This book discusses the theory of triangular norms and surveys several applied fields in which triangular norms play a significant part: probabilistic metric spaces, aggregation operators, many-valued logics, fuzzy logics, sets and control, and non-additive measures together with their corresponding integrals. It includes many graphical illustrations and gives a well-balanced picture of theory and applications. It is for mathematicians, computer scientists, applied computer scientists and engineers.

We welcome Volume 20, Formal Aspects of Context. Context has always been recognised as strongly relevant to models in language, philosophy, logic and artifi cial intelligence. In recent years theoretical advances in these areas and especially in logic have accelerated the study of context in the international community. An annual conference is held and many researchers have come to realise that many of the old puzzles should be reconsidered with proper attention to context. The volume editors and contributors are from among the most active front-line researchers in the area and the contents shows how wide and vigorous this area is. There are strong scientific connections with earlier volumes in the series. I am confident that the appearance of this book in our series will help secure the study of context as an important area of applied logic. D.M.Gabbay INTRODUCTION This book is a result of the First International and Interdisciplinary Con ference on Modelling and Using Context, which was organised in Rio de Janeiro in January 1997, and contains a selection of the papers presented there, refereed and revised through a process of anonymous peer review. The treatment of contexts as bona-fide objects of logical formalisation has gained wide acceptance in recent years, following the seminal impetus by McCarthy in his 'lUring award address."

We are happy to present the second volume of the Handbook of Defeasible Reasoning and Uncertainty Management Systems. Uncertainty pervades the real world and must therefore be addressed by every system that attempts to represent reality. The representation of un certainty is a major concern of philosophers, logicians, artificial intelligence researchers and computer sciencists, psychologists, statisticians, economists and engineers. The present Handbook volumes provide frontline coverage of this area. This Handbook was produced in the style of previous handbook series like the Handbook of Philosophical Logic, the Handbook of Logic in Computer Science, the Handbook of Logic in Artificial Intelligence and Logic Programming, and can be seen as a companion to them in covering the wide applications of logic and reasoning. We hope it will answer the needs for adequate representations of uncertainty. This Handbook series grew out of the ESPRIT Basic Research Project DRUMS II, where the acronym is made out of the Handbook series title. This project was financially supported by the European Union and regroups 20 major European research teams working in the general domain of uncer tainty. As a fringe benefit of the DRUMS project, the research community was able to create this Handbook series, relying on the DRUMS partici pants as the core of the authors for the Handbook together with external international experts."

Multi-Criteria Decision Making (MCDM) has been one of the fastest growing problem areas in many disciplines. The central problem is how to evaluate a set of alternatives in terms of a number of criteria. Although this problem is very relevant in practice, there are few methods available and their quality is hard to determine. Thus, the question Which is the best method for a given problem?' has become one of the most important and challenging ones. This is exactly what this book has as its focus and why it is important. The author extensively compares, both theoretically and empirically, real-life MCDM issues and makes the reader aware of quite a number of surprising abnormalities' with some of these methods. What makes this book so valuable and different is that even though the analyses are rigorous, the results can be understood even by the non-specialist. Audience: Researchers, practitioners, and students; it can be used as a textbook for senior undergraduate or graduate courses in business and engineering.

Fixed point theory in probabilistic metric spaces can be considered as a part of Probabilistic Analysis, which is a very dynamic area of mathematical research. A primary aim of this monograph is to stimulate interest among scientists and students in this fascinating field. The text is self-contained for a reader with a modest knowledge of the metric fixed point theory. Several themes run through this book. The first is the theory of triangular norms (t-norms), which is closely related to fixed point theory in probabilistic metric spaces. Its recent development has had a strong influence upon the fixed point theory in probabilistic metric spaces. In Chapter 1 some basic properties of t-norms are presented and several special classes of t-norms are investigated. Chapter 2 is an overview of some basic definitions and examples from the theory of probabilistic metric spaces. Chapters 3, 4, and 5 deal with some single-valued and multi-valued probabilistic versions of the Banach contraction principle. In Chapter 6, some basic results in locally convex topological vector spaces are used and applied to fixed point theory in vector spaces. Audience: The book will be of value to graduate students, researchers, and applied mathematicians working in nonlinear analysis and probabilistic metric spaces.

This is the first of two volumes comprising the papers submitted for publication by the invited participants to the Tenth International Congress of Logic, Methodology and Philosophy of Science, held in Florence, August 1995. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science. The invited lectures published in the two volumes demonstrate much of what goes on in the fields of the Congress and give the state of the art of current research. The two volumes cover the traditional subdisciplines of mathematical logic and philosophical logic, as well as their interfaces with computer science, linguistics and philosophy. Philosophy of science is broadly represented, too, including general issues of natural sciences, social sciences and humanities. The papers in Volume One are concerned with logic, mathematical logic, the philosophy of logic and mathematics, and computer science.

The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory.
The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment.
The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.

This book, in some sense, began to be written by the first author in 1983, when optional lectures on Abelian groups were held at the Fac ulty of Mathematics and Computer Science, 'Babes-Bolyai' University in Cluj-Napoca, Romania. From 1992, these lectures were extended to a twosemester electivecourse on abelian groups for undergraduate stu dents, followed by a twosemester course on the same topic for graduate students in Algebra. All the other authors attended these two years of lectures and are now Assistants to the Chair of Algebra of this Fac ulty. The first draft of this collection, including only exercises solved by students as home works, the last ten years, had 160pages. We felt that there is a need for a book such as this one, because it would provide a nice bridge between introductory Abelian Group Theory and more advanced research problems. The book InfiniteAbelianGroups, published by LaszloFuchsin two volumes 1970 and 1973 willwithout doubt last as the most important guide for abelian group theorists. Many exercises are selected from this source but there are plenty of other bibliographical items (see the Bibliography) which were used in order to make up this collection. For some of the problems stated, recent developments are also given. Nevertheless, there are plenty of elementary results (the so called 'folklore') in Abelian Group Theory whichdo not appear in any written material. It is also one purpose of this book to complete this gap."

This monograph is a continuation of several themes presented in my previous books [146, 149]. In those volumes, I was concerned primarily with the properties of semirings. Here, the objects of investigation are sets of the form RA, where R is a semiring and A is a set having a certain structure. The problem is one of translating that structure to RA in some "natural" way. As such, it tries to find a unified way of dealing with diverse topics in mathematics and theoretical com puter science as formal language theory, the theory of fuzzy algebraic structures, models of optimal control, and many others. Another special case is the creation of "idempotent analysis" and similar work in optimization theory. Unlike the case of the previous work, which rested on a fairly established mathematical foundation, the approach here is much more tentative and docimastic. This is an introduction to, not a definitative presentation of, an area of mathematics still very much in the making. The basic philosphical problem lurking in the background is one stated suc cinctly by Hahle and Sostak [185]: ". . . to what extent basic fields of mathematics like algebra and topology are dependent on the underlying set theory?" The conflicting definitions proposed by various researchers in search of a resolution to this conundrum show just how difficult this problem is to see in a proper light.

The nationwide research project Deduktion', funded by the Deutsche Forschungsgemeinschaft (DFG)' for a period of six years, brought together almost all research groups within Germany engaged in the field of automated reasoning. Intensive cooperation and exchange of ideas led to considerable progress both in the theoretical foundations and in the application of deductive knowledge. This three-volume book covers these original contributions moulded into the state of the art of automated deduction. The three volumes are intended to document and advance a development in the field of automated deduction that can now be observed all over the world. Rather than restricting the interest to purely academic research, the focus now is on the investigation of problems derived from realistic applications. In fact industrial applications are already pursued on a trial basis. In consequence the emphasis of the volumes is not on the presentation of the theoretical foundations of logical deduction as such, as in a handbook; rather the books present the concepts and methods now available in automated deduction in a form which can be easily accessed by scientists working in applications outside of the field of deduction. This reflects the strong conviction that automated deduction is on the verge of being fully included in the evolution of technology. Volume I focuses on basic research in deduction and on the knowledge on which modern deductive systems are based. Volume II presents techniques of implementation and details about system building. Volume III deals with applications of deductive techniques mainly, but not exclusively, to mathematics and the verification of software. Each chapter was read by two referees, one an international expert from abroad and the other a knowledgeable participant in the national project. It has been accepted for inclusion on the basis of these review reports. Audience: Researchers and developers in software engineering, formal methods, certification, verification, validation, specification of complex systems and software, expert systems, natural language processing.

Fuzzy Control of Industrial Systems: Theory and Applications presents the basic theoretical framework of crisp and fuzzy set theory, relating these concepts to control engineering based on the analogy between the Laplace transfer function of linear systems and the fuzzy relation of a nonlinear fuzzy system. Included are generic aspects of fuzzy systems with an emphasis on the many degrees of freedom and its practical design implications, modeling and systems identification techniques based on fuzzy rules, parametrized rules and relational equations, and the principles of adaptive fuzzy and neurofuzzy systems. Practical design aspects of fuzzy controllers are covered by the detailed treatment of fuzzy and neurofuzzy software design tools with an emphasis on iterative fuzzy tuning, while novel stability limit testing methods and the definition and practical examples of the new concept of collaborative control systems are also given. In addition, case studies of successful applications in industrial automation, process control, electric power technology, electric traction, traffic engineering, wastewater treatment, manufacturing, mineral processing and automotive engineering are also presented, in order to assist industrial control systems engineers in recognizing situations when fuzzy and neurofuzzy would offer certain advantages over traditional methods, particularly in controlling highly nonlinear and time-variant plants and processes.

This book is based on lectures delivered at Harvard in the Spring of 1991 and at the University of Utah during the academic year 1992-93. Formally, the book assumes only general algebraic knowledge (rings, modules, groups, Lie algebras, functors etc.). It is helpful, however, to know some basics of algebraic geometry and representation theory. Each chapter begins with its own introduction, and most sections even have a short overview. The purpose of what follows is to explain the spirit of the book and how different parts are linked together without entering into details. The point of departure is the notion of the left spectrum of an associative ring, and the first natural steps of general theory of noncommutative affine, quasi-affine, and projective schemes. This material is presented in Chapter I. Further developments originated from the requirements of several important examples I tried to understand, to begin with the first Weyl algebra and the quantum plane. The book reflects these developments as I worked them out in reallife and in my lectures. In Chapter 11, we study the left spectrum and irreducible representations of a whole lot of rings which are of interest for modern mathematical physics. The dasses of rings we consider indude as special cases: quantum plane, algebra of q-differential operators, (quantum) Heisenberg and Weyl algebras, (quantum) enveloping algebra ofthe Lie algebra sl(2) , coordinate algebra of the quantum group SL(2), the twisted SL(2) of Woronowicz, so called dispin algebra and many others.

without a properly developed inconsistent calculus based on infinitesimals, then in consistent claims from the history of the calculus might well simply be symptoms of confusion. This is addressed in Chapter 5. It is further argued that mathematics has a certain primacy over logic, in that paraconsistent or relevant logics have to be based on inconsistent mathematics. If the latter turns out to be reasonably rich then paraconsistentism is vindicated; while if inconsistent mathematics has seri ous restriytions then the case for being interested in inconsistency-tolerant logics is weakened. (On such restrictions, see this chapter, section 3. ) It must be conceded that fault-tolerant computer programming (e. g. Chapter 8) finds a substantial and important use for paraconsistent logics, albeit with an epistemological motivation (see this chapter, section 3). But even here it should be noted that if inconsistent mathematics turned out to be functionally impoverished then so would inconsistent databases. 2. Summary In Chapter 2, Meyer's results on relevant arithmetic are set out, and his view that they have a bearing on G8del's incompleteness theorems is discussed. Model theory for nonclassical logics is also set out so as to be able to show that the inconsistency of inconsistent theories can be controlled or limited, but in this book model theory is kept in the background as much as possible. This is then used to study the functional properties of various equational number theories."

The A-calculus was invented by Church in the 1930s with the purpose of sup plying a logical foundation for logic and mathematics 25]. Its use by Kleene as a coding for computable functions makes it the first programming lan guage, in an abstract sense, exactly as the Thring machine can be considered the first computer machine 57]. The A-calculus has quite a simple syntax (with just three formation rules for terms) and a simple operational seman tics (with just one operation, substitution), and so it is a very basic setting for studying computation properties. The first contact between A-calculus and real programming languages was in the years 1956-1960, when McCarthy developed the LISP programming language, inspired from A-calculus, which is the first "functional" program ming language, Le., where functions are first-dass citizens 66]. But the use of A-calculus as an abstract paradigm for programming languages started later as the work of three important scientists: Strachey, Landin and B6hm."

The present volume has its origins in a pair of informal workshops held at the Free University of Brussels, in June of 1998 and May of 1999, named "Current Research 1 in Operational Quantum Logic." These brought together mathematicians and physicists working in operational quantum logic and related areas, as well as a number of interested philosophers of science, for a rare opportunity to discuss recent developments in this field. After some discussion, it was decided that, rather than producing a volume of conference proceedings, we would try to organize the conferees to produce a set of comprehensive survey papers, which would not only report on recent developments in quantum logic, but also provide a tutorial overview of the subject suitable for an interested non-specialist audience. The resulting volume provides an overview of the concepts and methods used in current research in quantum logic, viewed both as a branch of mathemati cal physics and as an area of pure mathematics. The first half of the book is concerned with the algebraic side of the subject, and in particular the theory of orthomodular lattices and posets, effect algebras, etc. In the second half of the book, special attention is given to categorical methods and to connections with theoretical computer science. At the 1999 workshop, we were fortunate to hear three excellent lectures by David J. Foulis, represented here by two contributions. Dave's work, spanning 40 years, has helped to define, and continues to reshape, the field of quantum logic."

The present monograph is a slightly revised version of my Habilitations schrift Proof-theoretic Aspects of Intensional and Non-Classical Logics, successfully defended at Leipzig University, November 1997. It collects work on proof systems for modal and constructive logics I have done over the last few years. The main concern is display logic, a certain refinement of Gentzen's sequent calculus developed by Nuel D. Belnap. This book is far from offering a comprehensive presentation of generalized sequent systems for modal logics broadly conceived. The proof-theory of non-classical logics is a rapidly developing field, and even the generalizations of the ordinary notion of sequent listed in Chapter 1 can hardly be presented in great detail within a single volume. In addition to further investigating the various approaches toward generalized Gentzen systems, it is important to compare them and to discuss their relative advantages and disadvantages. An initial attempt at bringing together work on different kinds of proof systems for modal logics has been made in [188]. Another step in the same direction is [196]. Since Chapter 1 contains introductory considerations and, moreover, every remaining chapter begins with some surveying or summarizing remarks, in this preface I shall only emphasize a relation to philosophy that is important to me, register the sources of papers that have entered this book in some form or another, and acknowledge advice and support.

In this volume specialists in mathematics, physics, and linguistics present the first comprehensive analysis of the ideas and influence of Hermann G. Grassmann (1809-1877), the remarkable universalist whose work recast the foundations of these disciplines and shaped the course of their modern development.

Reasoning under uncertainty is always based on a specified language or for malism, including its particular syntax and semantics, but also on its associated inference mechanism. In the present volume of the handbook the last aspect, the algorithmic aspects of uncertainty calculi are presented. Theory has suffi ciently advanced to unfold some generally applicable fundamental structures and methods. On the other hand, particular features of specific formalisms and ap proaches to uncertainty of course still influence strongly the computational meth ods to be used. Both general as well as specific methods are included in this volume. Broadly speaking, symbolic or logical approaches to uncertainty and nu merical approaches are often distinguished. Although this distinction is somewhat misleading, it is used as a means to structure the present volume. This is even to some degree reflected in the two first chapters, which treat fundamental, general methods of computation in systems designed to represent uncertainty. It has been noted early by Shenoy and Shafer, that computations in different domains have an underlying common structure. Essentially pieces of knowledge or information are to be combined together and then focused on some particular question or domain. This can be captured in an algebraic structure called valuation algebra which is described in the first chapter. Here the basic operations of combination and focus ing (marginalization) of knowledge and information is modeled abstractly subject to simple axioms."

The present collection of papers derives from a philosophy conference organised in the Sicilian town of M ussomeli in September 1991. The con ference aimed at providing an analysis of certain aspects of the thought of Michael Dummett, whose contributions have been very influential in several aspects of the philosophical debate continuing within the analyt ical tradition. Logic, the philosophy of mathematics, the interpretation of Frege's philosophy, and metaphysics are only some of the areas within which Dummett's ideas have been fruitful over the years. The papers contained in this book, and Dummett's replies, will, it is hoped, not merely offer a partial reconstruction of a philosopher's life work, but provide an exciting and challenging vantage point from which to look at some of the main problems of contemporary philosophy. The First International Philosophy Conference of M ussomeli - this is what the conference was called - was an extraordinary event in many ways. The quality of the papers presented, the international reputa tion of many of the participants, the venue itself, together with the unavoidable, and sometimes quite funny, organisational hiccups, made that meeting memorable. Perhaps principally memorable was the warmth and sympathy of the people of Mussomeli who strongly supported and encouraged this initia tive. A special mention is also due to the City Council Administrators, who spared no effort to make the Conference a success."

Constructibility and complexity play central roles in recent research in computer science, mathematics and physics. For example, scientists are investigating the complexity of computer programs, constructive proofs in mathematics and the randomness of physical processes. But there are different approaches to the explication of these concepts. This volume presents important research on the state of this discussion, especially as it refers to quantum mechanics. This foundational debate' in computer science, mathematics and physics was already fully developed in 1930 in the Vienna Circle. A special section is devoted to its real founder Hans Hahn, referring to his contribution to the history and philosophy of science. The documentation section presents articles on the early Philipp Frank and on the Vienna Circle in exile. Reviews cover important recent literature on logical empiricism and related topics.

1. BASIC CONCEPTS OF INTERACTIVE THEOREM PROVING Interactive Theorem Proving ultimately aims at the construction of powerful reasoning tools that let us (computer scientists) prove things we cannot prove without the tools, and the tools cannot prove without us. Interaction typi cally is needed, for example, to direct and control the reasoning, to speculate or generalize strategic lemmas, and sometimes simply because the conjec ture to be proved does not hold. In software verification, for example, correct versions of specifications and programs typically are obtained only after a number of failed proof attempts and subsequent error corrections. Different interactive theorem provers may actually look quite different: They may support different logics (first-or higher-order, logics of programs, type theory etc.), may be generic or special-purpose tools, or may be tar geted to different applications. Nevertheless, they share common concepts and paradigms (e.g. architectural design, tactics, tactical reasoning etc.). The aim of this chapter is to describe the common concepts, design principles, and basic requirements of interactive theorem provers, and to explore the band width of variations. Having a 'person in the loop', strongly influences the design of the proof tool: proofs must remain comprehensible, - proof rules must be high-level and human-oriented, - persistent proof presentation and visualization becomes very important."

The main aim of this book is to present recent ideas in logic centered around the notion of a consequence operation. We wish to show these ideas in a factually and materially connected way, i.e., in the form of a consistent theory derived from several simple assumptions and definitions. These ideas have arisen in many research centers. The thorough study of their history can certainly be an exciting task for the historian of logic; in the book this aspect of the theory is being played down. The book belongs to abstract algebraic logic, the area of research that explores to a large extent interconnections between algebra and logic. The results presented here concern logics defined in zero-order languages (Le., quantifier-free sentential languages without predicate symbols). The reach of the theory expounded in the book is, in fact, much wider. The theory is also valid for logics defined in languages of higer orders. The problem of transferring the theory to the level of first-order languages has been satisfactorily solved and new ideas within this area have been put forward in the work of Blok and Pigozzi [1989].

At first glance, Robinson's original form of nonstandard analysis appears nonconstructive in essence, because it makes a rather unrestricted use of classical logic and set theory and, in particular, of the axiom of choice. Recent developments, however, have given rise to the hope that the distance between constructive and nonstandard mathematics is actually much smaller than it appears. So the time was ripe for the first meeting dedicated simultaneously to both ways of doing mathematics and to the current and future reunion of these seeming opposites.

Consisting of peer-reviewed research and survey articles written on the occasion of such an event, this volume offers views of the continuum from various standpoints. Including historical and philosophical issues, the topics of the contributions range from the foundations, the practice, and the applications of constructive and nonstandard mathematics, to the interplay of these areas and the development of a unified theory. This book will be of interest to mathematicians, logicians, and philosophers, as well as theoretical computer scientists, physicists, and economists who are interested in theories of the continuum and in constructive or nonstandard mathematics. The major part is accessible for the non-expert professional reader, from graduate student to academic level. "

Semiorder is probably one of the most frequently ordered structures in science. It naturally appears in fields like psychometrics, economics, decision sciences, linguistics and archaeology. It explicitly takes into account the inevitable imprecisions of scientific instruments by allowing the replacement of precise numbers by intervals. The purpose of this book is to dissect this structure and to study its fundamental properties. The main subjects treated are the numerical representations of semiorders, the generalizations of the concept to valued relations, the aggregation of semiorders and their basic role in a general theoretical framework for multicriteria decision-aid methods. Audience: This volume is intended for students and researchers in the fields of decision analysis, management science, operations research, discrete mathematics, classification, social choice theory, and order theory, as well as for practitioners in the design of decision tools.

Quantifiers: Logics, Models and Computation is the first concentrated effort to give a systematic presentation of the main research results on the subject, since the modern concept was formulated in the late '50s and early '60s. The majority of the papers are in the nature of a handbook. All of them are self-contained, at various levels of difficulty. The Introduction surveys the main ideas and problems encountered in the logical investigation of quantifiers. The Prologue, written by Per Lindstrom, presents the early history of the concept of generalised quantifiers. The volume then continues with a series of papers surveying various research areas, particularly those that are of current interest. Together they provide introductions to the subject from the points of view of mathematics, linguistics, and theoretical computer science. The present volume has been prepared in parallel with Quantifiers: Logics, Models and Computation, Volume Two. Contributions, which contains a collection of research papers on the subject in areas that are too fresh to be summarised. The two volumes are complementary. For logicians, mathematicians, philosophers, linguists and computer scientists. Suitable as a text for advanced undergraduate and graduate specialised courses in logic. "