Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics are covered.

This book provides a concise introduction to the mathematical foundations of time series analysis, with an emphasis on mathematical clarity. The text is reduced to the essential logical core, mostly using the symbolic language of mathematics, thus enabling readers to very quickly grasp the essential reasoning behind time series analysis. It appeals to anybody wanting to understand time series in a precise, mathematical manner. It is suitable for graduate courses in time series analysis but is equally useful as a reference work for students and researchers alike.

This monograph offers a critical introduction to current theories of how scientific models represent their target systems. Representation is important because it allows scientists to study a model to discover features of reality. The authors provide a map of the conceptual landscape surrounding the issue of scientific representation, arguing that it consists of multiple intertwined problems. They provide an encyclopaedic overview of existing attempts to answer these questions, and they assess their strengths and weaknesses. The book also presents a comprehensive statement of their alternative proposal, the DEKI account of representation, which they have developed over the last few years. They show how the account works in the case of material as well as non-material models; how it accommodates the use of mathematics in scientific modelling; and how it sheds light on the relation between representation in science and art. The issue of representation has generated a sizeable literature, which has been growing fast in particular over the last decade. This makes it hard for novices to get a handle on the topic because so far there is no book-length introduction that would guide them through the discussion. Likewise, researchers may require a comprehensive review that they can refer to for critical evaluations. This book meets the needs of both groups.

This book is an exploration and defense of the coherence of classical theism's doctrine of divine aseity in the face of the challenge posed by Platonism with respect to abstract objects. A synoptic work in analytic philosophy of religion, the book engages discussions in philosophy of mathematics, philosophy of language, metaphysics, and metaontology. It addresses absolute creationism, non-Platonic realism, fictionalism, neutralism, and alternative logics and semantics, among other topics. The book offers a helpful taxonomy of the wide range of options available to the classical theist for dealing with the challenge of Platonism. It probes in detail the diverse views on the reality of abstract objects and their compatibility with classical theism. It contains a most thorough discussion, rooted in careful exegesis, of the biblical and patristic basis of the doctrine of divine aseity. Finally, it challenges the influential Quinean metaontological theses concerning the way in which we make ontological commitments.

The aim of this volume is to collect original contributions by the best specialists from the area of proof theory, constructivity, and computation and discuss recent trends and results in these areas. Some emphasis will be put on ordinal analysis, reductive proof theory, explicit mathematics and type-theoretic formalisms, and abstract computations. The volume is dedicated to the 60th birthday of Professor Gerhard Jager, who has been instrumental in shaping and promoting logic in Switzerland for the last 25 years. It comprises contributions from the symposium "Advances in Proof Theory", which was held in Bern in December 2013. Proof theory came into being in the twenties of the last century, when it was inaugurated by David Hilbert in order to secure the foundations of mathematics. It was substantially influenced by Goedel's famous incompleteness theorems of 1930 and Gentzen's new consistency proof for the axiom system of first order number theory in 1936. Today, proof theory is a well-established branch of mathematical and philosophical logic and one of the pillars of the foundations of mathematics. Proof theory explores constructive and computational aspects of mathematical reasoning; it is particularly suitable for dealing with various questions in computer science.

This edited book presents the state-of-the-art of applying fuzzy logic to managerial decision-making processes in areas such as fuzzy-based portfolio management, recommender systems, performance assessment and risk analysis, among others. Presenting the latest research, with a strong focus on applications and case studies, it is a valuable resource for researchers, practitioners, project leaders and managers wanting to apply or improve their fuzzy-based skills.

This book provides a general survey of the main concepts, questions and results that have been developed in the recent interactions between quantum information, quantum computation and logic. Divided into 10 chapters, the books starts with an introduction of the main concepts of the quantum-theoretic formalism used in quantum information. It then gives a synthetic presentation of the main "mathematical characters" of the quantum computational game: qubits, quregisters, mixtures of quregisters, quantum logical gates. Next, the book investigates the puzzling entanglement-phenomena and logically analyses the Einstein-Podolsky-Rosen paradox and introduces the reader to quantum computational logics, and new forms of quantum logic. The middle chapters investigate the possibility of a quantum computational semantics for a language that can express sentences like "Alice knows that everybody knows that she is pretty", explore the mathematical concept of quantum Turing machine, and illustrate some characteristic examples that arise in the framework of musical languages. The book concludes with an analysis of recent discussions, and contains a Mathematical Appendix which is a survey of the definitions of all main mathematical concepts used in the book.

The contributions in this book survey results on combinations of probabilistic and various other classical, temporal and justification logical systems. Formal languages of these logics are extended with probabilistic operators. The aim is to provide a systematic overview and an accessible presentation of mathematical techniques used to obtain results on formalization, completeness, compactness and decidability. The book will be of value to researchers in logic and it can be used as a supplementary text in graduate courses on non-classical logics.

This book addresses mechanisms for reducing model heterogeneity induced by the absence of explicit semantics expression in the formal techniques used to specify design models. More precisely, it highlights the advances in handling both implicit and explicit semantics in formal system developments, and discusses different contributions expressing different views and perceptions on the implicit and explicit semantics. The book is based on the discussions at the Shonan meeting on this topic held in 2016, and includes contributions from the participants summarising their perspectives on the problem and offering solutions. Divided into 5 parts: domain modelling, knowledge-based modelling, proof-based modelling, assurance cases, and refinement-based modelling, and offers inspiration for researchers and practitioners in the fields of formal methods, system and software engineering, domain knowledge modelling, requirement analysis, and explicit and implicit semantics of modelling languages.

This contributed volume explores the ways logical skills have been perceived over the course of history. The authors approach the topic from the lenses of philosophy, anthropology, sociology, and history to examine two opposing perceptions of logic: the first as an innate human ability and the second as a skill that can be learned and mastered. Chapters focus on the social and political dynamics of the use of logic throughout history, utilizing case studies and critical analyses. Specific topics covered include: the rise of logical skills problems concerning medieval notions of idiocy and rationality decolonizing natural logic natural logic and the course of time Logical Skills: Social-Historical Perspectives will appeal to undergraduate and graduate students, as well as researchers in the fields of history, sociology, philosophy, and logic. Psychology and colonial studies scholars will also find this volume to be of particular interest.

This book has been designed to deal with the topics which are indispensable in the advanced age of computer science. The first three chapters cover mathematical logic, sets, relations and function. Next come the chapters on ordered sets, Boolean albegra and switching circuits and matrices. Finally there are individual chapters on combinatorics, discrete numeric functions, generating functinos, recurrence relations, algebraic structures and graph theory; Graphs are binary trees. The purpose of this book is to present principles and concepts of discrete structures as relevant to student learning. The matter has been presented in as simple and lucid manner as possible and a large number of solved examples to understand the concept and principle of the theory have been introduced.

This volume presents essays by pioneering thinkers including Tyler Burge, Gregory Chaitin, Daniel Dennett, Barry Mazur, Nicholas Humphrey, John Searle and Ian Stewart. Together they illuminate the Map/Territory Distinction that underlies at the foundation of the scientific method, thought and the very reality itself. It is imperative to distinguish Map from the Territory while analyzing any subject but we often mistake map for the territory. Meaning for the Reference. Computational tool for what it computes. Representations are handy and tempting that we often end up committing the category error of over-marrying the representation with what is represented, so much so that the distinction between the former and the latter is lost. This error that has its roots in the pedagogy often generates a plethora of paradoxes/confusions which hinder the proper understanding of the subject. What are wave functions? Fields? Forces? Numbers? Sets? Classes? Operators? Functions? Alphabets and Sentences? Are they a part of our map (theory/representation)? Or do they actually belong to the territory (Reality)? Researcher, like a cartographer, clothes (or creates?) the reality by stitching multitudes of maps that simultaneously co-exist. A simple apple, for example, can be analyzed from several viewpoints beginning with evolution and biology, all the way down its microscopic quantum mechanical components. Is there a reality (or a real apple) out there apart from these maps? How do these various maps interact/intermingle with each other to produce a coherent reality that we interact with? Or do they not? Does our brain uses its own internal maps to facilitate "physicist/mathematician" in us to construct the maps about the external territories in turn? If so, what is the nature of these internal maps? Are there meta-maps? Evolution definitely fences our perception and thereby our ability to construct maps, revealing to us only those aspects beneficial for our survival. But the question is, to what extent? Is there a way out of the metaphorical Platonic cave erected around us by the nature? While "Map is not the territory" as Alfred Korzybski remarked, join us in this journey to know more, while we inquire on the nature and the reality of the maps which try to map the reality out there. The book also includes a foreword by Sir Roger Penrose and an afterword by Dagfinn Follesdal.

This book presents an intuitive picture-oriented approach to the formative processes technique and to its applications. In the first part the authors introduce basic set-theoretic terminology and properties, the decision problem in set theory, and formative processes. The second part of the book is devoted to applications of the technique of formative processes to decision problems. All chapters contain exercises and the book is appropriate for researchers and graduate students in the area of computer science logic.

This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.

This monograph introduces and explores the notions of a commutator equation and the equationally-defined commutator from the perspective of abstract algebraic logic. An account of the commutator operation associated with equational deductive systems is presented, with an emphasis placed on logical aspects of the commutator for equational systems determined by quasivarieties of algebras. The author discusses the general properties of the equationally-defined commutator, various centralization relations for relative congruences, the additivity and correspondence properties of the equationally-defined commutator and its behavior in finitely generated quasivarieties. Presenting new and original research not yet considered in the mathematical literature, The Equationally-Defined Commutator will be of interest to professional algebraists and logicians, as well as graduate students and other researchers interested in problems of modern algebraic logic.

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.

This book is dedicated to the life and work of the mathematician Joachim Lambek (1922-2014). The editors gather together noted experts to discuss the state of the art of various of Lambek's works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist (notably in ring theory). In the 1960s, he began to work in category theory, categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard's linear logic. In a surprising development (2000), he introduced a novel and deeper algebraic framework (which he called pregroup grammars) for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists.

This book offers a multifaceted perspective on fuzzy set theory, discussing its developments over the last 50 years. It reports on all types of fuzzy sets, from ordinary to hesitant fuzzy sets, with each one explained by its own developers, authoritative scientists well known for their previous works. Highlighting recent theorems and proofs, the book also explores how fuzzy set theory has come to be extensively used in almost all branches of science, including the health sciences, decision science, earth science and the social sciences alike. It presents a wealth of real-world sample applications, from routing problem to robotics, and from agriculture to engineering. By offering a comprehensive, timely and detailed portrait of the field, the book represents an excellent reference guide for researchers, lecturers and postgraduate students pursuing research on new fuzzy set extensions.

Alfred Tarski was one of the two giants of the twentieth-century development of logic, along with Kurt Goedel. The four volumes of this collection contain all of Tarski's papers and abstracts published during his lifetime, as well as a comprehensive bibliography. Here will be found many of the works, spanning the period 1921 through 1979, which are the bedrock of contemporary areas of logic, whether in mathematics or philosophy. These areas include the theory of truth in formalized languages, decision methods and undecidable theories, foundations of geometry, set theory, and model theory, algebraic logic, and universal algebra.

This book gathers together selected contributions presented at the 3rd Moroccan Andalusian Meeting on Algebras and their Applications, held in Chefchaouen, Morocco, April 12-14, 2018, and which reflects the mathematical collaboration between south European and north African countries, mainly France, Spain, Morocco, Tunisia and Senegal. The book is divided in three parts and features contributions from the following fields: algebraic and analytic methods in associative and non-associative structures; homological and categorical methods in algebra; and history of mathematics. Covering topics such as rings and algebras, representation theory, number theory, operator algebras, category theory, group theory and information theory, it opens up new avenues of study for graduate students and young researchers. The findings presented also appeal to anyone interested in the fields of algebra and mathematical analysis.

This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Groebner bases) and geometry (via quiver theory). Groebner bases serve as effective models for computation in algebras of various types. Although the theory of Groebner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced - with big impact - in the 1990s. Divided into two parts, the book first discusses the theory of Groebner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Groebner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

This text and software package introduces readers to automated theorem proving, while providing two approaches implemented as easy-to-use programs. These are semantic-tree theorem proving and resolution-refutation theorem proving. The early chapters introduce first-order predicate calculus, well-formed formulae, and their transformation to clauses. Then the author goes on to show how the two methods work and provides numerous examples for readers to try their hand at theorem-proving experiments. Each chapter comes with exercises designed to familiarise the readers with the ideas and with the software, and answers to many of the problems.

This unique collection of research papers offers a comprehensive and up-to-date guide to algebraic approaches to rough sets and reasoning with vagueness. It bridges important gaps, outlines intriguing future research directions, and connects algebraic approaches to rough sets with those for other forms of approximate reasoning. In addition, the book reworks algebraic approaches to axiomatic granularity. Given its scope, the book offers a valuable resource for researchers and teachers in the areas of rough sets and algebras of rough sets, algebraic logic, non classical logic, fuzzy sets, possibility theory, formal concept analysis, computational learning theory, category theory, and other formal approaches to vagueness and approximate reasoning. Consultants in AI and allied fields will also find the book to be of great practical value.