diamonds on P (M Shioya); Rigidity and Biinterpretability in the Hyperdegrees (R A Shore); Some Fundamentals Issues Concerning Degrees or Unsolvability (S G Simpson); A tt Version of the Poner-Robinson Theorem (W H Woodin); and other papers.

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Book SynopsisThis book contains the material for a first course in pure model theory with applications to differentially closed fields. Topics covered in this book include saturated model criteria for model completeness and elimination of quantifiers; Morley rank and degree of element types; categoricity in power; two-cardinal theorems; existence and uniqueness of prime model extensions of substructures of models of totally transcendental theories; and homogeneity of models of ϖ1-categorical theories.Table of ContentsOrdinals and Diagrams; Similarity Types of Structures; Elementary Equivalence; Model Completeness; Skolemization of Structures; Saturated Structures; Omitting a Type; Homogeneous Structures; Inverse Systems of Compact Hausdorff Spaces; The Morley Derivative; Prime Model Extensions; Order Indiscernibles; The Baldwin-Lachlan Theorem; and other sections.

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Book SynopsisThe present monograph on matrix partial orders, the first on this topic, makes a unique presentation of many partial orders on matrices that have fascinated mathematicians for their beauty and applied scientists for their wide-ranging application potential. Except for the Löwner order, the partial orders considered are relatively new and came into being in the late 1970s. After a detailed introduction to generalized inverses and decompositions, the three basic partial orders — namely, the minus, the sharp and the star — and the corresponding one-sided orders are presented using various generalized inverses. The authors then give a unified theory of all these partial orders as well as study the parallel sums and shorted matrices, the latter being studied at great length. Partial orders of modified matrices are a new addition. Finally, applications are given in statistics and electrical network theory.∗DeceasedTable of ContentsIntroduction; Decompositions and Generalized Inverses; Minus Order; Sharp Order; Star Order; One-Sided Orders; Lowner Order and Majorization; Unified Theory of Matrix Partial Orders through Generalized Inverses; Parallel Sums; Schur Complements and Shorted Operators; Shorted Operators II; Supremum and Infimum for a Pair of Matrices; Partial Orders for Modified Matrices; Statistics; Electrical Network Theory.

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Book Synopsis'Points, questions, stories, and occasional rants introduce the 24 chapters of this engaging volume. With a focus on mathematics and peppered with a scattering of computer science settings, the entries range from lightly humorous to curiously thought-provoking. Each chapter includes sections and sub-sections that illustrate and supplement the point at hand. Most topics are self-contained within each chapter, and a solid high school mathematics background is all that is needed to enjoy the discussions. There certainly is much to enjoy here.'CHOICEEver notice how people sometimes use math words inaccurately? Or how sometimes you instinctively know a math statement is false (or not known)?Each chapter of this book makes a point like those above and then illustrates the point by doing some real mathematics through step-by-step mathematical techniques.This book gives readers valuable information about how mathematics and theoretical computer science work, while teaching them some actual mathematics and computer science through examples and exercises. Much of the mathematics could be understood by a bright high school student. The points made can be understood by anyone with an interest in math, from the bright high school student to a Field's medal winner.

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Book SynopsisThis book is about “diamond”, a logic of paradox. In diamond, a statement can be true yet false; an “imaginary” state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued Boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown “modulator”; and Metamathematical Dilemma, which relates diamond to Gödelian metamathematics and dilemma games.Table of ContentsParadox: Russell's Paradox; Santa Sentences; Antistrephon; Game Paradoxes; Paradox of the Boundary; Diamond: Diamond Values; Harmonic Functions; Diamond Circuits; Brownian Forms; Diamond Algebra: Bracket Algebra; Laws; Normal Forms; Completeness; Self-Reference: Re-entrance and Fixedpoints; Phase Order; The Outer Fixedpoints; Fixedpoint Lattices: Relative Lattices; Seeds and Spirals; Shared Fixed-Points; Limit Logic: Limit Fixedpoints; the Halting Theorem; Paradox Resolved: Russell's Paradox; Santa Sentences; Antistrephon; Game Paradoxes; The Continuum: Cantor's Paradox; Dedekind Splices; Zeno's Theorem; Fuzzy Chaos; Clique Theory: Cliques Defined; Clique Axioms; Graph Cliques; Clique Circuits; Orthogonal Logics: Analytic Functions; Star Logic; Harmonic Projection; Interferometry: Quadrature; Diffraction; Buzzers and Toggles; How to Count to Two: Brownian and Kauffman Modulators; Diffracting the Modulators; Rotors, Pumps and Tapes; the Ganglion; Metamathematics: Godelian Quanta; Meta-Logic; Dialectical Dilemma; Dilemma: Milo's Trick; Prisoner's Dilemma; Dilemma Diamond; Banker's Dilemma; The Unexpected Departure.

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Book SynopsisThis volume consists of papers selected from the presentations at the workshop and includes mainly recent developments in the fields of formal languages, automata theory and algebraic systems related to the theoretical computer science and informatics. It covers the areas such as automata and grammars, languages and codes, combinatorics on words, cryptosystems, logics and trees, Grobner bases, minimal clones, zero-divisor graphs, fine convergence of functions, and others.Table of ContentsAutomata and Grammars; Languages and Codes; Combinatorics on Words; Cryptosystems; Logics and Trees; Grobner Bases; Minimal Clones; Zero-Divisor Graphs; Fine Convergence of Functions; and other papers.

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Book SynopsisThis is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.Starting with the basics of set theory, induction and computability, it covers propositional and first-order logic — their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert's and Gentzen's systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.Unlike in most comparable texts, presentation of syntactic reasoning systems precedes the semantic explanations. The simplicity of syntactic constructions and rules — of a high, though often neglected, pedagogical value — aids students in approaching more complex semantic issues. This order of presentation also brings forth the relative independence of syntax from the semantics, helping to appreciate the importance of the purely symbolic systems, like those underlying computers.An overview of the history of logic precedes the main text, while informal analogies precede introduction of most central concepts. These informal aspects are kept clearly apart from the technical ones. Together, they form a unique text which may be appreciated equally by lecturers and students occupied with mathematical precision, as well as those interested in the relations of logical formalisms to the problems of computability and the philosophy of logic.Table of ContentsHistory of Logic; Basic Set Theory and Induction; Turing Machines and Computability; Propositional Logic: Hilbert's and Gentzen's Proof Systems; Boolean and Set-Valued Semantics; DNF and CNF; Soundness and Completeness; First-Order Logic: Hilbert's and Gentzen's Proof Systems; Tarski's Semantics; Prenex NF; Term Models and Prolog; Soundness and Completeness; Exercises to Each Topic.

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Book SynopsisThis is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.Starting with the basics of set theory, induction and computability, it covers propositional and first-order logic — their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert's and Gentzen's systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.Unlike in most comparable texts, presentation of syntactic reasoning systems precedes the semantic explanations. The simplicity of syntactic constructions and rules — of a high, though often neglected, pedagogical value — aids students in approaching more complex semantic issues. This order of presentation also brings forth the relative independence of syntax from the semantics, helping to appreciate the importance of the purely symbolic systems, like those underlying computers.An overview of the history of logic precedes the main text, while informal analogies precede introduction of most central concepts. These informal aspects are kept clearly apart from the technical ones. Together, they form a unique text which may be appreciated equally by lecturers and students occupied with mathematical precision, as well as those interested in the relations of logical formalisms to the problems of computability and the philosophy of logic.Table of ContentsHistory of Logic; Basic Set Theory and Induction; Turing Machines and Computability; Propositional Logic: Hilbert's and Gentzen's Proof Systems; Boolean and Set-Valued Semantics; DNF and CNF; Soundness and Completeness; First-Order Logic: Hilbert's and Gentzen's Proof Systems; Tarski's Semantics; Prenex NF; Term Models and Prolog; Soundness and Completeness; Exercises to Each Topic.

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Book SynopsisThe Asian Logic Conference is part of the series of logic conferences inaugurated in Singapore in 1981. It is normally held every three years and rotates among countries in the Asia-Pacific region. The 11th Asian Logic Conference was held at the National University of Singapore, in honor of Professor Chong Chitat on the occasion of his 60th birthday. The conference is on the broad area of logic, including theoretical computer science. It is considered a major event in this field and is regularly sponsored by the Association of Symbolic Logic. This volume contains papers from this meeting.Table of ContentsLimitwise Monotonic Functions and Their Applications (R G Downey et al.); Provably Δ02 and Weakly Descending Chains (T Arai); K-Trivials are NCR (G Barmpalias et al.); A Dichotomy for the Mackey Borel Structure (I Farah); Computable Dowd-Type Generic Oracles (M Kumabe & T Suzuki); Cappable CEA Sets and Ramsey's Theorem (A M Kach et al.); Amalgamation, Absoluteness, and Categoricity (J T Baldwin); Models of Long Sentences I (G Sacks); On Automatic Families (S Jain et al.); A Universally-Free Modal Logic (S C-M Yang).

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Book SynopsisThis penultimate volume contains numerous original, elegant, and surprising results in 1-dimensional cellular automata. Perhaps the most exciting, if not shocking, new result is the discovery that only 82 local rules, out of 256, suffice to predict the time evolution of any of the remaining 174 local rules from an arbitrary initial bit-string configuration. This is contrary to the well-known folklore that 256 local rules are necessary, leading to the new concept of quasi-global equivalence.Another surprising result is the introduction of a simple, yet explicit, infinite bit string called the super string S, which contains all random bit strings of finite length as sub-strings. As an illustration of the mathematical subtlety of this amazing discrete testing signal, the super string S is used to prove mathematically, in a trivial and transparent way, that rule 170 is as chaotic as a coin toss.Yet another unexpected new result, among many others, is the derivation of an explicit basin tree generation formula which provides an analytical relationship between the basin trees of globally-equivalent local rules. This formula allows the symbolic, rather than numerical, generation of the time evolution of any local rule corresponding to any initial bit-string configuration, from one of the 88 globally-equivalent local rules.But perhaps the most provocative idea is the proposal for adopting rule 137, over its three globally-equivalent siblings, including the heretofore more well-known rule 110, as the prototypical universal Turing machine.Table of ContentsPeriod-2 Rules; Period-3 Rules, Period-6 Rules, and Permutive Rules.

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Book SynopsisThe Asian Logic Conference is the most significant logic meeting outside of North America and Europe, and this volume represents work presented at, and arising from the 12th meeting. It collects a number of interesting papers from experts in the field. It covers many areas of logic.

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Book SynopsisThis book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Gödel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, in particular the Dialectica interpretation and its generalization to finite types, the Diller-Nahm interpretation, are studied on Heyting as well as Peano arithmetic in finite types and extended to functional interpretations of constructive as well as classical systems of analysis and set theory. Kreisel's modified realization and Troelstra's hybrids of it are presented as interpretations of Heyting arithmetic and extended to constructive set theory, both in finite types. They serve as background for the construction of hybrids of the Diller-Nahm interpretation of Heyting arithmetic and constructive set theory, again in finite types. All these functional interpretations yield relative consistency results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations.Table of ContentsArithmetic: Primitive Recursive Functionals; Λ- (Diller - Nahm) Interpretation of Heyting Arithmetic in Finite Types; The Dialectica Interpretation and Equality Functionals; Simultaneous Recursions in Linear Types; Computability, Consistency, Continuity; Modified Realization and its Hybrids; Hybrids of the Λ-Interpretation; N-Interpretations; Interpretations of Classical Arithmetic; Extensionality and Majorizability; Analysis: Bar Recursive Functionals; Λ- and Dialectica Interpretation of Bar Induction by Bar Recursion; Functional Interpretations of Classical Analysis; Computability of Bar Recursive Functionals; Set Theory: Constructive Set Functionals; Kripke - Platek Set Theory and Its Functional Interpretations; Constructive Set Theory and Its Λ-Interpretation; Modified Realizations of Constructive Set Theory; The Q-Hybrid of the Λ-Interpretation of Constructive Set Theory in Finite Types; Majorizability of Constructive Set Functionals.

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Book SynopsisEver since Paul Cohen's spectacular use of the forcing concept to prove the independence of the continuum hypothesis from the standard axioms of set theory, forcing has been seen by the general mathematical community as a subject of great intrinsic interest but one that is technically so forbidding that it is only accessible to specialists. In the past decade, a series of remarkable solutions to long-standing problems in C*-algebra using set-theoretic methods, many achieved by the author and his collaborators, have generated new interest in this subject. This is the first book aimed at explaining forcing to general mathematicians. It simultaneously makes the subject broadly accessible by explaining it in a clear, simple manner, and surveys advanced applications of set theory to mainstream topics.Table of ContentsPeano Arithmetic; Zermelo - Fraenkel Set Theory; Well-Ordered Sets; Ordinals; Cardinals; Relativization; Reflection; Forcing Posets; Generic Extensions; Forcing Equality; The Fundamental Theorem of Forcing; Forcing CH; Forcing not-CH; Families of Entire Functions; Self-Homeomorphisms of Beta N - N, I; Pure States on B(H); The Diamond Principle; Suslin's Problem, I; Naimark's problem; Product Forcing and Diamond^S; The Whitehead Problem, I; Two-Stage Iterated Forcing; Finite Support Iteration; Martin's Axiom; Suslin's Problem, II; The Whitehead Problem, II; The Open Coloring Axiom; Self-Homeomorphisms of Beta N - N, II; Automorphisms of the Calkin Algebra, I; Automorphisms of the Calkin Algebra, II; The Multiverse Interpretation.

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Book SynopsisThis volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters cover topics in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable within the Zermelo-Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progress in foundational studies.The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of interest to students, researchers and mathematicians concerned with issues in the foundations of mathematics.Table of ContentsSection I: Absoluteness, Truth, and Quotients (Ilijas Farah); A Multiverse Perspective on the Axiom of Constructiblity (Joel David Hamkins); Hilbert, Bourbaki and the Scorning of Logic (A R D Mathias); Toward Objectivity in Mathematics (Stephen G Simpson); Sort Logic and Foundations of Mathematics (Jouko Vaananen); Reasoning about Constructive Concepts (Nik Weaver); Perfect Infinites and Finite Approximation (Boris Zilber); Section II: An Objective Justification for Actual Infinity? (Stephen G Simpson); Oracle Questions (Theodore A Slaman and W Hugh Woodin).

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Book SynopsisIn the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach-Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions rather than on the relationship between different forcing axioms or their consistency strengths.Table of ContentsThe Baire Category Theorem and the Baire Category Numbers; Coding into the Reals; Descriptive Set-Theoretic Consequences; Measure-Theoretic Consequences; Variations on the Souslin Hypothesis; The S- and L-Space Problems; The Side-Condition Method; Ideal Dichotomies; Coherent and Lipschitz Trees; Applications to the S-Space Problem and the Von Neumann Problem; Biorthogonal Systems; Structure of Compact Spaces; Ramsey Theory on Ordinals; Five Cofinal Types; Five Linear Orderings; mm and Cardinal Arithmetic; Reflection Principles; Appendices: Basic Notions; Preserving Stationary Sets; Historical and Other Comments.

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Book SynopsisThis volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers and mathematicians.

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Book SynopsisThis volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2012 Asian Initiative for Infinity Logic Summer School. The major topics cover set-theoretic forcing, higher recursion theory, and applications of set theory to C*-algebra. This volume offers a wide spectrum of ideas and techniques introduced in contemporary research in the field of mathematical logic to students, researchers and mathematicians.

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Book SynopsisRaymond Smullyan presents a bombshell puzzle so startling that it seems incredible that there could be any solution at all! But there is indeed a solution — moreover, one that requires a chain of lesser puzzles to be solved first. The reader is thus taken on a journey through a maze of subsidiary problems that has all the earmarks of an entertaining detective story.This book leads the unwary reader into deep logical waters through seductively entertaining logic puzzles. One example is Boolean algebra with such weird looking equations as 1+1=0 — a subject which today plays a vital role, not only in mathematical systems, but also in computer science and artificial intelligence.Table of ContentsIt's All A Question of Logic: Puzzles or Monkey Tricks; Which Lady; Which Witch; Which Island; McGregor's Arithmetic Tricks; Ask Eldon White; Al, the Chemist; Sane or Mad; The Strange Case of McSnurd; The Knight-Knave Disease; Human or Android; Variable Lying and Paradox; The Magic Garden: George's Garden; Some Neighboring Gardens; The Grand Problem Solved!; Boolean Gardens and Variable Liars; Propositional Logic and Boolean Gardens; The Boolean Theory of Sets; Boolean Algebras in General; Boolean Gardens Revisited; Another Grand Problem; George Boole and Mathematical Logic;

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Book SynopsisThis volume provides a forum which highlights new achievements and overviews of recent developments of the thriving logic groups in the Asia-Pacific region. It contains papers by leading logicians and also some contributions in computer science logics and philosophic logics.Table of ContentsAn Analogy Between Cardinal Characteristics and Highness Properties of Oracles (Jorg Brendle, Andrew Brooke-Taylor, Keng Meng Ng and Andre Nies); Minimal Pairs in the C. E. Truth-Table Degrees (Rod Downey, Keng Meng Ng); Degree Spectra of Equivalence Relations (Liang Yu); Some Questions Concerning Ab Initio Generic Structures (Koichiro Ikeda); Model Complete Generic Structures (Koichiro Ikeda and Hirotaka Kikyo); Large Cardinals and Higher Degree Theory (Xianghui Shi); Characterization of the Second Homology Group of a Stationary Type in a Stable Theory (John Goodrick, Byunghan Kim and Alexei Kolesnikov); On Categorical Relationship Among Various Fuzzy Topological Systems, Fuzzy Topological Spaces and Related Algebraic Structures (Purbita Jana and Mihir K Chakraborty); Realizability and Existence Property of a Constructive Set Theory with Types (F Kachapova); A Non-Uniformly C-Productive Sequence & Non-Constructive Disjunctions (John Case, Michael Ralston and Yohji Akama); On Extensions of Basic Propositional Logic (Minghui Ma and Katsuhiko Sano); Goal-Directed Unbounded Coalitional Game and Its Complexity (Hu Liu); A Survey on Recent Results on Partial Learning (Ziyuan Gao, Sanjay Jain, Frank Stephan and Sandra Zilles);

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Book SynopsisRaymond Smullyan presents a bombshell puzzle so startling that it seems incredible that there could be any solution at all! But there is indeed a solution — moreover, one that requires a chain of lesser puzzles to be solved first. The reader is thus taken on a journey through a maze of subsidiary problems that has all the earmarks of an entertaining detective story.This book leads the unwary reader into deep logical waters through seductively entertaining logic puzzles. One example is Boolean algebra with such weird looking equations as 1+1=0 — a subject which today plays a vital role, not only in mathematical systems, but also in computer science and artificial intelligence.Table of ContentsIt's All A Question of Logic: Puzzles or Monkey Tricks; Which Lady; Which Witch; Which Island; McGregor's Arithmetic Tricks; Ask Eldon White; Al, the Chemist; Sane or Mad; The Strange Case of McSnurd; The Knight-Knave Disease; Human or Android; Variable Lying and Paradox; The Magic Garden: George's Garden; Some Neighboring Gardens; The Grand Problem Solved!; Boolean Gardens and Variable Liars; Propositional Logic and Boolean Gardens; The Boolean Theory of Sets; Boolean Algebras in General; Boolean Gardens Revisited; Another Grand Problem; George Boole and Mathematical Logic;

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Book SynopsisThis volume presents the lecture notes of short courses given by three leading experts in mathematical logic at the 2010 and 2011 Asian Initiative for Infinity Logic Summer Schools. The major topics covered set theory and recursion theory, with particular emphasis on forcing, inner model theory and Turing degrees, offering a wide overview of ideas and techniques introduced in contemporary research in the field of mathematical logic.

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Book SynopsisThis is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.Starting with the basics of set theory, induction and computability, it covers propositional and first order logic — their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert's and Gentzen's systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.Unlike in most comparable texts, presentation of syntactic reasoning systems precedes the semantic explanations. The simplicity of syntactic constructions and rules — of a high, though often neglected, pedagogical value — aids students in approaching more complex semantic issues. This order of presentation also brings forth the relative independence of syntax from the semantics, helping to appreciate the importance of the purely symbolic systems, like those underlying computers.An overview of the history of logic precedes the main text, while informal analogies precede introduction of most central concepts. These informal aspects are kept clearly apart from the technical ones. Together, they form a unique text which may be appreciated equally by lecturers and students occupied with mathematical precision, as well as those interested in the relations of logical formalisms to the problems of computability and the philosophy of logic.This revised edition contains also, besides many new exercises, a new chapter on semantic paradoxes. An equivalence of logical and graphical representations allows us to see vicious circularity as the odd cycles in the graphical representation and can be used as a simple tool for diagnosing paradoxes in natural discourse.Table of ContentsA History of Logic: Patterns of Reasoning; A Language and Its Meaning; A Symbolic Language; 1850-1950 - Mathematical Logic; Modern Symbolic Logic; Summary; Elements of Set Theory: Sets, Functions, Relations; Induction; Turing Machines: Computability and Decidability; Propositional Logic: Syntax and Proof Systems; Semantics of PL; Soundness and Completeness; Diagnosing Paradoxes; First Order Logic: Syntax and Proof Systems of FOL; Semantics of FOL; More Semantics; Soundness and Completeness; Why is First Order Logic "First Order"?;

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Book SynopsisThis is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.Starting with the basics of set theory, induction and computability, it covers propositional and first order logic — their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert's and Gentzen's systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.Unlike in most comparable texts, presentation of syntactic reasoning systems precedes the semantic explanations. The simplicity of syntactic constructions and rules — of a high, though often neglected, pedagogical value — aids students in approaching more complex semantic issues. This order of presentation also brings forth the relative independence of syntax from the semantics, helping to appreciate the importance of the purely symbolic systems, like those underlying computers.An overview of the history of logic precedes the main text, while informal analogies precede introduction of most central concepts. These informal aspects are kept clearly apart from the technical ones. Together, they form a unique text which may be appreciated equally by lecturers and students occupied with mathematical precision, as well as those interested in the relations of logical formalisms to the problems of computability and the philosophy of logic.This revised edition contains also, besides many new exercises, a new chapter on semantic paradoxes. An equivalence of logical and graphical representations allows us to see vicious circularity as the odd cycles in the graphical representation and can be used as a simple tool for diagnosing paradoxes in natural discourse.Table of ContentsA History of Logic: Patterns of Reasoning; A Language and Its Meaning; A Symbolic Language; 1850-1950 - Mathematical Logic; Modern Symbolic Logic; Summary; Elements of Set Theory: Sets, Functions, Relations; Induction; Turing Machines: Computability and Decidability; Propositional Logic: Syntax and Proof Systems; Semantics of PL; Soundness and Completeness; Diagnosing Paradoxes; First Order Logic: Syntax and Proof Systems of FOL; Semantics of FOL; More Semantics; Soundness and Completeness; Why is First Order Logic "First Order"?;

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Book Synopsis'A wealth of examples to which solutions are given permeate the text so the reader will certainly be active.'The Mathematical GazetteThis is the final book written by the late great puzzle master and logician, Dr. Raymond Smullyan.This book is a sequel to my Beginner's Guide to Mathematical Logic.The previous volume deals with elements of propositional and first-order logic, contains a bit on formal systems and recursion, and concludes with chapters on Gödel's famous incompleteness theorem, along with related results.The present volume begins with a bit more on propositional and first-order logic, followed by what I would call a 'fein' chapter, which simultaneously generalizes some results from recursion theory, first-order arithmetic systems, and what I dub a 'decision machine.' Then come five chapters on formal systems, recursion theory and metamathematical applications in a general setting. The concluding five chapters are on the beautiful subject of combinatory logic, which is not only intriguing in its own right, but has important applications to computer science. Argonne National Laboratory is especially involved in these applications, and I am proud to say that its members have found use for some of my results in combinatory logic.This book does not cover such important subjects as set theory, model theory, proof theory, and modern developments in recursion theory, but the reader, after studying this volume, will be amply prepared for the study of these more advanced topics.

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Book Synopsis

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