Information is precious. It reduces our uncertainty in making decisions. Knowledge about the outcome of an uncertain event gives the possessor an advantage. It changes the course of lives, nations, and history itself. Information is the food of Maxwell's demon. His power comes from know ing which particles are hot and which particles are cold. His existence was paradoxical to classical physics and only the realization that information too was a source of power led to his taming. Information has recently become a commodity, traded and sold like or ange juice or hog bellies. Colleges give degrees in information science and information management. Technology of the computer age has provided access to information in overwhelming quantity. Information has become something worth studying in its own right. The purpose of this volume is to introduce key developments and results in the area of generalized information theory, a theory that deals with uncertainty-based information within mathematical frameworks that are broader than classical set theory and probability theory. The volume is organized as follows."

This book presents a collection of contributions from related logics to applied paraconsistency. Moreover, all of them are dedicated to Jair Minoro Abe,on the occasion of his sixtieth birthday. He is one of the experts in Paraconsistent Engineering, who developed the so-called annotated logics. The book includes important contributions on foundations and applications of paraconsistent logics in connection with engineering, mathematical logic, philosophical logic, computer science, physics, economics, and biology. It will be of interest to students and researchers, who are working on engineering and logic.

This is the first book presenting cardinality theory of fuzzy sets with triangular norms, including its scalar and "fuzzy" streams. This theory constitutes not only a powerful basis but also a useful tool for modelling and processing vague and imprecise quantitative information. The multiple application areas of the theory encompass computer science, soft computing, computing with words, and decision-making. Starting with a presentation of the fundamentals of triangular norms and fuzzy set theory, the book offers a self-contained, concise and systematic exposition of cardinalities of fuzzy sets that includes many examples.

Many philosophers have considered logical reasoning as an inborn ability of mankind and as a distinctive feature in the human mind; but we all know that the distribution of this capacity, or at any rate its development, is very unequal. Few people are able to set up a cogent argument; others are at least able to follow a logical argument and even to detect logical fallacies. Nevertheless, even among educated persons there are many who do not even attain this relatively modest level of development. According to my personal observations, lack of logical ability may be due to various circumstances. In the first place, I mention lack of general intelligence, insufficient power of concentration, and absence of formal education. Secondly, however, I have noticed that many people are unable, or sometimes rather unwilling, to argue ex hypothesi; such persons cannot, or will not, start from premisses which they know or believe to be false or even from premisses whose truth is not, in their opinion, sufficient ly warranted. Or, if they agree to start from such premisses, they sooner or later stray away from the argument into attempts first to settle the truth or falsehood of the premisses. Presumably this attitude results either from lack of imagination or from undue moral rectitude. On the other hand, proficiency in logical reasoning is not in itself a guarantee for a clear theoretic insight into the principles and foundations of logic."

This book features a unique approach to the teaching of mathematical logic by putting it in the context of the puzzles and paradoxes of common language and rational thought. It serves as a bridge from the author 's puzzle books to his technical writing in the fascinating field of mathematical logic. Using the logic of lying and truth-telling, the author introduces the readers to informal reasoning preparing them for the formal study of symbolic logic, from propositional logic to first-order logic, a subject that has many important applications to philosophy, mathematics, and computer science. The book includes a journey through the amazing labyrinths of infinity, which have stirred the imagination of mankind as much, if not more, than any other subject.

This book is concerned with advances in serial-data computa tional architectures, and the CAD tools for their implementation in silicon. The bit-serial tradition at Edinburgh University (EU) stretches back some 6 years to the conception of the FIRST silicon compiler. FIRST owes much of its inspiration to Dick Lyon, then at Xerox P ARC, who proposed a 'structured-design' methodology for construction of signal processing systems from bit-serial building blocks. Based on an nMOS cell-library, FIRST automates much of Lyon's physical design process. More recently, we began to feel that FIRST should be able to exploit more modern technologies. Before this could be achieved, we were faced with a massive manual re-design task, i. e. the porting of FIRST cell-library to a new technology. As it was to avoid such tasks that FIRST was conceived in the first place, we decided to move the level of user-specification much nearer to the silicon level (while still hiding details of transistor circuit design, place and route etc., from the user), and by so doing, enable the specification of more functionally powerful libraries in technology-free form. The results of this work are in evidence as advances in serial-data design techniques, and the SECOND silicon compiler, introduced later in this book. These achievements could not have been accomplished without help from various sources. We take this opportunity to thank Profs."

The main idea of statistical convergence is to demand convergence only for a majority of elements of a sequence. This method of convergence has been investigated in many fundamental areas of mathematics such as: measure theory, approximation theory, fuzzy logic theory, summability theory, and so on. In this monograph we consider this concept in approximating a function by linear operators, especially when the classical limit fails. The results of this book not only cover the classical and statistical approximation theory, but also are applied in the fuzzy logic via the fuzzy-valued operators. The authors in particular treat the important Korovkin approximation theory of positive linear operators in statistical and fuzzy sense. They also present various statistical approximation theorems for some specific real and complex-valued linear operators that are not positive. This is the first monograph in Statistical Approximation Theory and Fuzziness. The chapters are self-contained and several advanced courses can be taught. The research findings will be useful in various applications including applied and computational mathematics, stochastics, engineering, artificial intelligence, vision and machine learning. This monograph is directed to graduate students, researchers, practitioners and professors of all disciplines.

Towards the end of the nineteenth century, Frege gave us the abstraction principles and the general notion of functions. Self-application of functions was at the heart of Russell's paradox. This led Russell to introduce type theory in order to avoid the paradox. Since, the twentieth century has seen an amazing number of theories concerned with types and functions and many applications. Progress in computer science also meant more and more emphasis on the use of logic, types and functions to study the syntax, semantics, design and implementation of programming languages and theorem provers, and the correctness of proofs and programs. The authors of this book have themselves been leading the way by providing various extensions of type theory which have been shown to bring many advantages. This book gathers much of their influential work and is highly recommended for anyone interested in type theory. The main emphasis is on:
- Types: from Russell to Ramsey, to Church, to the modern Pure Type Systems and some of their extensions.
- Functions: from Frege, to Russell to Church, to Automath and the use of functions in mathematics, programming languages and theorem provers.
- The role of types in logic: Kripke's notion of truth, the evolution and role of the propositions as types concept and its use in logical frameworks.
- The role of types in computation: extensions of type theories which can better model proof checkers and programming languages are given.
The first part of the book is historical, yet at the same time, places historical systems (like Russell's RTT) in the modern setting. The second part deals with modern type theory as it developed since the 1940s, andwith the role of propositions as types (or proofs as terms), but at the same time, places another historical system (the proof checker Automath) in the modern setting. The third part uses this bridging in the first two parts between historical and modern systems to propose new systems that bring more advantages together. This book has much to offer to mathematicians, logicians and to computer scientists in general. It will have considerable influence for many years to come.' - Henk Barendregt

Mathematics is often considered as a body of knowledge that is essen tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Mathe matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The language dependency of verisimilitude; 3) The proof of the weak and strong anti inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The language dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposi tions and theories."

This is a comprehensive introduction into the method of inverse spectra - a powerful method successfully employed in various branches of topology.

The notion of an inverse sequence and its limits, first appeared in the well-known memoir by Alexandrov where a special case of inverse spectra - the so-called projective spectra - were considered. The concept of an inverse spectrum in its present form was first introduced by Lefschetz. Meanwhile, Freudental, had introduced the notion of a morphism of inverse spectra. The foundations of the entire method of inverse spectra were laid down in these basic works.

Subsequently, inverse spectra began to be widely studied and applied, not only in the various major branches of topology, but also in functional analysis and algebra. This is not surprising considering the categorical nature of inverse spectra and the extraordinary power of the related techniques.

Updated surveys (including proofs of several statements) of the Hilbert cube and Hilbert space manifold theories are included in the book. Recent developments of the Menger and Nobeling manifold theories are also presented.

This work significantly extends and updates the author's previously published book and has been completely rewritten in order to incorporate new developments in the field.

Constructive mathematics – mathematics in which 'there exists' always means 'we can construct' – is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches.

This book gives a thorough and self contained presentation of H, its known isomorphic invariants and a complete classification of H on spaces of homogeneous type. The necessary background is developed from scratch. This includes a detailed discussion of the Haar system, together with the operators that can be built from it. Complete proofs are given for the classical martingale inequalities, and for large deviation inequalities. Complex interpolation is treated. Througout, special attention is given to the combinatorial methods developed in the field. An entire chapter is devoted to study the combinatorics of coloured dyadic Intervals.

The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES."

The work of which this is an English translation appeared originally in French as Precis de logique mathematique. In 1954 Dr. Albert Menne brought out a revised and somewhat enlarged edition in German (Grund riss der Logistik, F. Schoningh, Paderborn). In making my translation I have used both editions. For the most part I have followed the original French edition, since I thought there was some advantage in keeping the work as short as possible. However, I have included the more extensive historical notes of Dr. Menne, his bibliography, and the two sections on modal logic and the syntactical categories ( 25 and 27), which were not in the original. I have endeavored to correct the typo graphical errors that appeared in the original editions and have made a few additions to the bibliography. In making the translation I have profited more than words can tell from the ever-generous help of Fr. Bochenski while he was teaching at the University of Notre Dame during 1955-56. OTTO BIRD Notre Dame, 1959 I GENERAL PRINCIPLES O. INTRODUCTION 0. 1. Notion and history. Mathematical logic, also called 'logistic', .symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last century with the aid of an artificial notation and a rigorously deductive method."

Absolute Space, Absolute Time, and Absolute Motion exist. These are shown to be facts through an investigation of the nature of infinitesimals. Knowledge of that nature also makes the irrational magnitudes within the unit comprehensible. The number line is shown to be cognitively superior to set theory; furthermore, non-Euclidean geometry is shown to be a mere manipulation of symbols and not an expression of a "parallel universe." Inside, the reader will also learn about a hitherto unknown number system locked within _-1. He will also discover in the infinitesimal calculus a hidden key to a level of reality beneath that of nano-technology.. The foundation of science is not some vague generality, but the exercise of reason as originating from the human sensorium. There is no difference between mathematical and ordinary inductive reasoning.

In this book the authors present new results on interpolation for nonmonotonic logics, abstract (function) independence, the Talmudic Kal Vachomer rule, and an equational solution of contrary-to-duty obligations. The chapter on formal construction is the conceptual core of the book, where the authors combine the ideas of several types of nonmonotonic logics and their analysis of 'natural' concepts into a formal logic, a special preferential construction that combines formal clarity with the intuitive advantages of Reiter defaults, defeasible inheritance, theory revision, and epistemic considerations. It is suitable for researchers in the area of computer science and mathematical logic.

Propositional Logics presents the history, philosophy, and mathematics of the major systems of propositional logic. Classical logic, modal logics, many-valued logics, intuitionism, paraconsistent logics, and dependent implication are examined in separate chapters. Each begins with a motivation in the originators' own terms, followed by the standard formal semantics, syntax, and completeness theorem. The chapters on the various logics are largely self-contained so that the book can be used as a reference. An appendix summarizes the formal semantics and axiomatizations of the logics. The view that unifies the exposition is that propositional logics comprise a spectrum. As the aspect of propositions under consideration varies, the logic varies. Each logic is shown to fall naturally within a general framework for semantics. A theory of translations between logics is presented that allows for further comparisons, and necessary conditions are given for a translation to preserve meaning. For this third edition the material has been re-organized to make the text easier to study, and a new section on paraconsistent logics with simple semantics has been added which challenges standard views on the nature of consequence relations. The text includes worked examples and hundreds of exercises, from routine to open problems, making the book with its clear and careful exposition ideal for courses or individual study.

This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. It aims to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named 'fuzzy inference' can be naturally understood as logical deduction. It is for mathematicians, logicians, computer scientists, specialists in artificial intelligence and knowledge engineering, and developers of fuzzy logic.

This volume, the 7th volume in the DRUMS Handbook series, is part of the aftermath of the successful ESPRIT project DRUMS (Defeasible Reasoning and Uncertainty Management Systems) which took place in two stages from 1989- 1996. In the second stage (1993-1996) a work package was introduced devoted to the topics Reasoning and Dynamics, covering both the topics of "Dynamics of Reasoning," where reasoning is viewed as a process, and "Reasoning about Dynamics," which must be understood as pertaining to how both designers of and agents within dynamic systems may reason about these systems. The present volume presents work done in this context extended with some work done by outstanding researchers outside the project on related issues. While the previous volume in this series had its focus on the dynamics of reasoning pro cesses, the present volume is more focused on "reasoning about dynamics', viz. how (human and artificial) agents reason about (systems in) dynamic environments in order to control them. In particular we consider modelling frameworks and generic agent models for modelling these dynamic systems and formal approaches to these systems such as logics for agents and formal means to reason about agent based and compositional systems, and action & change more in general. We take this opportunity to mention that we have very pleasant recollections of the project, with its lively workshops and other meetings, with the many sites and researchers involved, both within and outside our own work package."