Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments.

We who live in this post-modern late twentieth century culture are still children of dualism. For a variety of rather complex reasons we continue to split apart and treat as radical opposites body and spirit, medicine and religion, sacred and secular, private and public, love and justice, men and women. Though this is still our strong tendency, we are beginning to discover both the futility and the harm of such dualistic splitting. Peoples of many ancient cultures might smile at the belatedness of our discovery concerning the commonalities of medicine and religion. A cur sory glance back at ancient Egypt, Samaria, Babylonia, Persia, Greece, and Rome would disclose a common thread - the close union of religion and medicine. Both were centrally concerned with healing, health, and wholeness. The person was understood as a unity of body, mind, and spirit. The priest and the physician frequently were combined in the same individual. One of the important contributions of this significant volume of essays is the sustained attack upon dualism. From a variety of vantage points, virtually all of the authors unmask the varied manifestations of dualism in religion and medicine, urging a more holistic approach. Since the editor has provided an excellent summary of each article, I shall not attempt to comment on specific contributions. Rather, I wish to highlight three 1 broad themes which I find notable for theological ethics."

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.

Modal Logic is a branch of logic with applications in many related disciplines such as computer science, philosophy, linguistics and artificial intelligence. Over the last twenty years, in all of these neighbouring fields, modal systems have been developed that we call multi-dimensional. (Our definition of multi-dimensionality in modal logic is a technical one: we call a modal formalism multi-dimensional if, in its intended semantics, the universe of a model consists of states that are tuples over some more basic set.) This book treats such multi-dimensional modal logics in a uniform way, linking their mathematical theory to the research tradition in algebraic logic. We will define and discuss a number of systems in detail, focusing on such aspects as expressiveness, definability, axiomatics, decidability and interpolation. Although the book will be mathematical in spirit, we take care to give motivations from the disciplines mentioned earlier on.

"The questions this book deals with deserve close investigation. The book offers a rare opportunity to think about the limitations of our standard logic and semantic analysis from an overall perspective." - Notre Dame Philosophical Reviews In this challenging and provocative book, Dale Jacquette argues that contemporary philosophy labours under a number of historically inherited delusions about the nature of logic and the philosophical significance of certain formal properties of specific types of logical constructions. Exposing some of the key misconceptions about formal symbolic logic and its relation to thought, language and the world, Jacquette clears the ground of some very well-entrenched philosophical doctrines about the nature of logic, including some of the most fundamental seldom-questioned parts of elementary propositional and predicate-quantificational logic. Having presented difficulties for conventional ways of thinking about truth functionality, the metaphysics of reference and predication, the role of a concept of truth in a theory of meaning, among others, Jacquette proceeds to reshape the network of ideas about traditional logic that philosophy has acquired along with modern logic itself. In so doing Jacquette is able to offer a new perspective on a number of existing problems in logic and philosophy of logic.

The analytic/synthetic distinction looks simple. It is a distinction between two different kinds of sentence. Synthetic sentences are true in part because of the way the world is, and in part because of what they mean. Analytic sentences - like all bachelors are unmarried and triangles have three sides - are different. They are true in virtue of meaning, so no matter what the world is like, as long as the sentence means what it does, it will be true.
This distinction seems powerful because analytic sentences seem to be knowable in a special way. One can know that all bachelors are unmarried, for example, just by thinking about what it means. But many twentieth-century philosophers, with Quine in the lead, argued that there were no analytic sentences, that the idea of analyticity didn't even make sense, and that the analytic/synthetic distinction was therefore an illusion. Others couldn't see how there could fail to be a distinction, however ingenious the arguments of Quine and his supporters.
But since the heyday of the debate, things have changed in the philosophy of language. Tools have been refined, confusions cleared up, and most significantly, many philosophers now accept a view of language - semantic externalism - on which it is possible to see how the distinction could fail. One might be tempted to think that ultimately the distinction has fallen for reasons other than those proposed in the original debate.
In Truth in Virtue of Meaning, Gillian Russell argues that it hasn't. Using the tools of contemporary philosophy of language, she outlines a view of analytic sentences which is compatible with semantic externalism and defends that view against the old Quineanarguments. She then goes on to draw out the surprising epistemological consequences of her approach.

The modern discussion on the concept of truthlikeness was started in 1960. In his influential Word and Object, W. V. O. Quine argued that Charles Peirce's definition of truth as the limit of inquiry is faulty for the reason that the notion 'nearer than' is only "defined for numbers and not for theories." In his contribution to the 1960 International Congress for Logic, Methodology, and Philosophy of Science at Stan ford, Karl Popper defended the opposite view by defining a compara tive notion of verisimilitude for theories. was originally introduced by the The concept of verisimilitude Ancient sceptics to moderate their radical thesis of the inaccessibility of truth. But soon verisimilitudo, indicating likeness to the truth, was confused with probabilitas, which expresses an opiniotative attitude weaker than full certainty. The idea of truthlikeness fell in disrepute also as a result of the careless, often confused and metaphysically loaded way in which many philosophers used - and still use - such concepts as 'degree of truth', 'approximate truth', 'partial truth', and 'approach to the truth'. Popper's great achievement was his insight that the criticism against truthlikeness - by those who urge that it is meaningless to speak about 'closeness to truth' - is more based on prejudice than argument."

This is an outline of a coherence theory of law. Its basic ideas are: reasonable support and weighing of reasons. All the rest is commentary.

These words at the beginning of the preface of this book perfectly indicate what On Law and Reason is about. It is a theory about the nature of the law which emphasises the role of reason in the law and which refuses to limit the role of reason to the application of deductive logic.

In 1989, when the first edition of On Law and Reason appeared, this book was ground breaking for several reasons. It provided a rationalistic theory of the law in the language of analytic philosophy and based on a thorough understanding of the results, including technical ones, of analytic philosophy. That was not an obvious combination at the time of the book s first appearance and still is not. The result is an analytical rigor that is usually associated with positivist theories of the law, combined with a philosophical position that is not natural law in a strict sense, but which shares with it the emphasis on the role of reason in determining what the law is. If only for this rare combination, On Law and Reason still deserves careful study.

On Law and Reason also foreshadowed and influenced a development in the field of Legal Logic that would take place in the nineties of the 20th century, namely the development of non-monotonic ( defeasible ) logics for the analysis of legal reasoning. In the new Introduction to this second edition, this aspect is explored in some more detail."

Aristotle's "On Interpretation", a centrepiece of his logic, studies the relationship between conflicting pairs of statements. The first eight chapters, studied here, explain what statements are; they start from their basic components, the words, and work up to the character of opposed affirmations and negations. The 15,000 pages of the ancient Greek commentators on Aristotle, written mainly between 200 and 500 AD, constitute the largest corpus of extant Greek philosophical writing not translated into English or other European languages. This new series of translations, planned in 60 volumes, fills an important gap in the history of European thought.

Labelled deduction is an approach to providing frameworks for presenting and using different logics in a uniform and natural way by enriching the language of a logic with additional information of a semantic proof-theoretical nature. Labelled deduction systems often possess attractive properties, such as modularity in the way that families of related logics are presented, parameterised proofs of metatheoretic properties, and ease of mechanisability. It is thus not surprising that labelled deduction has been applied to problems in computer science, AI, mathematical logic, cognitive science, philosophy and computational linguistics - for example, formalizing and reasoning about dynamic state oriented' properties such as knowledge, belief, time, space, and resources.

Doing Worlds with Words throws light on the problem of meaning as the meeting point of linguistics, logic and philosophy, and critically assesses the possibilities and limitations of elucidating the nature of meaning by means of formal logic, model theory and model-theoretical semantics. The main thrust of the book is to show that it is misguided to understand model theory metaphysically and so to try to base formal semantics on something like formal metaphysics; rather, the book states that model theory and similar tools of the analysis of language should be understood as capturing the semantically relevant, especially inferential, structure of language. From this vantage point, the reader gains a new light on many of the traditional concepts and problems of logic and philosophy of language, such as meaning, reference, truth and the nature of formal logic.

Hilbert's Program was founded on a concern for the phenomenon of paradox in mathematics. To Hilbert, the paradoxes, which are at once both absurd and irresistible, revealed a deep philosophical truth: namely, that there is a discrepancy between the laws accord ing to which the mind of homo mathematicus works, and the laws governing objective mathematical fact. Mathematical epistemology is, therefore, to be seen as a struggle between a mind that naturally works in one way and a reality that works in another. Knowledge occurs when the two cooperate. Conceived in this way, there are two basic alternatives for mathematical epistemology: a skeptical position which maintains either that mind and reality seldom or never come to agreement, or that we have no very reliable way of telling when they do; and a non-skeptical position which holds that there is significant agree ment between mind and reality, and that their potential discrepan cies can be detected, avoided, and thus kept in check. Of these two, Hilbert clearly embraced the latter, and proposed a program designed to vindicate the epistemological riches represented by our natural, if non-literal, ways of thinking. Brouwer, on the other hand, opted for a position closer (in Hilbert's opinion) to that of the skeptic. Having decided that epistemological purity could come only through sacrifice, he turned his back on his classical heritage to accept a higher calling."

It is with great pleasure that we are presenting to the community the second edition of this extraordinary handbook. It has been over 15 years since the publication of the first edition and there have been great changes in the landscape of philosophical logic since then. The first edition has proved invaluable to generations of students and researchers in formal philosophy and language, as well as to consumers of logic in many applied areas. The main logic article in the Encyclopaedia Britannica 1999 has described the first edition as 'the best starting point for exploring any of the topics in logic'. We are confident that the second edition will prove to be just as good, The first edition was the second handbook published for the logic commu nity. It followed the North Holland one volume Handbook of Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook of Philosophical Logic, published 1983-1989 came at a fortunate temporal junction at the evolution of logic. This was the time when logic was gaining ground in computer science and artificial intelligence circles. These areas were under increasing commercial pressure to provide devices which help and/or replace the human in his daily activity. This pressure required the use of logic in the modelling of human activity and organisa tion on the one hand and to provide the theoretical basis for the computer program constructs on the other."

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.

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.

Alexander Broadie explains the theories of truth and validity of the great logicians of the fourteenth century. He shows how far they advanced beyond the logic of Aristotle, and reveals that much of what seems characteristically twentieth-century logic was familiar long ago. He has extensively revised his text for this second edition, while preserving the character of the first. There are now fuller accounts of supposition of intentional contexts, and of medieval syllogistics, and the conclusion has been substantially expanded.

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."

First published in 2005. Routledge is an imprint of Taylor & Francis, an informa company.

The relation between logic and knowledge provoked a heated debate since the 1960s. The epistemic approaches, found their formal argument in the mathematics of Brouwer and intuitionistic logic. And following Michael Dummett - started to call themselves: antirealists'. Others persisted with the formal background of the Frege-Tarski tradition, where Cantorian set theory is linked via model theory to classical logic. Jaakko Hintikka tried to join both traditions by means of what is now known as explicit epistemic logic'. Here the epistemic content is introduced into the object language as an operator which yield propositions from propositions rather than as metalogical constraint on the notion of inference. The Realism-Antirealism debate had thus three players: classical logicians, intuitionists and explicit epistemic logicians. The editors of the present volume think that in these days and age of Alternative Logics, where manifold developments in logic happen in a breathtaking pace, this debate should be revisited. Using the most recent logical and epistemological tools, this book provides a novel and refreshing view on the most important topics of the Realism vs. Antirealism debate. Its general scope is to show the most recent developments in philosophical logic to deal with problems inherited from this debate. It is meant for researcher and advanced students in philosophy, logic, formal methods. It's complete collection with a variety of approaches, it is written by leading authors in the fields, every chapter is self-contained.

This book is intended to serve as an advanced text and reference work on modal logic, a subject of growing importance which has applications to philosophy and linguistics. Although it is based mainly on research which I carried out during the years 1969-1973, it also includes some related results obtained by other workers in the field (see the refer ences in Part 7). Parts 0, 1 and 2, can be used as the basis of a one year graduate course in modal logic. The material which they contain has been taught in such courses at Stanford since 1970. The remaining parts of the book contain more than enough material for a second course in modal logic. The exercises supplement the text and are usually difficult. I wish to thank Stanford University and Bar-Han University for making it possible for me to continue and finish this work, and A. Ungar for correcting the typescript. Bar-Ilan University, Israel Dov M. GABBA Y PART 0 AN INTRODUCTION TO GENERAL INTENSIONAL LOGICS CHAPTER 0 CONSEQUENCE RELATIONS Motivation We introduce the notions of a consequence relation (which is a generalization of the notion of a logical system) and of a semantics. We show that every consequence relation is complete for a canonical semantics. We define the notion of one semantics being Dian in another and study the basic properties of this notion. The concepts of this chapter are generalizations of the various notions of logical system and possible world semantics found in the literature."

A classic of how to think clearly and critically and ahead of its time in anticipating the threats to democracy by poor argument and shoddy reasoning Engaging, clear and witty, it is a brilliant example of how philosophy can connect with the concerns with everyone and requires no knowledge of the subject Susan Stebbing was the first woman in the UK to be appointed a professor of philosophy, in 1933 A new foreword by Nigel Warburton and introduction by Peter West help to set Stebbing book in helpful context

This study looks to the work of Tarski's mentors Stanislaw Lesniewski and Tadeusz Kotarbinski, and reconsiders all of the major issues in Tarski scholarship in light of the conception of Intuitionistic Formalism developed: semantics, truth, paradox, logical consequence.

Reasoning is an integral part of intelligent systems in fields like databases, logic programming, robotics, knowledge engineering, human/computer interfaces, programming environments, etc. In reality any such system has to cope with a changing world and its dynamics. Hence it is of great importance that reasoning must account for coping with change in order to be truly useful in practice. The book comprises several contributions to current ways of approaching this problem. On the one hand it surveys and synthesizes recent research work, while on the other hand new research results are included. Among the topics treated are logics for reasoning about actions and planning, belief revision and the reconciliation of logically conflicting inputs, resolving of conflicts by merging of knowledge and issues in the evolution in object-oriented databases. The book is aimed at the researcher and advanced student active in this field.