Mathematical logic Books

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Book SynopsisO livro-texto popular de Dirk van Dalen, Lógica e Estrutura, agora em sua quinta edição, oferece uma introdução abrangente aos fundamentos da lógica clássica e da lógica intuicionística, da teoria dos modelos e do famoso teorema da incompletude de Gödel. A lógica proposicional e a lógica de predicados são apresentadas num estilo fácil de ler usando a dedução natural de Gentzen. O livro prossegue com alguns conceitos e fatos básicos da teoria dos modelos: uma discussão sobre compaccidade, Skolem-Löwenheim, modelos não-padrão e eliminação de quantificadores. A discussão da lógica clássica é encerrada com uma exposição concisa da lógica de segunda ordem. Em vista do crescente reconhecimento de métodos e princípios construtivos, a lógica intuicionística e a semântica de Kripke são cuidadosamente exploradas. Uma série de características construtivas específicas, como a separação e a igualdade, a tradução de Gödel, as propriedades da disjunção e da existência também foram incluídas. O último capítulo sobre o primeiro teorema da incompletude de Gödel é autocontido e fornece uma exposição sistemática da teoria da recursão necessária. Esta nova edição foi devidamente revisada e contém uma nova seção sobre ultraprodutos.

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Book SynopsisLogic and Representation brings together a collection of essays, written over a period of ten years, that apply formal logic and the notion of explicit representation of knowledge to a variety of problems in artificial intelligence, natural language semantics and the philosophy of mind and language. Particular attention is paid to modelling and reasoning about knowledge and belief, including reasoning about one's own beliefs, and the semantics of sentences about knowledge and belief. Robert C. Moore begins by exploring the role of logic in artificial intelligence, considering logic as an analytical tool, as a basis for reasoning systems, and as a programming language. He then looks at various logical analyses of propositional attitudes, including possible-world models, syntactic models, and models based on Russellian propositions. Next Moore examines autoepistemic logic, a logic for modelling reasoning about one's own beliefs. Rounding out the volume is a section on the semantics of natural language, including a survey of problems in semantic representation; a detailed study of the relations among events, situations, and adverbs; and a presentation of a unification-based approach to semantic interpretation. Robert C. Moore is principal scientist of the Artificial Intelligence Center of SRI International.Table of ContentsPart I. Methodological Arguments: 1. The role of logic in artificial intelligence: 2. A cognitivist reply to behaviourism; Part II. Propositional Attitudes: 3. A formal theory of knowledge and action; 4. Computational models of belief and the semantics of belief sentences; 5. Propositional attitudes and Russellian propositions; Part III. Autoepistemic Logic: 6. Semantical considerations on nonmonotonic logic; 7. Possible-world semantics for autoepistemic logic; Part IV. Semantics of Natural Language: 8. Events, situations, and adverbs; 9. Unification-based semantic interpretation; Index.

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Book SynopsisThe goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.Table of Contents- Part I Preliminaries. - Ultrafilters. - Nonstandard Analysis. - Hyperfinite Generators of Ultrafilters. - Many Stars: Iterated Nonstandard Extensions. - LoebMeasure. - Part II Ramsey Theory. - Ramsey’s Theorem. - The Theorems of van der Waerden and Hales-Jewett. - From Hindman to Gowers. - Partition Regularity of Equations. - Part III Combinatorial Number Theory. - Densities and Structural Properties. - Working in the Remote Realm. - Jin’s Sumset Theorem. - Sumset Configurations in Sets of Positive Density. - Near Arithmetic Progressions in Sparse Sets. - The Interval Measure Property. - Part IV Other Topics. - Triangle Removal and Szemerédi Regularity. - Approximate Groups. - Foundations of Nonstandard Analysis.

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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 SynopsisKeeping students involved and actively learning is challenging. Instructors in computer science are aware of the cognitive value of modelling puzzles and often use logical puzzles as an efficient pedagogical instrument to engage students and develop problem-solving skills. This unique book is a comprehensive resource that offers teachers and students fun activities to teach and learn logic. It provides new, complete, and running formalisation in Propositional and First Order Logic for over 130 logical puzzles, including Sudoku-like puzzles, zebra-like puzzles, island of truth, lady and tigers, grid puzzles, strange numbers, or self-reference puzzles. Solving puzzles with theorem provers can be an effective cognitive incentive to motivate students to learn logic. They will find a ready-to-use format which illustrates how to model each puzzle, provides running implementations, and explains each solution. This concise and easy-to-follow textbook is a much-needed support tool for students willing to explore beyond the introductory level of learning logic and lecturers looking for examples to heighten student engagement in their computer science courses. Trade Review“The purpose of this book is to introduce first-order logic (FOL) to newcomers. … The book is a treasure trove of puzzles like this. … All of these are motivated in an approachable, fun way. … the book is a hands-on guide to Prover9 and Mace4 … . It is quite valuable to have so many puzzles in a single book.” (Jesse Adam Alama, Mathematical Reviews, October, 2022)Table of ContentsPreface.- Getting Started with Prover9 and Mace4.- Micro Arithmetic Puzzles.- Strange Numbers.- Practical Puzzles.- Lady and Tigers.- Einstein Puzzles.- Island of Truth.- Love and Marriage.- Grid Puzzles.- Japanese Puzzles.- Russian Puzzles.- Polyomino Puzzles.- Self-reference and Other Puzzles.- Epigraph in Natural Language

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Book SynopsisMany believe mathematics is only about calculations, formulas, numbers, and strange letters. But mathematics is much more than just crunching numbers or manipulating symbols. Mathematics is about discovering patterns, uncovering hidden structures, finding counterexamples, and thinking logically. Mathematics is a way of thinking. It is an activity that is both highly creative and challenging. This book offers an introduction to mathematical reasoning for beginning university or college students, providing a solid foundation for further study in mathematics, computer science, and related disciplines. Written in a manner that directly conveys the sense of excitement and discovery at the heart of doing science, its 25 short and visually appealing chapters cover the basics of set theory, logic, proof methods, combinatorics, graph theory, and much more. In the book you will, among other things, find answers to: What is a proof? What is a counterexample? What does it mean to say that something follows logically from a set of premises? What does it mean to abstract over something? How can knowledge and information be represented and used in calculations? What is the connection between Morse code and Fibonacci numbers? Why could it take billions of years to solve Hanoi's Tower? Logical Methods is especially appropriate for students encountering such concepts for the very first time. Designed to ease the transition to a university or college level study of mathematics or computer science, it also provides an accessible and fascinating gateway to logical thinking for students of all disciplines.Trade Review"The definitions are followed by examples to help explain their meaning, along with counterexamples ... . Therefore, very little basic knowledge is required for this introduction to logical methods ... which is written in an accessible style ... . contained in the book are several hundred small figures; arrow, Venn, and Hasse diagrams; and simplifies visual representations ... . The author has also elected to use color to draw the reader's attention ... ." “From personal teaching experience, knowledge of these mathematical areas is necessary for disparate fields of CS and informatics. These foundations are needed for many fields, from database theory to various domains of information systems applications. The book’s presentation of topics and incentives for problem-solving, along with its exercises, is very useful for university-level instructors and students. The compact chapters contain clear explanations, diagrams, and brief descriptions of interesting facts.” (Bálint Molnár, Computing Reviews, July 27, 2021)Table of ContentsPreface.- 0 The Art of Thinking Abstractly and Mathematically.- 1 Basic Set Theory.- 2 Propositional Logic.- 3 Semantics from Propositional Logic.- 4 Concepts in Propositional Logic.- 5 Proofs, Conjectures, and Counterexamples.- 6 Relations.- 7 Functions.- 8 A Little More Set Theory.- 9 Closures and Inductively Defined Sets.- 10 Recursively Defined Functions.- 11 Mathematical Induction.- 12 Structural Induction.- 13 First-Order Languages.- 14 Representation of Quantified Statements.- 15 Interpretation in Models.- 16 Reasoning About Models.- 17 Abstraction with Equivalences and Partitions.- 18 Combinatorics.- 19 A Little More Combinatorics.- 20 A Bit of Abstract Algebra.- 21 Graph Theory.- 22 Walks in Graphs.- 23 Formal Languages and Grammars.- 24 Natural Deduction.- The Road Ahead.- Index. Symbols.

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Book SynopsisSince its inception, fuzzy logic has attracted an incredible amount of interest, and this interest continues to grow at an exponential rate. As such, scientists, researchers, educators and practitioners of fuzzy logic continue to expand on the applicability of what and how fuzzy can be utilised in the real-world. In this book, the authors present key application areas where fuzzy has had significant success. The chapters cover a plethora of application domains, proving credence to the versatility and robustness of a fuzzy approach. A better understanding of fuzzy will ultimately allow for a better appreciation of fuzzy. This book provides the reader with a varied range of examples to illustrate what fuzzy logic can be capable of and how it can be applied. The text will be ideal for individuals new to the notion of fuzzy, as well as for early career academics who wish to further expand on their knowledge of fuzzy applications. The book is also suitable as a supporting text for advanced undergraduate and graduate-level modules on fuzzy logic, soft computing, and applications of AI.Table of ContentsRecognising Handwritten Digits Using a Fuzzy Neural Network Joshua Reynolds and Tianhua Chen Fuzzy Assessment of Student Academic Performances Shangen Yang and Tianhua Chen A Hybrid Fuzzy Neural Network for Image Recognition Samaresh Nayak and Tianhua Chen A Fuzzy Diagnostic System for Heart Disease Siyue Song, Tianhua Chen, and Grigoris Antoniou Analysing Medical Notes using Fuzzy Logic Siyue Song, Tianhua Chen, and Grigoris Antoniou Fostering Positive Personalisation through Fuzzy Clustering Raymond Moodley Fuzzy Logic in Modern Information Retrieval Steve Wade Fuzzy Applied to Sentiment Analysis Orestes Appel Fuzzy Logic, a Logicians Perspective Patrick Fogarty Applications of Fuzzy Logic in an Automated Warehouse Patrick Fogarty Can Fuzzy Systems Assist with Project Planning? Daniel Maia and Arjab Khuman Fuzzy Logic in Autonomous Vehicles David McDougall and Arjab Khuman AI Spawning Fuzzy Logic Fuzzy Inference System Reece Carey and Arjab Khuman The Application of Fuzzy Logic on Intelligent Transportation Systems Nath Lloyd and Arjab Khuman Fuzzy Logic Applied to Water Processes Will Chapman and Arjab Khuman Applications of Fuzzy Logic in Autonomous Vehicles Sam Asquith and Arjab Khuman Predicting Cyber Threats using Fuzzy Logic Jarrad Morden and Arjab Khuman Implementations of Fuzzy Logic in Camera Systems Sophie Hughes and Arjab Khuman Application of a Fuzzy Logic Control System for Stock Market Prediction Based on Technical Indicators and Fundamental Analysis Humza Nazir and Arjab Khuman The Application of Fuzzy Logic in Determining Outcomes of Sporting Events Spencer Deane and Arjab Khuman Using Fuzzy Logic to Educate People on Phishing Harry Taylor and Arjab Khuman

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Book SynopsisThis introduction to first-order logic clearly works out the role of first-order logic in the foundations of mathematics, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines. It covers several advanced topics not commonly treated in introductory texts, such as Fraïssé's characterization of elementary equivalence, Lindström's theorem on the maximality of first-order logic, and the fundamentals of logic programming.Trade Review“This newest edition has been reclassified, fittingly, as a graduate text, and it is admirably suited to that role. … Those who are already well-versed in logic will find this text to be a valuable reference and a strong resource for teaching at the graduate level, while those who are new to the field will come to know not only how mathematical logic is studied but also, perhaps more importantly, why.” (Stephen Walk, MAA Reviews, January 6, 2023)Table of ContentsA.- I Introduction.- II Syntax of First-Order Languages.- III Semantics of First-Order Languages.- IV A Sequent Calculus.- V The Completeness Theorem.- VI The Löwenheim–Skolem and the Compactness Theorem.- VII The Scope of First-Order Logic.- VIII Syntactic Interpretations and Normal Forms.- B.- IX Extensions of First-Order Logic.- X Computability and Its Limitations.- XI Free Models and Logic Programming.- XII An Algebraic Characterization of Elementary Equivalence.- XIII Lindström’s Theorems.- References.- List of Symbols.- Subject Index.

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Book SynopsisThis volume is dedicated to Hiroakira Ono life’s work on substructural logics. Chapters, written by well-established academics, cover topics related to universal algebra, algebraic logic and the Full Lambek calculus; the book includes a short biography about Hiroakira Ono. The book starts with detailed surveys on universal algebra, abstract algebraic logic, topological dualities, and connections to computer science.It further contains specialised contributions on connections to formal languages (recognizability in residuated lattices and connections to the finite embedding property), covering systems for modal substructural logics, results on the existence and disjunction properties and finally a study of conservativity of expansions. This book will be primarily of interest to researchers working in algebraic and non-classical logic.Table of ContentsChapter 1. A scientific autobiography (Hiroakira Ono).- Part I: Expository and survey chapters.- Chapter 2. Universal algebraic methods for non-classical logics (James G. Raftery).- Chapter 3. Abstract algebraic logic - An introductory chapter (Josep Maria Font).- Chapter 4. Topological duality and algebraic completions (Mai Gehrke).- Chapter 5. An algebraic glimpse at bunched implications and separation logic (Peter Jipsen and Tadeusz Litak).- Part II: Special topics.- Chapter 6. Recognizability in Residuated Lattices (José Gil-Férez and Constantine Tsinakis).- Chapter 7. Finite embeddability property for residuated lattices via regular languages (Rostislav Horčík). Chapter 8. Cover systems for the modalities of linear logic (Robert Goldblatt).- Chapter 9. A negative solution to Ono’s Problem P52: Existence and disjunction properties in intermediate predicate logic (Nobu-Yuki Suzuki).- Chapter 10. Conservative expansions of substructural logics (Jacopo Amidei, Rodolfo C. Ertola-Biraben and Franco Montagna).

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Book SynopsisThis book gathers together a colorful set of problems on classical Mathematical Logic, selected from over 30 years of teaching. The initial chapters start with problems from supporting fields, like set theory (ultrafilter constructions), full-information game theory (strategies), automata, and recursion theory (decidability, Kleene’s theorems). The work then advances toward propositional logic (compactness and completeness, resolution method), followed by first-order logic, including quantifier elimination and the Ehrenfeucht– Fraïssé game; ultraproducts; and examples for axiomatizability and non-axiomatizability. The Arithmetic part covers Robinson’s theory, Peano’s axiom system, and Gödel’s incompleteness theorems. Finally, the book touches universal graphs, tournaments, and the zero-one law in Mathematical Logic. Instructors teaching Mathematical Logic, as well as students who want to understand its concepts and methods, can greatly benefit from this work. The style and topics have been specially chosen so that readers interested in the mathematical content and methodology could follow the problems and prove the main theorems themselves, including Gödel’s famous completeness and incompleteness theorems. Examples of applications on axiomatizability and decidability of numerous mathematical theories enrich this volume.Table of ContentsChapter 1 - Special Set Systems.- Chapter 2 - Games and Voting.- Chapter 3 - Formal languages and automata.- Chapter 4 - Recursion Theory.- Chapter 5 - Propositional Calculus.- Chapter 6 - First-order logic.- Chapter 7 - Fundamental Theorems.- Chapter 8 - Elementary Equivalence.- Chapter 9 - Ultraproducts.- Chapter 10 - Arithmetic.- Chapter 11 - Selected Applications.- Chapter 12 - Solutions.

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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 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 book explores the premise that a physical theory is an interpretation of the analytico–canonical formalism. Throughout the text, the investigation stresses that classical mechanics in its Lagrangian formulation is the formal backbone of theoretical physics. The authors start from a presentation of the analytico–canonical formalism for classical mechanics, and its applications in electromagnetism, Schrödinger's quantum mechanics, and field theories such as general relativity and gauge field theories, up to the Higgs mechanism.The analysis uses the main criterion used by physicists for a theory: to formulate a physical theory we write down a Lagrangian for it. A physical theory is a particular instance of the Lagrangian functional. So, there is already an unified physical theory. One only has to specify the corresponding Lagrangian (or Lagrangian density); the dynamical equations are the associated Euler–Lagrange equations. The theory of Suppes predicates as the main tool in the axiomatization and examples from the usual theories in physics. For applications, a whole plethora of results from logic that lead to interesting, and sometimes unexpected, consequences.This volume looks at where our physics happen and which mathematical universe we require for the description of our concrete physical events. It also explores if we use the constructive universe or if we need set–theoretically generic spacetimes.Trade Review“This book is a compilation, ‘an essay’, of the bulk of their work from 1990 to the present. This 191 page essay includes some historical background and lots of snippets and parts of da Costa and Doria’s work on the meta-mathematics of mathematical physics. It starts with a primer on graduate-level basic physics … ending with a consideration of hypercomputation.” (Deborah Konkowski, zbMATH 1494.00005, 2022)Table of ContentsForeword1. PreliminaryPart I. Physics: A Primer2. Classical mechanics3. Variational calculus4. Lagrangian formulation5. Hamilton’s equations6. Hamilton–Jacobi theory7. Where the action is8. From classical to quantum9. Field theory10. Electromagnetism11. Special relativity12. General relativity13. Gauge field theoriesPart II. Axiomatics14. Axiomatizations in ZFCPart III. Technicalities15. HierarchiesPart IV. More applications16. Arnol’d’s 1974 problems17. Forcing and gravitation18. Economics and ecology.Part V. Computer science19. Fast–growing functionsPart VI. Hypercomputation20. HypercomputationReferences

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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 textbook covers topics of undergraduate mathematics in abstract algebra, geometry, topology and analysis with the purpose of connecting the underpinning key ideas. It guides STEM students towards developing knowledge and skills to enrich their scientific education. In doing so it avoids the common mechanical approach to problem-solving based on the repetitive application of dry formulas. The presentation preserves the mathematical rigour throughout and still stays accessible to undergraduates. The didactical focus is threaded through the assortment of subjects and reflects in the book’s structure.Part 1 introduces the mathematical language and its rules together with the basic building blocks. Part 2 discusses the number systems of common practice, while the backgrounds needed to solve equations and inequalities are developed in Part 3. Part 4 breaks down the traditional, outdated barriers between areas, exploring in particular the interplay between algebra and geometry. Two appendices form Part 5: the Greek etymology of frequent terms and a list of mathematicians mentioned in the book. Abundant examples and exercises are disseminated along the text to boost the learning process and allow for independent work.Students will find invaluable material to shepherd them through the first years of an undergraduate course, or to complement previously learnt subject matters. Teachers may pick’n’mix the contents for planning lecture courses or supplementing their classes.Trade Review“The book being reviewed is a collection of what the author considers to be essential material for undergraduates … . it has to be said that many students will find that there is plenty to learn from this well-written book, which would also be a useful reference text had there been a properly compiled index.” (Peter Shiu, The Mathematical Gazette, Vol. 107 (570), November, 2023)Table of ContentsPart I: Basic Objects and Formalisation - Round-up of Elementary Logic.- Naive Set Theory.- Functions.- More Set Theory and Logic.- Boolean Algebras. Part 2: Numbers and Structures - Intuitive Arithmetics.- Real Numbers.- Totally Ordered Spaces.- Part 3: Elementary Real Functions - Real Polynomials.- Real Functions of One Real Variables.- Algebraic Functions.- Elementary Transcendental Functions.- Complex Numbers.- Enumerative Combinatorics.- Part 4: Geometry through Algebra - Vector Spaces.- Orthogonal Operators.- Actions & Representations.- Elementary Plane Geometry.- Metric Spaces.- Part 5: Appendices - Etymologies.- Index of names.- Main figures.- Glossary.- References.

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Book SynopsisThis textbook covers topics of undergraduate mathematics in abstract algebra, geometry, topology and analysis with the purpose of connecting the underpinning key ideas. It guides STEM students towards developing knowledge and skills to enrich their scientific education. In doing so it avoids the common mechanical approach to problem-solving based on the repetitive application of dry formulas. The presentation preserves the mathematical rigour throughout and still stays accessible to undergraduates. The didactical focus is threaded through the assortment of subjects and reflects in the book’s structure.Part 1 introduces the mathematical language and its rules together with the basic building blocks. Part 2 discusses the number systems of common practice, while the backgrounds needed to solve equations and inequalities are developed in Part 3. Part 4 breaks down the traditional, outdated barriers between areas, exploring in particular the interplay between algebra and geometry. Two appendices form Part 5: the Greek etymology of frequent terms and a list of mathematicians mentioned in the book. Abundant examples and exercises are disseminated along the text to boost the learning process and allow for independent work.Students will find invaluable material to shepherd them through the first years of an undergraduate course, or to complement previously learnt subject matters. Teachers may pick’n’mix the contents for planning lecture courses or supplementing their classes.Trade Review“The book being reviewed is a collection of what the author considers to be essential material for undergraduates … . it has to be said that many students will find that there is plenty to learn from this well-written book, which would also be a useful reference text had there been a properly compiled index.” (Peter Shiu, The Mathematical Gazette, Vol. 107 (570), November, 2023)Table of ContentsPart I: Basic Objects and Formalisation - Round-up of Elementary Logic.- Naive Set Theory.- Functions.- More Set Theory and Logic.- Boolean Algebras. Part 2: Numbers and Structures - Intuitive Arithmetics.- Real Numbers.- Totally Ordered Spaces.- Part 3: Elementary Real Functions - Real Polynomials.- Real Functions of One Real Variables.- Algebraic Functions.- Elementary Transcendental Functions.- Complex Numbers.- Enumerative Combinatorics.- Part 4: Geometry through Algebra - Vector Spaces.- Orthogonal Operators.- Actions & Representations.- Elementary Plane Geometry.- Metric Spaces.- Part 5: Appendices - Etymologies.- Index of names.- Main figures.- Glossary.- References.

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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 SynopsisThis open access book is the first ever collection of Karl Popper's writings on deductive logic.Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics.This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work.Table of Contents Part I: Articles.- Chapter 1. Introduction to Popper’s Articles on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 2. Are Contradictions Embracing? (1943) (Karl R. Popper).- Chapter 3. Logic without Assumptions (1947) (Karl R. Popper).- Chapter 4. New Foundations for Logic (1947) (Karl R. Popper).- Chapter 5. Functional Logic without Axioms or Primitive Rules of Inference (1947)(Karl R. Popper).- Chapter 6. On the Theory of Deduction, Part I. Derivation and its Generalizations (1948) (Karl R. Popper).- Chapter 7. On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation (1948) (Karl R. Popper).- Chapter 8. The Trivialization of Mathematical Logic (1949) (Karl R. Popper).- Chapter 9. A Note on Tarski’s Definition of Truth (1955) (Karl R. Popper).-Chapter 10. On a Proposed Solution of the Paradox of the Liar (1955) (Karl R. Popper).- Chapter 11. On Subjunctive Conditionals with Impossible Antecedents (1959) (Karl R. Popper).- Chapter 12. Lejewski’s Axiomatization of My Theory of Deducibility (1974) (Karl R. Popper).- Chapter 13. Reviews of Popper’s Articles on Logic (Wilhelm Ackermann et.al).- Part II: Manuscripts.- Chapter 14. Introduction to Popper’s Manuscripts on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 15. On Systems of Rules of Inference (Karl R. Popper and Paul Bernays).- Chapter 16. A General Theory of Inference (Karl R. Popper).- Chapter 17. On the Logic of Negation (Karl R. Popper).- Chapter 18. A Note on the Classical Conditional (Karl R. Popper).- Part III: Correspondence.- Chapter 19. Introduction to Popper’s Correspondence on Logic (David Binder, Thomas Piecha, and Peter Schroeder-Heister).- Chapter 20. Popper’s Correspondence with Paul Bernays (Karl R. Popper and Paul Bernays).- Chapter 21. Popper’s Correspondence with Luitzen Egbertus Jan Brouwer (Karl R. Popper and Luitzen E. J. Brouwer).- Chapter 22. Popper’s Correspondence with Rudolf Carnap (Karl R. Popper and Rudolf Carnap).- Chapter 23. Popper’s Correspondence with Alonzo Church (Karl R. Popper and Alonzo Church).- Chapter 24. Popper’s Correspondence with Kalman Joseph Cohen (Karl R. Popper and Kalman J. Cohen).- Chapter 25. Popper’s Correspondence with Henry George Forder (Karl R. Popper and Henry George Forder).- Chapter 26. Popper’s Correspondence with Harold Jeffreys (Karl R. Popper and Harold Jeffreys).- Chapter 27. Popper’s Correspondence with Stephen Cole Kleene (Karl R. Popper and Stephen C. Kleene).- Chapter 28. Popper’s Correspondence with William Calvert Kneale (Karl R. Popper and William C. Kneale).- Chapter 29. Popper’s Correspondence with Willard Van Orman Quine (Karl R. Popper and Willard V. O. Quine).- Chapter 30. Popper’s Correspondence with Heinrich Scholz (Karl R. Popper and Heinrich Scholz).- Chapter 31. Popper’s Correspondence with Peter Schroeder-Heister (Karl R. Popper and Peter Schroeder-Heister).- Concordances.- Bibliography.- Index.

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Book SynopsisThis open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems.

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Book SynopsisThis volume explores A.P. Morse’s (1911-1984) development of a formal language for writing mathematics, his application of that language in set theory and mathematical analysis, and his unique perspective on mathematics. The editor brings together a variety of Morse’s works in this compilation, including Morse's book A Theory of Sets, Second Edition (1986), in addition to material from another of Morse’s publications, Web Derivatives, and notes for a course on analysis from the early 1950's. Because Morse provided very little in the way of explanation in his written works, the editor’s commentary serves to outline Morse’s goals, give informal explanations of Morse’s formal language, and compare Morse’s often unique approaches to more traditional approaches. Minor corrections to Morse’s previously published works have also been incorporated into the text, including some updated axioms, theorems, and definitions. The editor’s introduction thoroughly details the corrections and changes made and provides readers with valuable insight on Morse’s methods.A.P. Morse’s Set Theory and Analysis will appeal to graduate students and researchers interested in set theory and analysis who also have an interest in logic. Readers with a particular interest in Morse’s unique perspective and in the history of mathematics will also find this book to be of interest.Table of ContentsPreface.- Editor's Introduction.- Language and Inference.- Logic.- Set Theory.- Elementary Analysis.- Metrics.- Measure.- Linear Measure and Total Variation.- Integration.- Product Measures.- Web Derivatives.- Classical Differentiation.- The Construction of Definition.- The Consistency of the Axiom of Size.- Suggested Reading.- Publications of A.P. Morse.- Errata to A Theory of Sets, Second Edition.- Integration with Respect to Addor Functions.- The Henstock-Kurzweil Integral.

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Book SynopsisThis book constitutes revised selected papers from the refereed proceedings of the 4th International Workshop on Dynamic Logic, DaLí 2022, held in Haifa, Israel, in July/August 2022.The 8 full papers presented in this volume were carefully reviewed and selected from 22 submissions. They deal with new trends and applications in the area of Dynamic Logic. Table of ContentsFirst steps in updating knowing how.- Parametrized modal logic II: the unidimensional case.- Relating Kleene algebras.- Dynamic epistemic logic for budget-constrained agents.- Action models for coalition logic.- Quantum logic for observation of physical quantities.- Cautious distributed belief.- A STIT logic of intentionality.

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Book SynopsisEdited in collaboration with FoLLI, this book constitutes the refereed proceedings of the 10th Indian Conference on Logic and Its Applications, ICLA 2023, which was held in Indore, India, in March 2023.Besides 6 invited papers presented in this volume, there are 9 contributed full papers which were carefully reviewed and selected from 18 submissions. The volume covers a wide range of topics. These topics are related to modal and temporal logics, intuitionistic connexive and imperative logics, systems for reasoning with vagueness and rough concepts, topological quasi-Boolean logic and quasi-Boolean based rough set models, and first-order definability of path functions of graphs.Table of ContentsA Note on the Ontology of Mathematics.- Boolean Functional Synthesis: From Under the Hood of Solvers.- Labelled Calculi for Lattice-based Modal Logics.- Two Ways to Scare a Gruffalo.- Determinacy Axioms and Large Cardinals.- Big ideas from logic for mathematics and computing education.- Modal Logic of Generalized Separated Topological Spaces.- Multiple-valued Semantics for Metric Temporal Logic.- Segment transit function of the induced path function of graphs and its first-order definability.- Fuzzy Free Logic with Dual Domain Semantics.- A New Dimension of Imperative Logic. -Quasi-Boolean based models in Rough Set theory: A case of Covering.- Labelled calculi for the logics of rough concepts.- An Infinity of Intuitionistic Connexive Logics.- Relational Semantics for Normal Topological Quasi-Boolean Logic.

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Book SynopsisThis book constitutes the proceedings of the 5th International Workshop on Formal Methods Teaching, FMTea 2023, which was held in Lübeck, Germany, in March 2023.The 7 full papers presented in this volume were carefully reviewed and selected from 10 submissions. FMTea 2023 aim is to support a worldwide improvement in learning Formal Methods, mainly by teaching but also via self-learning.Table of ContentsAutomated Exercise Generation for Satisfiability Checking.- Graphical Loop Invariant Based Programming.- A Gentle Introduction to Verification of Parameterized Reactive Systems.- Model Checking Concurrent Programs for Autograding in pseuCo Book.- Teaching TLA+ to Engineers at Microsoft.- Teaching and Training in Formalisation with B.- Teaching low-code Formal Methods with Coloured Petri Nets.

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Book SynopsisFrom the ‘punctuated equilibrium' of Eldrege and Gould, through Lewontin's ‘triple helix' and the various visions and revisions of the Extended Evolutionary Synthesis (EES) of Laland and others, both data and theory have demanded an opening-up of the 1950's Evolutionary Synthesis that so firmly wedded evolutionary theory to the mathematics of gene frequency analysis. It can, however, be argued that a single deep and comprehensive mathematical theory may simply not be possible for the almost infinite varieties of evolutionary process active at and across the full range of scales of biological, social, institutional, and cultural phenomena. Indeed, the case history of 'meme theory' should have raised a red flag that narrow gene-centered models of evolutionary process may indeed have serious limitations. What is attempted here is less grand, but still broader than a gene-centered analysis. Following the instruction of Maturana and Varela that all living systems are cognitive, in a certain sense, and that living as a process is a process of cognition, the asymptotic limit theorems of information and control theories that bound all cognition provide a basis for constructing an only modestly deep but wider-ranging series of probability models that might be converted into useful statistical tools for the analysis of observational and experimental data related to evolutionary process. The line of argument in this series of interrelated essays proves to be surprisingly direct.Table of Contents1 Onthemajortransitions1.1 Introduction1.2 Symmetryandsymmetry-breaking1.3 Resources1.4 Cognitioninnonergodicsystems1.5 Theprebiotic`bigbang'1.6 Biological`recombinationtransparency'1.7 Asimpleapplication1.8 Specializationandcooperation:multipleworkspaces1.9 Discussion1. MathematicalAppendix1. References2 OntheExtendedEvolutionarySynthesis2.1 Introduction2.2 Firstnotions2.3 Thebasictheory2.4 Examples2.5 Moretheory:selectionpressureasshadowprice2.6 Extendingthemodels2.7 Discussion2.8 MathematicalAppendix2.9 References3O nregulation3.1 Introduction3.2 Theory3.3 Applications3.4 Discussion3.5 MathematicalAppendix3.6 References4 Punctuatedregulationasanevolutionarymechanism4.1 Introduction4.2 FisherZerosreconsidered4.3 ExtinctionI:Simplenoise-inducedtransitions4.4 ExtinctionII:Morecomplicatednoise-inducedtransitions4.5 ExtinctionIII:Environmentalshadowprice4.6 Discussion4.7 MathematicalAppendix4.8 References5 Institutionaldynamicsunderselectionpressureanduncertainty5.1 Introduction5.2 ARateDistortionTheoremmodelofcontrol5.3 Selectionpressuredynamics5.4 Destabilizationbydelay5.5 ExtendingtheDataRateTheorem5.6 Movingon5.7 Reconsideringcognition\textit{AnSich5.8 Changingtheviewpoint5.9 Discussion5. References6O n`Speciation':Fragmentsizeininformationsystemphasetransitions6.1 Introduction6.2`Simple'phasetransition6.3 Phasetransitionsinnetworksofinformation-exchangemodules6.4 Discussion6.5 MathematicalAppendix:`Biological'renormalizations6.6 References7 Adaptingcognitionmodelstobiomolecularcondensatedynamics7.1 Introduction7.2 Resources7.3 Cognition7.4 PhasetransitionsI:Fisherzeros7.5 Cognitive`reactionrate'7.6 PhasetransitionsII:Signaltransductionandnoise7.7 Discussion7.8 MathematicalAppendix:Groupoids7.9 References8 EvolutionaryExaptation:Sharedinterbrainactivityinsocialcommunication8.1 Introduction8.2 Correlation8.3 Cognition8.4 Dynamics8.5 Cognitionrate8.6 Anexample8.7 Cooperation:Multipleworkspaces8.8 Networktopologyisimportant8.9 Timeandresourceconstraintsareimportant8.10 Furthertheoreticaldevelopment8.11 Discussion8.12 MathematicalAppendix8.13 References9 Afterward

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Book SynopsisThe main aim of this book is to provide a compact self-contained presentation of the forcing technique devised by Cohen to establish the independence of the continuum hypothesis from the axioms of set theory. The book follows the approach to the forcing technique via Boolean valued semantics independently introduced by Vopenka and Scott/Solovay; it develops out of notes I prepared for several master courses on this and related topics and aims to provide an alternative (and more compact) account of this topic with respect to the available classical textbooks. The aim of the book is to take up a reader with familiarity with logic and set theory at the level of an undergraduate course on both topics (e.g., familiar with most of the content of introductory books on first-order logic and set theory) and bring her/him to page with the use of the forcing method to produce independence (or undecidability results) in mathematics. Familiarity of the reader with general topology would also be quite helpful; however, the book provides a compact account of all the needed results on this matter. Furthermore, the book is organized in such a way that many of its parts can also be read by scholars with almost no familiarity with first-order logic and/or set theory. The book presents the forcing method outlining, in many situations, the intersections of set theory and logic with other mathematical domains. My hope is that this book can be appreciated by scholars in set theory and by readers with a mindset oriented towards areas of mathematics other than logic and a keen interest in the foundations of mathematics.

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Book SynopsisThe starting point for this monograph is the previously unknown connection between the Continuum Hypothesis and the saturation of the non-stationary ideal on ω1; and the principle result of this monograph is the identification of a canonical model in which the Continuum Hypothesis is false. This is the first example of such a model and moreover the model can be characterized in terms of maximality principles concerning the universal-existential theory of all sets of countable ordinals. This model is arguably the long sought goal of the study of forcing axioms and iterated forcing but is obtained by completely different methods, for example no theory of iterated forcing whatsoever is required. The construction of the model reveals a powerful technique for obtaining independence results regarding the combinatorics of the continuum, yielding a number of results which have yet to be obtained by any other method. This monograph is directed to researchers and advanced graduate students in Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of Ω-logic and related matters.

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Book SynopsisMathematical Logic: An Introduction is a textbook that uses mathematical tools to investigate mathematics itself. In particular, the concepts of proof and truth are examined. The book presents the fundamental topics in mathematical logic and presents clear and complete proofs throughout the text. Such proofs are used to develop the language of propositional logic and the language of first-order logic, including the notion of a formal deduction. The text also covers Tarski’s definition of truth and the computability concept. It also provides coherent proofs of Godel’s completeness and incompleteness theorems. Moreover, the text was written with the student in mind and thus, it provides an accessible introduction to mathematical logic. In particular, the text explicitly shows the reader how to prove the basic theorems and presents detailed proofs throughout the book. Most undergraduate books on mathematical logic are written for a reader who is well-versed in logical notation and mathematical proof. This textbook is written to attract a wider audience, including students who are not yet experts in the art of mathematical proof.

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Book SynopsisThis book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.

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Book SynopsisThe two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability.Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs.Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.Trade Review“This monograph by the outstanding Czech logician Pavel Pudlák provides a broad but also deep survey of work in logic and computer science relevant to foundational issues, interpreted in a wide sense. … This is a fine overview of logic and complexity theory that can be confidently recommended to anybody who would like to orient themselves in an increasingly intricate and difficult field.” (Alasdair Urquhart, Philosophia Mathematica, Vol. 23 (3), October, 2015)“For the non-expert it offers indeed a ‘gentle introduction’ to logic that is well selected and excellently explained. And for the logician it certainly offers some of the best introductions to those topics outside their area of direct expertise. … it contains plenty of informal explanations, intuition and motivation. … It is truly a gift to the logic and wider communities … . This book is very enjoyable to read and I wish it all success.” (Olaf Beyersdorff, Mathematical Reviews, August, 2014)“It spans the historical, logical, and at times philosophical underpinnings of the theory of computational complexity. Students of mathematics seeking a transition to higher mathematics will find it helpful, as will mathematicians with expertise in other areas. … an excellent choice for a first text in studying complexity, or as a clarifying adjunct to any assigned text in this area. … a compact guide for graduate students with a need for or interest in computational complexity and its foundations.” (Tom Schulte, MAA Reviews, July, 2014)“This book, exactly as indicated by its title, deals with the main philosophical, historical, logical and mathematical aspects … in a quite approachable and attractive way. … the prospective readers of this book are mathematicians with an interest in the foundations, philosophers with a good background in mathematics, and also philosophically minded scientists. Due to the author’s nice style, the book will be a very good choice for the first text in studying this subject.” (Branislav Boričić, zbMATH, Vol. 1270, 2013)Table of Contents​​​​​​​​​​Mathematician’s world.- Language, logic and computations.- Set theory.- Proofs of impossibility.- The complexity of computations.- Proof complexity.- Consistency, Truth and Existence.- References.

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Book SynopsisWhile most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself.By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics.Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.Trade Review“This is a book of both analysis and set theory, and the analysis begins at an elementary level with the necessary treatment of completeness of the reals. … the analysis makes it valuable to the serious student, say a senior or first-year graduate student. … Stillwell’s book can work well as a text for the course in foundations, with its good treatment of the cardinals and ordinals. … This enjoyable book makes the connection clear.” (James M. Cargal, The UMAP Journal, Vol. 38 (1), 2017)“This book is an interesting introduction to set theory and real analysis embedded in properties of the real numbers. … The 300-plus problems are frequently challenging and will interest both upper-level undergraduate students and readers with a strong mathematical background. … A list of approximately 100 references at the end of the book will help students to further explore the topic. … Summing Up: Recommended. Lower-division undergraduates.” (D. P. Turner, Choice, Vol. 51 (11), August, 2014)“This is an informal look at the nature of the real numbers … . There are extensive historical notes about the evolution of real analysis and our understanding of real numbers. … Stillwell has deliberately set out to provide a different sort of construction where you understand what the foundation is supporting and why it is important. I think this is very successful, and his book … is much more informative and enjoyable.” (Allen Stenger, MAA Reviews, February, 2014)“This book will be fully appreciated by either professional mathematicians or those students, who already have passed a course in analysis or set theory. … The book contains a quantity of motivation examples, worked examples and exercises, what makes it suitable also for self-study.” (Vladimír Janiš, zbMATH, 2014)“The book offers a rigorous foundation of the real number system. It is intended for senior undergraduates who have already studied calculus, but a wide range of readers will find something interesting, new, or instructive in it. … This is an extremely reader-friendly book. It is full of interesting examples, very clear explanations, historical background, applications. Each new idea comes after proper motivation.” (László Imre Szabó, Acta Scientiarum Mathematicarum (Szeged), Vol. 80 (1-2), 2014)Table of ContentsThe Fundamental Questions.- From Discrete to Continuous.- Infinite Sets.- Functions and Limits.- Open Sets and Continuity.- Ordinals.- The Axiom of Choice.- Borel Sets.- Measure Theory.- Reflections.- Bibliography.- Index.

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Book SynopsisThis is a comprehensive book on the life and works of Leon Henkin (1921–2006), an extraordinary scientist and excellent teacher whose writings became influential right from the beginning of his career with his doctoral thesis on “The completeness of formal systems” under the direction of Alonzo Church. Upon the invitation of Alfred Tarski, Henkin joined the Group in Logic and the Methodology of Science in the Department of Mathematics at the University of California Berkeley in 1953. He stayed with the group until his retirement in 1991. This edited volume includes both foundational material and a logic perspective. Algebraic logic, model theory, type theory, completeness theorems, philosophical and foundational studies are among the topics covered, as well as mathematical education. The work discusses Henkin’s intellectual development, his relation to his predecessors and contemporaries and his impact on the recent development of mathematical logic. It offers a valuable reference work for researchers and students in the fields of philosophy, mathematics and computer science.Table of ContentsPart I Biographical Studies.- Leon Henkin.- Lessons from Leon.- Tracing back “Logic in Wonderland” to my work with Leon Henkin.- Henkin and the Suit.- A Fortuitous Year with Leon Henkin.- Leon Henkin and a Life of Service.- Part II Henkin‘s Contribution to XX Century Logic.- Leon Henkin and Cylindric Algebras.- A Bit of History Related to Logic Based on Equality.- Pairing Logical and Pedagogical Foundations for the Theory of Positive Rational Numbers. Henkin‘s unfinished work.- Leon Henkin the Reviewer.- Henkin‘s Theorem in Textbooks.- Henkin on Completeness.- Part III Extensions and Perspectives in Henkin‘s Work.- The Countable Henkin Principle.- Reflections on a Theorem of Henkin.- Henkin‘s Completeness Proof and Glivenko‘s Theorem.- From Classical to Fuzzy Type Theory.- The Henkin Sentence.- April the 19th.- Henkin and Hybrid Logic.- Changing a Semantics: Oportunism or Courage?.- Appendix Curriculum Vitae: Leon Henkin.

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Book SynopsisThis volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three “dogmas of proof-theoretic semantics” are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.Trade Review“Swedish logician and philosopher Dag Prawitz and his distinguished contributions to philosophical and mathematical logic are the focus of this book. … This is an excellent book, celebrating not only Prawitz’s career, but also a movement in the contrary direction of W. V. O Quine’s views against the so-called (somehow prejudicially) ‘deviant’ logics, and I cannot forbear from congratulating the editor for the distinctive choice of topics and for the general tone of the book.” (Walter Carnielli, Computing Reviews, May, 2015)Table of ContentsPrawitz, proofs, and meaning; Wansing, Heinrich.- A short scientific autobiography; Prawitz, Dag.- Explaining deductive inference; Prawitz, Dag.- Necessity of Thought; Cozzo, Cesare.- On the Motives for Proof Theory; Detlefsen, Michael.- Inferential Semantics; Došen, Kosta.- Cut elimination, substitution and normalization; Dyckhoff, Roy.- Inversion principles and introduction rules; Milne, Peter.- Intuitionistic Existential Instantiation and Epsilon Symbol; Mints, Grigori.- Meaning in Use; Negri, Sara and von Plato, Jan.- Fusing Quantifiers and Connectives: Is Intuitionistic Logic Different?; Pagin, Peter.- On constructive fragments of Classical Logic; Pereira; Luiz Carlos and Haeusler, Edward Hermann.- General-Elimination Harmony and Higher-Level Rules; Read, Stephen.- Hypothesis-discharging rules in atomic bases; Sandqvist, Tor.- Harmony in proof-theoretic semantics: A reductive analysis; Schroeder-Heister, Peter.- First-order Logic without bound variables: Compositional Semantics; Tait, William W.- On Gentzen’s Structural Completeness Proof; Tennant, Neil.- A Notion of C-Justification for Empirical Statements; Usberti, Gabriele.

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Book SynopsisGerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus, central in many areas of logic and computer science today. Another noteworthy achievement is his resolution of the embarrassing situation created by Gödel's incompleteness results, especially the second one about the unprovability of consistency of elementary arithmetic. After these successes, Gentzen dedicated the rest of his short life to the main problem of Hilbert's proof theory, the question of the consistency of analysis. He was arrested in the summer of 1945 with other professors of the German University of Prague and died soon afterward of starvation in a prison cell. Attempts at locating his lost manuscripts failed at the time, but several decades later, two slim folders of shorthand notes were found. In this volume, Jan von Plato gives an overview of Gentzen's life and scientific achievements, based on detailed archival and systematic studies, and essential for placing the translations of shorthand manuscripts that follow in the right setting. The materials in this book are singular in the way they show the birth and development of Gentzen's central ideas and results, sometimes in a well-developed form, and other times as flashes into the anatomy of the workings of a unique mind.Trade Review“This book is obviously indispensable to historians of logic in the immediate wake of Gödel’s 1931 incompleteness theorems. … Saved from the Cellar is also valuable for less specialist readers (like myself ) who wish to understand the broader outlines of what proof theory has meant to several of its leading creators.” (Colin McLarty, Isis, Vol. 111 (1), 2020)“The book contains translations of shorthand notes which survived in the Nachlass of the mathematical logician Gerhard Gentzen. ... The book is valuable source for the history of modern logic; the editor did an excellent work in getting the shorthand notes, first transcribed in normal German text, and then translating it to English.” (Reinhard Kahle, zbMath 1414.03002, 2019)“Every general reader interested in modern logic and its history, … may find a source of inspiration in Genzen’s unpublished notes of the thirties, as well as for the philosopher concerned with epistemological aspects of modern logic.” (Adrian Rezus, Studia Logica, Vol. 107, 2019)“This is an account and transcription of two slim folders of stenographic material in Gerhard Gentzen's handwriting that were found in 1984. … this book is a valuable contribution to the history of the development of mathematical logic in the first half of the twentieth century.” (Henry Africk, Mathematical Reviews, December, 2017)Table of ContentsPart I: A Sketch of Gentzen's Life and Work.- 1. Overture.- 2. Gentzen's years of study.- Dr. Gentzen's arduous years in Nazi Germany.- 4. The scientific accomplishments.- 5. Loose ends.- 6. Gentzen's genuis.- Part II: Overview of the Shorthand Notes.- 1. Gentzen's series of stenographic manuscripts.- 2. The items in this collection.- Practical remarks on the manuscripts.- Manuscript illustrations.- The German alphabet in Latin, Sutterlin, and Fraktur Type.- Bibliography for parts I and II.- Index of names for Parts I and II.- Part III: The Original Writings.- 1. Reduction of number-theoretic problems to predicate logic.- 2. Replacement of functions by predicates.- 3. The formation of abstract concepts.- 4. Five different forms of natural calculi.- 5. Formal conception of correctness in arithmetic I.- 6. Investigations into logical inferences.- 7. Reduction of classical to intuitionistic logic.- 8. CV of the candidate Gerhard Gentzen.-0 9. Letters to Heyting.- 10. Formal conception of correctness in arithmetic II.- 11. Proof theory of number theory.- 12. Consistency of artihmetic, for publication.- 13. Correspondence with Paul Bernays.- 14. Forms of type theory.- 15. Predicate logic.- 16. Propositional logic.- 17. Foundational research in mathematics.- Table of cross-references in the Gentzen papers.- Index of names in the Gentzen papers.- Index of subjects in the Gentzen papers.

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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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