This monograph develops projective geometries and provides a systematic treatment of morphisms. It introduces a new fundamental theorem and its applications describing morphisms of projective geometries in homogeneous coordinates by semilinear maps. Other topics treated include three equivalent definitions of projective geometries and their correspondence with certain lattices; quotients of projective geometries and isomorphism theorems; and recent results in dimension theory.

This volume, the 6th volume in the DRUMS Handbook series, is part of the after math of the successful ESPRIT project DRUMS (Defeasible Reasoning and Un certainty 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 Rea soning', 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. This work has an emphasis on modelling and formal techniques in the investigation of the topic "Reasoning and Dynamics," but it is not mere theory that occupied us. Rather research was aimed at bridging the gap between theory and practice. Therefore also real-life applications of the modelling techniques were considered, and we hope this also shows in this volume, which is focused on the dynamics of reasoning processes. In order to give the book a broader perspective, we have invited a number of well-known researchers outside the project but working on similar topics to contribute as well. 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."

One can distinguish, roughly speaking, two different approaches to the philosophy of mathematics. On the one hand, some philosophers (and some mathematicians) take the nature and the results of mathematicians' activities as given, and go on to ask what philosophical morals one might perhaps find in their story. On the other hand, some philosophers, logicians and mathematicians have tried or are trying to subject the very concepts which mathematicians are using in their work to critical scrutiny. In practice this usually means scrutinizing the logical and linguistic tools mathematicians wield. Such scrutiny can scarcely help relying on philosophical ideas and principles. In other words it can scarcely help being literally a study of language, truth and logic in mathematics, albeit not necessarily in the spirit of AJ. Ayer. As its title indicates, the essays included in the present volume represent the latter approach. In most of them one of the fundamental concepts in the foundations of mathematics and logic is subjected to a scrutiny from a largely novel point of view. Typically, it turns out that the concept in question is in need of a revision or reconsideration or at least can be given a new twist. The results of such a re-examination are not primarily critical, however, but typically open up new constructive possibilities. The consequences of such deconstructions and reconstructions are often quite sweeping, and are explored in the same paper or in others.

Proof theory and category theory were first drawn together by Lambek some 30 years ago but, until now, the most fundamental notions of category theory (as opposed to their embodiments in logic) have not been explained systematically in terms of proof theory. Here it is shown that these notions, in particular the notion of adjunction, can be formulated in such as way as to be characterised by composition elimination. Among the benefits of these composition-free formulations are syntactical and simple model-theoretical, geometrical decision procedures for the commuting of diagrams of arrows. Composition elimination, in the form of Gentzen's cut elimination, takes in categories, and techniques inspired by Gentzen are shown to work even better in a purely categorical context than in logic. An acquaintance with the basic ideas of general proof theory is relied on only for the sake of motivation, however, and the treatment of matters related to categories is also in general self contained. Besides familiar topics, presented in a novel, simple way, the monograph also contains new results. It can be used as an introductory text in categorical proof theory.

Fuzzy Sets, Logics and Reasoning about Knowledge reports recent results concerning the genuinely logical aspects of fuzzy sets in relation to algebraic considerations, knowledge representation and commonsense reasoning. It takes a state-of-the-art look at multiple-valued and fuzzy set-based logics, in an artificial intelligence perspective. The papers, all of which are written by leading contributors in their respective fields, are grouped into four sections. The first section presents a panorama of many-valued logics in connection with fuzzy sets. The second explores algebraic foundations, with an emphasis on MV algebras. The third is devoted to approximate reasoning methods and similarity-based reasoning. The fourth explores connections between fuzzy knowledge representation, especially possibilistic logic and prioritized knowledge bases. Readership: Scholars and graduate students in logic, algebra, knowledge representation, and formal aspects of artificial intelligence.

This volume contains a selection of papers presented at a Seminar on Intensional Logic held at the University of Amsterdam during the period September 1990-May 1991. Modal logic, either as a topic or as a tool, is common to most of the papers in this volume. A number of the papers are con cerned with what may be called well-known or traditional modal systems, but, as a quick glance through this volume will reveal, this by no means implies that they walk the beaten tracks. In deed, such contributions display new directions, new results, and new techniques to obtain familiar results. Other papers in this volume are representative examples of a current trend in modal logic: the study of extensions or adaptations of the standard sys tems that have been introduced to overcome various shortcomings of the latter, especially their limited expressive power. Finally, there is another major theme that can be discerned in the vol ume, a theme that may be described by the slogan 'representing changing information. ' Papers falling under this heading address long-standing issues in the area, or present a systematic approach, while a critical survey and a report contributing new techniques are also included. The bulk of the papers on pure modal logic deal with theoreti calor even foundational aspects of modal systems."

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."

The present monograph intends to establish a solid link among three fields: fuzzy set theory, information retrieval, and cluster analysis. Fuzzy set theory supplies new concepts and methods for the other two fields, and provides a common frame work within which they can be reorganized. Four principal groups of readers are assumed: researchers or students who are interested in (a) application of fuzzy sets, (b) theory of information retrieval or bibliographic databases, (c) hierarchical clustering, and (d) application of methods in systems science. Readers in group (a) may notice that the fuzzy set theory used here is very simple, since only finite sets are dealt with. This simplification enables the max min algebra to deal with fuzzy relations and matrices as equivalent entities. Fuzzy graphs are also used for describing theoretical properties of fuzzy relations. This assumption of finite sets is sufficient for applying fuzzy sets to information retrieval and cluster analysis. This means that little theory, beyond the basic theory of fuzzy sets, is required. Although readers in group (b) with little background in the theory of fuzzy sets may have difficulty with a few sections, they will also find enough in this monograph to support an intuitive grasp of this new concept of fuzzy information retrieval. Chapter 4 provides fuzzy retrieval without the use of mathematical symbols. Also, fuzzy graphs will serve as an aid to the intuitive understanding of fuzzy relations."

This book presents a specific and unified approach to Knowledge Discovery and Data Mining, termed IFN for Information Fuzzy Network methodology. Data Mining (DM) is the science of modelling and generalizing common patterns from large sets of multi-type data. DM is a part of KDD, which is the overall process for Knowledge Discovery in Databases. The accessibility and abundance of information today makes this a topic of particular importance and need. The book has three main parts complemented by appendices as well as software and project data that are accessible from the book's web site (http: //www.eng.tau.ac.iV-maimonlifn-kdg ). Part I (Chapters 1-4) starts with the topic of KDD and DM in general and makes reference to other works in the field, especially those related to the information theoretic approach. The remainder of the book presents our work, starting with the IFN theory and algorithms. Part II (Chapters 5-6) discusses the methodology of application and includes case studies. Then in Part III (Chapters 7-9) a comparative study is presented, concluding with some advanced methods and open problems. The IFN, being a generic methodology, applies to a variety of fields, such as manufacturing, finance, health care, medicine, insurance, and human resources. The appendices expand on the relevant theoretical background and present descriptions of sample projects (including detailed results)."

"In case you are considering to adopt this book for courses with over 50 students, please contact ""[email protected]"" for more information. "

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability.

The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory.

Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises.

"Audience: " This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. "

Internal logic is the logic of content. The content is here arithmetic and the emphasis is on a constructive logic of arithmetic (arithmetical logic). Kronecker's general arithmetic of forms (polynomials) together with Fermat's infinite descent is put to use in an internal consistency proof. The view is developed in the context of a radical arithmetization of mathematics and logic and covers the many-faceted heritage of Kronecker's work, which includes not only Hilbert, but also Frege, Cantor, Dedekind, Husserl and Brouwer.

The book will be of primary interest to logicians, philosophers and mathematicians interested in the foundations of mathematics and the philosophical implications of constructivist mathematics. It may also be of interest to historians, since it covers a fifty-year period, from 1880 to 1930, which has been crucial in the foundational debates and their repercussions on the contemporary scene.

Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof procedures. The volume is of immense importance for the interdisciplinary fields of logic, knowledge representation, and automated deduction.

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations.

Homological Mirror Symmetry, the study of dualities of certain quantum field theories in a mathematically rigorous form, has developed into a flourishing subject on its own over the past years. The present volume bridges a gap in the literature by providing a set of lectures and reviews that both introduce and representatively review the state-of-the art in the field from different perspectives. With contributions by K. Fukaya, M. Herbst, K. Hori, M. Huang, A. Kapustin, L. Katzarkov, A. Klemm, M. Kontsevich, D. Page, S. Quackenbush, E. Sharpe, P. Seidel, I. Smith and Y. Soibelman, this volume will be a reference on the topic for everyone starting to work or actively working on mathematical aspects of quantum field theory.

In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation."

Mathematics has stood as a bridge between the Humanities and the Sciences since the days of classical antiquity. For Plato, mathematics was evidence of Being in the midst of Becoming, garden variety evidence apparent even to small children and the unphilosophical, and therefore of the highest educational significance. In the great central similes of The Republic it is the touchstone ofintelligibility for discourse, and in the Timaeus it provides in an oddly literal sense the framework of nature, insuring the intelligibility ofthe material world. For Descartes, mathematical ideas had a clarity and distinctness akin to the idea of God, as the fifth of the Meditations makes especially clear. Cartesian mathematicals are constructions as well as objects envisioned by the soul; in the Principles, the work ofthe physicist who provides a quantified account ofthe machines of nature hovers between description and constitution. For Kant, mathematics reveals the possibility of universal and necessary knowledge that is neither the logical unpacking ofconcepts nor the record of perceptual experience. In the Critique ofPure Reason, mathematics is one of the transcendental instruments the human mind uses to apprehend nature, and by apprehending to construct it under the universal and necessary lawsofNewtonian mechanics.

This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Curves and Surfaces, held in Avignon, in June 2010. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications." The 39 revised full papers presented together with 9 invited talks were carefully reviewed and selected from 114 talks presented at the conference. The topics addressed by the papers range from mathematical foundations to practical implementation on modern graphics processing units and address a wide area of topics such as computer-aided geometric design, computer graphics and visualisation, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, scattered data processing and learning theory and subdivision, wavelets and multi-resolution methods.

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.

The Institute Vienna Circle held a conference in Vienna in 2003, Cambridge and Vienna Frank P. Ramsey and the Vienna Circle, to commemorate the philosophical and scientific work of Frank Plumpton Ramsey (1903 1930). This Ramsey conference provided not only historical and biographical perspectives on one of the most gifted thinkers of the Twentieth Century, but also new impulses for further research on at least some of the topics pioneered by Ramsey, whose interest and potential are greater than ever.

Ramsey did pioneering work in several fields, practitioners of which rarely know of his important work in other fields: philosophy of logic and theory of language, foundations of mathematics, mathematics, probability theory, methodology of science, philosophy of psychology, and economics. There was a focus on the one topic which was of strongest mutual concern to Ramsey and the Vienna Circle, namely the question of foundations of mathematics, in particular the status of logicism.

Although the major scientific connection linking Ramsey with Austria is his work on logic, to which the Vienna Circle dedicated several meetings, certainly the connection which is of greater general interest concerns Ramsey's visits and discussions with Wittgenstein. Ramsey was the only important thinker to actually visit Wittgenstein during his school-teaching career in Puchberg and Ottertal in the 1920s, in Lower Austria; and later, Ramsey was instrumental in getting Wittgenstein positions at Cambridge. "

The notion of complexity is an important contribution of logic to theoretical computer science and mathematics. This volume attempts to approach complexity in a holistic way, investigating mathematical properties of complexity hierarchies at the same time as discussing algorithms and computational properties. A main focus of the volume is on some of the new paradigms of computation, among them Quantum Computing and Infinitary Computation. The papers in the volume are tied together by an introductory article describing abstract properties of complexity hierarchies.

This volume will be of great interest to both mathematical logicians and theoretical computer scientists, providing them with new insights into the various views of complexity and thus shedding new light on their own research.

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.

In this monograph, questions of extensions and relaxations are consid ered. These questions arise in many applied problems in connection with the operation of perturbations. In some cases, the operation of "small" per turbations generates "small" deviations of basis indexes; a corresponding stability takes place. In other cases, small perturbations generate spas modic change of a result and of solutions defining this result. These cases correspond to unstable problems. The effect of an unstability can arise in extremal problems or in other related problems. In this connection, we note the known problem of constructing the attainability domain in con trol theory. Of course, extremal problems and those of attainability (in abstract control theory) are connected. We exploit this connection here (see Chapter 5). However, basic attention is paid to the problem of the attainability of elements of a topological space under vanishing perturba tions of restrictions. The stability property is frequently missing; the world of unstable problems is of interest for us. We construct regularizing proce dures. However, in many cases, it is possible to establish a certain property similar to partial stability. We call this property asymptotic nonsensitivity or roughness under the perturbation of some restrictions. The given prop erty means the following: in the corresponding problem, it is the same if constraints are weakened in some "directions" or not. On this basis, it is possible to construct a certain classification of constraints, selecting "di rections of roughness" and "precision directions.""

During the last few decades the ideas, methods, and results of the theory of Boolean algebras have played an increasing role in various branches of mathematics and cybernetics. This monograph is devoted to the fundamentals of the theory of Boolean constructions in universal algebra. Also considered are the problems of presenting different varieties of universal algebra with these constructions, and applications for investigating the spectra and skeletons of varieties of universal algebras. For researchers whose work involves universal algebra and logic.

This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets."

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.