This book is intended to be a survey of the most important results in mathematical logic for philosophers. It is a survey of results which have philosophical significance and it is intended to be accessible to philosophers. I have assumed the mathematical sophistication acquired. in an introductory logic course or in reading a basic logic text. In addition to proving the most philosophically significant results in mathematical logic, I have attempted to illustrate various methods of proof. For example, the completeness of quantification theory is proved both constructively and non-constructively and relative ad vantages of each type of proof are discussed. Similarly, constructive and non-constructive versions of Godel's first incompleteness theorem are given. I hope that the reader. will develop facility with the methods of proof and also be caused by reflect on their differences. I assume familiarity with quantification theory both in under standing the notations and in finding object language proofs. Strictly speaking the presentation is self-contained, but it would be very difficult for someone without background in the subject to follow the material from the beginning. This is necessary if the notes are to be accessible to readers who have had diverse backgrounds at a more elementary level. However, to make them accessible to readers with no background would require writing yet another introductory logic text. Numerous exercises have been included and many of these are integral parts of the proofs."

While post-Fregean logicians tend to ignore or even denigrate the traditional logic of Aristotle and the Scholastics, new work in recent years has shown the viability of a renewed, extended, and strengthened logic of terms that shares fundamental features of the old syllogistic. A number of logicians, following the lead of Fred Sommers, have built just such a term logic. It is a system of formal logic that not only matches the expressive and inferential powers of today's standard logic, but surpasses it and is far simpler and more natural. This book aims to substantiate this claim by exhibiting just how the term logic can shed need light on a variety of challenges that face any system of formal logic.

Philosophical Dimensions of Logic and Science is a collection of outstanding contributed papers presented at the 11th International Congress of Logic, Methodology, and Philosophy of Science held in Krakow in 1999. The Congress was a follow-up to the series of meetings, initiated once by Alfred Tarski, which aimed to provide an interdisciplinary forum for scientists, philosophers and logicians. The articles selected for publication in the book comply with that idea and innovatively address current issues in logic, metamathematics, philosophy of language, philosophy of science, and cognitive science, as well as philosophical problems of biology, chemistry and physics.
The volume will be of interest to philosophers, logicians and scientists interested in foundational problems of their disciplines."

"Necessity is the mother of invention. " Part I: What is in this book - details. There are several different types of formal proof procedures that logicians have invented. The ones we consider are: 1) tableau systems, 2) Gentzen sequent calculi, 3) natural deduction systems, and 4) axiom systems. We present proof procedures of each of these types for the most common normal modal logics: S5, S4, B, T, D, K, K4, D4, KB, DB, and also G, the logic that has become important in applications of modal logic to the proof theory of Peano arithmetic. Further, we present a similar variety of proof procedures for an even larger number of regular, non-normal modal logics (many introduced by Lemmon). We also consider some quasi-regular logics, including S2 and S3. Virtually all of these proof procedures are studied in both propositional and first-order versions (generally with and without the Barcan formula). Finally, we present the full variety of proof methods for Intuitionistic logic (and of course Classical logic too). We actually give two quite different kinds of tableau systems for the logics we consider, two kinds of Gentzen sequent calculi, and two kinds of natural deduction systems. Each of the two tableau systems has its own uses; each provides us with different information about the logics involved. They complement each other more than they overlap. Of the two Gentzen systems, one is of the conventional sort, common in the literature.

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 weIl 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 0/ Mathematical Logic, published in 1977, edited by the late Jon Barwise. The four volume Handbook 0/ 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."

Dynamic Epistemic Logic is the logic of knowledge change. This book provides various logics to support such formal specifications, including proof systems. Concrete examples and epistemic puzzles enliven the exposition. The book also offers exercises with answers. It is suitable for graduate courses in logic. Many examples, exercises, and thorough completeness proofs and expressivity results are included. A companion web page offers slides for lecturers and exams for further practice.

The present study of sense and reference in the logic of Frege represents the first fruits of several years of dealing with the work of this great German logician. In the preparation of this work, which was presented as a dissertation to the Faculty of Philosophy of the Friedrich-Alexander University in Erlangen/Nuremberg, assistance came from many quarters. lowe most to Professor R. Zocher, who directed this dissertation with understanding counsel and unflagging interest. I must also thank Professor P. Lorenzen, whose courses and seminars provided more inspiration than might be immediately apparent in the book. Professor W. Britzelmayr of Munich was so kind as to provide copies of important fragments of Frege's works. These texts are reproduced with the permission of Professor H. Hermes, Director of the 'Institut fUr mathematische Logik und Grund lagenforschung' in Munster, where Frege's works and letters are being prepared for publication. The preparation of this work was greatly facilitated by a two-year grant from the Fritz Thyssen Foundation. CHRISTIAN THIEL Nuremberg, February 1965 v TRANSLATOR'S NOTE In the difficult matter of Fregean terminology we have taken Ignacio Angelelli's translation of Two Soviet Studies on Frege as the model. Both Professor Angelelli and Dr. Thiel have been so kind as to read over the translation before publication."

This book is unusual in many respects. It was written by a prolific author whose tragic untimely death did not allow to finish this and many other of his undertakings. It was assembled from numerous excerpts, notes, and fragments according to his initial plans. Zilberman's legacy still awaits its true discovery and this book is a second installment to it after The Birth of Meaning in Hindu Thought (Kluwer, 1988). Zilberman's treatment of analogy is unique in its approach, scope, and universality for Western philosophical thought. Constantly compared to eastern and especially classical Indian interpretations, analogy is presented by Zilberman as an important and in many ways primary method of philosophizing or philosophy-building. Due to its universality, this method can be also applied in linguistics, logic, social analysis, as well as historical and anthropological research. These applications are integral part of Zilberman's book. A prophetic leap to largely uncharted territories, this book could be of considerable interest for experts and novices in the field of analogy alike.

This volume is a collection of my essays on philosophy of logic from a phenomenological perspective. They deal with the four kinds of logic I have been concerned with: formal logic, transcendental logic, speculative logic and hermeneutic logic. Of these, only one, the essay on Hegel, touches upon 'speculative logic', and two, those on Heidegger and Konig, are concerned with hermeneutic logic. The rest have to do with Husser and Kant. I have not tried to show that the four logics are compatible. I believe, they are--once they are given a phenomenological underpinning. The original plan of writing an Introduction in which the issues would have to be formulated, developed and brought together, was abandoned in favor of writing an Introductory Essay on the 'origin'- in the phenomenological sense -of logic. J.N.M. Philadelphia INTRODUCTION: THE ORIGIN OF LOGIC The question of the origin of logic may pertain to historical origin (When did it all begin? Who founded the science of logic?), psychological origin (When, in the course of its mental development, does the child learn logical operations?), cultural origin (What cultural - theological, metaphysical and linguisti- conditions make such a discipline as logic possible?), or transcendental constitutive origin (What sorts of acts and/or practices make logic possible?)."

The first and most thoroughly developed audience-oriented argumentation text has been updated to its 8th edition: Argumentation and Critical Decision Making presents argumentation as a cooperative, communicative process. This text examines the general principles of argument in a rigorous yet readable manner and then applies those principles to different spheres of life -- law, science, religion, business, government, and politics -- to explore how conventions of argument change when applied to these real-world arenas. Focusing on the dynamics of decision making and using real-life examples to illustrate principles, Argumentation and Critical Decision Making aims to help readers develop practical argumentation skills within the world of their daily lives.

This book presents and discusses the varying and seminal role which colour plays in the development of Wittgenstein's philosophy. Having once said that "Colours spur us to philosophize", the theme of colour was one to which Wittgenstein returned constantly throughout his career. Ranging from his Notebooks, 1914-1916 and the Tractatus Logico-Philosophicus to the posthumously published Remarks on Colours and On Certainty, this book explores how both his view of philosophical problems generally and his view on colours specifically changed considerably over time. Paying particular attention to his so-called intermediary period, it takes a case-based approach to the presentation of colour in texts from this period, from Some Remarks on Logical Form and Philosophical Remarks to his Big Typescript.

The papers collected in this volume were written over a period of some eight or nine years, with some still earlier material incorporated in one of them. Publishing them under the same cover does not make a con tinuous book of them. The papers are thematically connected with each other, however, in a way which has led me to think that they can naturally be grouped together. In any list of philosophically important concepts, those falling within the range of application of modal logic will rank high in interest. They include necessity, possibility, obligation, permission, knowledge, belief, perception, memory, hoping, and striving, to mention just a few of the more obvious ones. When a satisfactory semantics (in the sense of Tarski and Carnap) was first developed for modal logic, a fascinating new set of methods and ideas was thus made available for philosophical studies. The pioneers of this model theory of modality include prominently Stig Kanger and Saul Kripke. Several others were working in the same area independently and more or less concurrently. Some of the older papers in this collection, especially 'Quantification and Modality' and 'Modes of Modality', serve to clarify some of the main possibilities in the semantics of modal logics in general."

My impulse when I decided to collect into a single volume the essays on topics in logical theory and related subjects that I have written in the last fifteen years was to borrow from the title of a work by Sextus Empiricus, and call my collection "Against the Logicians." Although the essays address a variety of problems that interest me, the thread that runs through them is a scepticism about how logicians see things. So, the title appealed to me. However, I had second thoughts and chose instead a title of one of my own essays, "In Defense of Informal Logic," which emphasizes my support for other approaches. Although my criticisms of logical theory are designed to cut deeply, I do not want to be unresponsive to the needs that it is supposed to satisfy. However, my position that we have adequate resources for critically analyzing a particular argument and 00 not need a theory of argumentation, will not completely satisfy those who think that there is a need for it. So, I want them to know that I am taking their concerns seriously.

This edited volume focuses on the work of Professor Larisa Maksimova, providing a comprehensive account of her outstanding contributions to different branches of non-classical logic. The book covers themes ranging from rigorous implication, relevance and algebraic logic, to interpolation, definability and recognizability in superintuitionistic and modal logics. It features both her scientific autobiography and original contributions from experts in the field of non-classical logics. Professor Larisa Maksimova's influential work involved combining methods of algebraic and relational semantics. Readers will be able to trace both influences on her work, and the ways in which her work has influenced other logicians. In the historical part of this book, it is possible to trace important milestones in Maksimova's career. Early on, she developed an algebraic semantics for relevance logics and relational semantics for the logic of entailment. Later, Maksimova discovered that among the continuum of superintuitionisitc logics there are exactly three pretabular logics. She went on to obtain results on the decidability of tabularity and local tabularity problems for superintuitionistic logics and for extensions of S4. Further investigations by Maksimova were aimed at the study of fundamental properties of logical systems (different versions of interpolation and definability, disjunction property, etc.) in big classes of logics, and on decidability and recognizability of such properties. To this end she determined a powerful combination of algebraic and semantic methods, which essentially determine the modern state of investigations in the area, as can be seen in the later chapters of this book authored by leading experts in non-classical logics. These original contributions bring the reader up to date on the very latest work in this field.

This book covers work written by leading scholars from different schools within the research area of paraconsistency. The authors critically investigate how contemporary paraconsistent logics can be used to better understand human reasoning in science and mathematics. Offering a variety of perspectives, they shed a new light on the question of whether paraconsistent logics can function as the underlying logics of inconsistent but useful scientific and mathematical theories. The great variety of paraconsistent logics gives rise to various, interrelated questions, such as what are the desiderata a paraconsistent logic should satisfy, is there prospect of a universal approach to paraconsistent reasoning with axiomatic theories, and to what extent is reasoning about sets structurally analogous to reasoning about truth. Furthermore, the authors consider paraconsistent logic's status as either a normative or descriptive discipline (or one which falls in between) and which inconsistent but non-trivial axiomatic theories are well understood by which types of paraconsistent approaches. This volume addresses such questions from different perspectives in order to (i) obtain a representative overview of the state of the art in the philosophical debate on paraconsistency, (ii) come up with fresh ideas for the future of paraconsistency, and most importantly (iii) provide paraconsistent logic with a stronger philosophical foundation, taking into account the developments within the different schools of paraconsistency.

The aim of the book is to introduce the reader to some new areas oflogic which have yet to find their way into the bulk of modern logic books written from the more orthodox direction of the mainstream of develop ments. Such a work seems to me much needed, both because of the in trinsic value and increasing prominence of the nonstandard sector of logic, and because this particular sector is of the greatest interest from the standpoint of philosophical implications and applications. This book unites a series of studies in philosophical logic, drawing for the most part on material which I have contributed to the journal liter ature of the subject over the past ten years. Despite the fact that some of these essays have been published in various journals at different times, they possess a high degree of thematic and methodological unity. All of these studies deal with material of substantial current interest in philo sophical logic and embody a fusion of the modern techniques of logical and linguistic-philosophical analysis for the exploration of areas of logic that are of substantial philosophical relevance."

First published in 1952, professor 's Strawson 's highly influential Introduction to Logical Theory provides a detailed examination of the relationship between the behaviour of words in common language and the behaviour of symbols in a logical system. He seeks to explain both the exact nature of the discipline known as Formal Logic, and also to reveal something of the intricate logical structure of ordinary unformalised discourse.

The media bombard us with claims that are often strange, unclear, and even upsetting. Quantum physicists claim that "vacuum nothingness" is not really nothing, because it teems with energy and virtual particles. Psychological research suggests that most of our neighbors suffer from some degree of mental disorder. Social scientists assure us that science itself is simply a cultural myth. Can anyone sort out fact from fiction in today's world? The answer, thankfully, is "Yes " But first, you must make a radical shift in your approach, because serious thinking about reality involves serious thinking about fiction, not only in your everyday mind, but also in the scholarly and technical realms. For anyone who has ever wondered-and you should wonder-whether there really are such things as government, society, the economy, or even marriage, the deeply philosophical and utterly practical "Logical Fictions" shows you how a solid grounding in logic and language can help you avoid getting trapped by the ideological fictions prevalent in today's sophisticated world. Consider yourself warned: humorous and filled with entertaining examples, this book will stretch your brain and provoke your thoughts. Your view of the world may never be the same.

P. A. Schilpp's 'Library of Living Philosophers' is the series which introduced to the philosophical community the format of a volume of essays on the work of a distinguished philosopher, combined with replies to the essays by the philosopher targeted. The format proved attracti ve to a discipline which has always placed a high premium on debate. But the Schilpp series has shown itself unenterprising in its choice of subjects, concentrating on end-of-year reports on philosophers who are of undoubted distinction, but whose contribution to the subject can be regarded as rather definitely over. Which leaves a gap, which the present series is designed to fill, for volumes of a similar format aiming at assessment of philosophers who have distinguished themselves already by making a substan tial impact on their discipline, but whose further work too is awaited with eager anticipation. Michael Dummett is an ideal subject for a series with this goal of mid term assessment. His writings to date have permanently altered philosophy's conception of what is at issue between realism and idealism (and its paler cousin, anti-realism); and this has been achieved by way of a supplementary clarification of a host of issues in the philosophy of language and of mathematics, and of the Frege/Wittgenstein historical tradition from which such issues are typically approached in contemporary philosophy."

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

The present publication forms part of a projected book that F. P. Ramsey drafted but never completed. It survived among his papers and ultimately came into the possession of the University of Pittsburgh in the circumstances detailed in the Editor's Introduction. Our hope in issuing this work at this stage - some sixty years after Ramsey's premature death at the age of 26 - is both to provide yet another token of his amazing philosophical creativity, and also to make available an important datum for the still to be written history of the development of philosophical analysis. This is a book whose appearance will, we hope and expect, be appreciated both by those interested in linguistic philosophy itself and by those concerned for its historical development in the present century. EDITORS'INTRODUCTION 1. THE RAMSEY COLLECTION Frank Plump ton Ramsey (22 February 1903 -19 January 1930) was an extra ordinary scholarly phenomenon. Son of a distinguished mathematician and President of Magdalene College, Cambridge and brother of Arthur Michael, eventual Archbishop of Canterbury, Ramsey was closely connected with Cambridge throughout his life, ultimately becoming lecturer in Mathematics in the University. Notwithstanding his great mathematical talent, it was primarily logic and philosophy that engaged his interests, and he wrote original and important contributions to logic, semantics, epistomology, probability theory, philosophy of science, and economics, in addition to seminal work in the foundations of mathematics."