Originally published in 1995 Time and Logic examines understanding and application of temporal logic, presented in computational terms. The emphasis in the book is on presenting a broad range of approaches to computational applications. The techniques used will also be applicable in many cases to formalisms beyond temporal logic alone, and it is hoped that adaptation to many different logics of program will be facilitated. Throughout, the authors have kept implementation-orientated solutions in mind. The book begins with an introduction to the basic ideas of temporal logic. Successive chapters examine particular aspects of the temporal theoretical computing domain, relating their applications to familiar areas of research, such as stochastic process theory, automata theory, established proof systems, model checking, relational logic and classical predicate logic. This is an essential addition to the library of all theoretical computer scientists. It is an authoritative work which will meet the needs both of those familiar with the field and newcomers to it.

The study of lattice sums began when early investigators wanted to go from mechanical properties of crystals to the properties of the atoms and ions from which they were built (the literature of Madelung's constant). A parallel literature was built around the optical properties of regular lattices of atoms (initiated by Lord Rayleigh, Lorentz and Lorenz). For over a century many famous scientists and mathematicians have delved into the properties of lattices, sometimes unwittingly duplicating the work of their predecessors. Here, at last, is a comprehensive overview of the substantial body of knowledge that exists on lattice sums and their applications. The authors also provide commentaries on open questions, and explain modern techniques which simplify the task of finding new results in this fascinating and ongoing field. Lattice sums in one, two, three, four and higher dimensions are covered.

Written by two well-known scholars in the field, Combinatorial Reasoning: An Introduction to the Art of Counting presents a clear and comprehensive introduction to the concepts and methodology of beginning combinatorics. Focusing on modern techniques and applications, the book develops a variety of effective approaches to solving counting problems. Balancing abstract ideas with specific topical coverage, the book utilizes real world examples with problems ranging from basic calculations that are designed to develop fundamental concepts to more challenging exercises that allow for a deeper exploration of complex combinatorial situations. Simple cases are treated first before moving on to general and more advanced cases. Additional features of the book include: Approximately 700 carefully structured problems designed for readers at multiple levels, many with hints and/or short answers Numerous examples that illustrate problem solving using both combinatorial reasoning and sophisticated algorithmic methods A novel approach to the study of recurrence sequences, which simplifies many proofs and calculations Concrete examples and diagrams interspersed throughout to further aid comprehension of abstract concepts A chapter-by-chapter review to clarify the most crucial concepts covered Combinatorial Reasoning: An Introduction to the Art of Counting is an excellent textbook for upper-undergraduate and beginning graduate-level courses on introductory combinatorics and discrete mathematics.

Algebra & Geometry: An Introduction to University Mathematics, Second Edition provides a bridge between high school and undergraduate mathematics courses on algebra and geometry. The author shows students how mathematics is more than a collection of methods by presenting important ideas and their historical origins throughout the text. He incorporates a hands-on approach to proofs and connects algebra and geometry to various applications. The text focuses on linear equations, polynomial equations, and quadratic forms. The first few chapters cover foundational topics, including the importance of proofs and a discussion of the properties commonly encountered when studying algebra. The remaining chapters form the mathematical core of the book. These chapters explain the solutions of different kinds of algebraic equations, the nature of the solutions, and the interplay between geometry and algebra. New to the second edition Several updated chapters, plus an all-new chapter discussing the construction of the real numbers by means of approximations by rational numbers Includes fifteen short 'essays' that are accessible to undergraduate readers, but which direct interested students to more advanced developments of the material Expanded references Contains chapter exercises with solutions provided online at www.routledge.com/9780367563035

The theory of Boolean algebras was created in 1847 by the English mat- matician George Boole. He conceived it as a calculus (or arithmetic) suitable for a mathematical analysis of logic. The form of his calculus was rather di?erent from the modern version, which came into being during the - riod 1864-1895 through the contributions of William Stanley Jevons, Aug- tus De Morgan, Charles Sanders Peirce, and Ernst Schr. oder. A foundation of the calculus as an abstract algebraic discipline, axiomatized by a set of equations, and admitting many di?erent interpretations, was carried out by Edward Huntington in 1904. Only with the work of Marshall Stone and Alfred Tarski in the 1930s, however, did Boolean algebra free itself completely from the bonds of logic and become a modern mathematical discipline, with deep theorems and - portantconnections toseveral otherbranchesofmathematics, includingal- bra,analysis, logic, measuretheory, probability andstatistics, settheory, and topology. For instance, in logic, beyond its close connection to propositional logic, Boolean algebra has found applications in such diverse areas as the proof of the completeness theorem for ?rst-order logic, the proof of the Lo ' s conjecture for countable ? rst-order theories categorical in power, and proofs of the independence of the axiom of choice and the continuum hypothesis ? in set theory. In analysis, Stone's discoveries of the Stone-Cech compac- ?cation and the Stone-Weierstrass approximation theorem were intimately connected to his study of Boolean algebras.

This book offers a defense against non-classical approaches to the paradoxes. The author argues that, despite appearances, the paradoxes give no reason at all to reject classical logic. In fact, he believes classical solutions fare better than non-classical ones with respect to key tests like Curry's Paradox, a Liar-like paradox that dialetheists are forced to solve in a way totally disjoint from their solution to the Liar. Graham Priest's In Contradiction was the first major work that advocated the use of non-classical approaches. Since then, these views have moved into the philosophical mainstream. Much of this movement is fueled by a widespread sense that these logically heterodox solutions get to the real nub of the issue. They lack the ad hoc feel of many other solutions to the paradoxes. The author believes that it's long past time for a response to these attacks against classical orthodoxy. He presents a non-logically-revisionary solution to the paradoxes. This title offers a literal way of cashing out the disquotation metaphor. While the details of the view are novel, the idea has a pre-history in the relevant literature. The author examines objections in detail. He rejects each in turn and concludes by comparing the virtues of his logically orthodox approach with those of the paraconsistent and paracomplete competition.

This collection brings together exciting new works that address today's key challenges for a feminist power-sensitive approach to knowledge and scientific practice. Taking up such issues as the role of contextualism in epistemology, democracy and dissent in knowledge practices, and epistemic agency under conditions of oppression, the essays build upon well-established work in feminist epistemology and philosophy of science such as standpoint theory and contextual empiricism, offering new interpretations and applications. Many contributions capture the current engagement of feminist epistemologists with the insights and programs of nonfeminist epistemologists, while others focus on the intersections between feminist epistemology and other fields of feminist inquiry such as feminist ethics and metaphysics. *see remarks below for remainder of text*

Originally published in 1967. An introduction to the literature of nonstandard logic, in particular to those nonstandard logics known as many-valued logics. Part I expounds and discusses implicational calculi, modal logics and many-valued logics and their associated calculi. Part II considers the detailed development of various many-valued calculi, and some of the important metathereoms which have been proved for them. Applications of the calculi to problems in the philosophy are also surveyed. This work combines criticism with exposition to form a comprehensive but concise survey of the field.

Originally published in 1966. An introduction to current studies of kinds of inference in which validity cannot be determined by ordinary deductive models. In particular, inductive inference, predictive inference, statistical inference, and decision making are examined in some detail. The last chapter discusses the relationship of these forms of inference to philosophical notions of rationality. Special features of the monograph include a discussion of the legitimacy of various criteria for successful predictive inference, the development of an intuitive model which exhibits the difficulties of choosing probability measures over infinite sets, and a comparison of rival views on the foundations of probability in terms of the amount of information which the members of these schools believe suitable for fruitful formalization. The bibliographies include articles by statisticians accessible to students of symbolic logic.

This book addresses the argument in the history of the philosophy of science between the positivists and the anti-positivists. The author starts from a point of firm conviction that all science and philosophy must start with the given... But that the range of the given is not definite. He begins with an examination of science from the outside and then the inside, explaining his position on metaphysics and attempts to formulate the character of operational acts before a general theory of symbolism is explored. The last five chapters constitute a treatise to show that the development from one stage of symbolismto the next is inevitable, consequently that explanatory science represents the culmination of knowledge.

Originally published in 1964. This book is concerned with general arguments, by which is meant broadly arguments that rely for their force on the ideas expressed by all, every, any, some, none and other kindred words or phrases. A main object of quantificational logic is to provide methods for evaluating general arguments. To evaluate a general argument by these methods we must first express it in a standard form. Quantificational form is dealt with in chapter one and in part of chapter three; in the remainder of the book an account is given of methods by which arguments when formulated quantificationally may be tested for validity or invalidity. Some attention is also paid to the logic of identity and of definite descriptions. Throughout the book an attempt has been made to give a clear explanation of the concepts involved and the symbols used; in particular a step-by-step and partly mechanical method is developed for translating complicated statements of ordinary discourse into the appropriate quantificational formulae. Some elementary knowledge of truth-functional logic is presupposed.

Introduction to the Theory of Optimization in Euclidean Space is intended to provide students with a robust introduction to optimization in Euclidean space, demonstrating the theoretical aspects of the subject whilst also providing clear proofs and applications. Students are taken progressively through the development of the proofs, where they have the occasion to practice tools of differentiation (Chain rule, Taylor formula) for functions of several variables in abstract situations. Throughout this book, students will learn the necessity of referring to important results established in advanced Algebra and Analysis courses. Features Rigorous and practical, offering proofs and applications of theorems Suitable as a textbook for advanced undergraduate students on mathematics or economics courses, or as reference for graduate-level readers Introduces complex principles in a clear, illustrative fashion

This book is an introduction to a functorial model theory based on infinitary language categories. The author introduces the properties and foundation of these categories before developing a model theory for functors starting with a countable fragment of an infinitary language. He also presents a new technique for generating generic models with categories by inventing infinite language categories and functorial model theory. In addition, the book covers string models, limit models, and functorial models.

This book provides a detailed exposition of one of the most practical and popular methods of proving theorems in logic, called Natural Deduction. It is presented both historically and systematically. Also some combinations with other known proof methods are explored. The initial part of the book deals with Classical Logic, whereas the rest is concerned with systems for several forms of Modal Logics, one of the most important branches of modern logic, which has wide applicability.

This book contains fundamental concepts on discrete mathematical structures in an easy to understand style so that the reader can grasp the contents and explanation easily. The concepts of discrete mathematical structures have application to computer science, engineering and information technology including in coding techniques, switching circuits, pointers and linked allocation, error corrections, as well as in data networking, Chemistry, Biology and many other scientific areas. The book is for undergraduate and graduate levels learners and educators associated with various courses and progammes in Mathematics, Computer Science, Engineering and Information Technology. The book should serve as a text and reference guide to many undergraduate and graduate programmes offered by many institutions including colleges and universities. Readers will find solved examples and end of chapter exercises to enhance reader comprehension. Features Offers comprehensive coverage of basic ideas of Logic, Mathematical Induction, Graph Theory, Algebraic Structures and Lattices and Boolean Algebra Provides end of chapter solved examples and practice problems Delivers materials on valid arguments and rules of inference with illustrations Focuses on algebraic structures to enable the reader to work with discrete structures

Originally published in 1962. This book gives an account of the concepts and methods of a basic part of logic. In chapter I elementary ideas, including those of truth-functional argument and truth-functional validity, are explained. Chapter II begins with a more comprehensive account of truth-functionality; the leading characteristics of the most important monadic and dyadic truth-functions are described, and the different notations in use are set forth. The main part of the book describes and explains three different methods of testing truth-functional aguments and agument forms for validity: the truthtable method, the deductive method and the method of normal forms; for the benefit mainly of readers who have not acquired in one way or another a general facility in the manipulation of symbols some of the procedures have been described in rather more detail than is common in texts of this kind. In the final chapter the author discusses and rejects the view, based largely on the so called paradoxes of material implication, that truth-functional logic is not applicable in any really important way to arguments of ordinary discourse.

Physically Unclonable Functions (PUFs) translate unavoidable variations in certain parameters of materials, waves, or devices into random and unique signals. They have found many applications in the Internet of Things (IoT), authentication systems, FPGA industry, several other areas in communications and related technologies, and many commercial products. Statistical Trend Analysis of Physically Unclonable Functions first presents a review on cryptographic hardware and hardware-assisted cryptography. The review highlights PUF as a mega trend in research on cryptographic hardware design. Afterwards, the authors present a combined survey and research work on PUFs using a systematic approach. As part of the survey aspect, a state-of-the-art analysis is presented as well as a taxonomy on PUFs, a life cycle, and an established ecosystem for the technology. In another part of the survey, the evolutionary history of PUFs is examined, and strategies for further research in this area are suggested. In the research side, this book presents a novel approach for trend analysis that can be applied to any technology or research area. In this method, a text mining tool is used which extracts 1020 keywords from the titles of the sample papers. Then, a classifying tool classifies the keywords into 295 meaningful research topics. The popularity of each topic is then numerically measured and analyzed over the course of time through a statistical analysis on the number of research papers related to the topic as well as the number of their citations. The authors identify the most popular topics in four different domains; over the history of PUFs, during the recent years, in top conferences, and in top journals. The results are used to present an evolution study as well as a trend analysis and develop a roadmap for future research in this area. This method gives an automatic popularity-based statistical trend analysis which eliminates the need for passing personal judgments about the direction of trends, and provides concrete evidence to the future direction of research on PUFs. Another advantage of this method is the possibility of studying a whole lot of existing research works (more than 700 in this book). This book will appeal to researchers in text mining, cryptography, hardware security, and IoT.

Emphasizes the computer science aspects of the subject.

Details applications in databases, complexity theory, and formal languages, as well as other branches of computer science.

This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham's theorem on simplicial complexes. In addition, Sullivan's results on computing the rational homotopy type from forms is presented. New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma *Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theory With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.

Features Over sixty paper stars, all made without cutting, gluing or decorating using the modular origami technique Hundreds of clear step-by-step instructions show you how, based on the technique of folding a small number of simple units and joining them together as a satisfying puzzle Secrets tips to make new shapes just by varying a few lengths and angles Suitable for teaching and learning art, geometry and mathematics. Teachers will appreciate the practical advice to succeed in using origami for education.

Originally published in 1973. This book presents a valid mode of reasoning that is different to mathematical probability. This inductive logic is investigated in terms of scientific investigation. The author presents his criteria of adequacy for analysing inductive support for hypotheses and discusses each of these criteria in depth. The chapters cover philosophical problems and paradoxes about experimental support, probability and justifiability, ending with a system of logical syntax of induction. Each section begins with a summary of its contents and there is a glossary of technical terms to aid the reader.

Blockchain is a technology that has attracted the attention of all types of businesses. Cryptocurrency such as Bitcoin has gained the most attention, but now companies are applying Blockchain technology to develop solutions improving traditional applications and securing all types of transactions. Robust and innovative, this technology is being combined with other well-known technologies including Cloud Computing, Big Data, and IoT to revolutionize outcomes in all verticals. Unlike books focused on financial applications, Essential Enterprise Blockchain Concepts and Applications is for researchers and practitioners who are looking for secure, viable, low-cost, and workable applications to solve a broad range of business problems. The book presents research that rethinks how to incorporate Blockchain with existing technology. Chapters cover various applications based on Blockchain technology including: Digital voting Smart contracts Supply chain management Internet security Logistics management Identity management Securing medical devices Asset management Blockchain plays a significant role in providing security for data operations. It defines how trusted transactions can be carried out and addresses Internet vulnerability problems. Blockchain solves the security fault line between AI and IoT in smart systems as well as in other systems using devices connected to each other through public networks. Linear and permanent indexed records are maintained by Blockchain to face the vulnerability issues in a wide variety applications. In addition to applications, the book also covers consensus algorithms and protocols and performance of Blockchain algorithms.

Logicism, as put forward by Bertrand Russell, was predicated on a belief that all of mathematics can be deduced from a very small number of fundamental logical principles. In Logicism Renewed, the author revisits this concept in light of advances in mathematical logic and the need for languages that can be understood by both humans and computers that require distinguishing between the intension and extension of predicates. Using Intensional Type Theory (ITT) the author provides a unified foundation for mathematics and computer science, yielding a much simpler foundation for recursion theory and the semantics of computer programs than that currently provided by category theory.

The aim of this book is to present the fundamental theoretical results concerning inference rules in deductive formal systems. Primary attention is focused on:

- admissible or permissible inference rules

- the derivability of the admissible inference rules

- the structural completeness of logics

- the bases for admissible and valid inference rules.

There is particular emphasis on propositional non-standard logics (primary, superintuitionistic and modal logics) but general logical consequence relations and classical first-order theories are also considered.

The book is basically self-contained and special attention has been made to present the material in a convenient manner for the reader. Proofs of results, many of which are not readily available elsewhere, are also included.

The book is written at a level appropriate for first-year graduate students in mathematics or computer science. Although some knowledge of elementary logic and universal algebra are necessary, the first chapter includes all the results from universal algebra and logic that the reader needs. For graduate students in mathematics and computer science the book is an excellent textbook.

We do not perceive the present as it is and in totality, nor do we infer the future from the present with any high degree of dependability, nor yet do we accurately know the consequences of our own actions. In addition, there is a fourth source of error to be taken into account, for we do not execute actions in the precise form in which they are imaged and willed. Frank H. Knight [R4.34, p. 202] The "degree" of certainty of confidence felt in the conclusion after it is reached cannot be ignored, for it is of the greatest practical signi- cance. The action which follows upon an opinion depends as much upon the amount of confidence in that opinion as it does upon fav- ableness of the opinion itself. The ultimate logic, or psychology, of these deliberations is obscure, a part of the scientifically unfathomable mystery of life and mind. Frank H. Knight [R4.34, p. 226-227] With some inaccuracy, description of uncertain consequences can be classified into two categories, those which use exclusively the language of probability distributions and those which call for some other principle, either to replace or supplement.