This is an open access title available under the terms of a CC BY-NC-ND 4.0 International licence. It is free to read at Oxford Scholarship Online and offered as a free PDF download from OUP and selected open access locations. We need to understand the impossible. Francesco Berto and Mark Jago start by considering what the concepts of meaning, information, knowledge, belief, fiction, conditionality, and counterfactual supposition have in common. They are all concepts which divide the world up more finely than logic does. Logically equivalent sentences may carry different meanings and information and may differ in how they're believed. Fictions can be inconsistent yet meaningful. We can suppose impossible things without collapsing into total incoherence. Yet for the leading philosophical theories of meaning, these phenomena are an unfathomable mystery. To understand these concepts, we need a metaphysical, logical, and conceptual grasp of situations that could not possibly exist: Impossible Worlds. This book discusses the metaphysics of impossible worlds and applies the concept to a range of central topics and open issues in logic, semantics, and philosophy. It considers problems in the logic of knowledge, the meaning of alternative logics, models of imagination and mental simulation, the theory of information, truth in fiction, the meaning of conditional statements, and reasoning about the impossible. In all these cases, impossible worlds have an essential role to play.

This volume examines the entire logical and philosophical production of Nicolai A. Vasil'ev, studying his life and activities as a historian and man of letters. Readers will gain a comprehensive understanding of this influential Russian logician, philosopher, psychologist, and poet. The author frames Vasil'ev's work within its historical and cultural context. He takes into consideration both the situation of logic in Russia and the state of logic in Western Europe, from the end of the 19th century to the beginning of the 20th. Following this, the book considers the attempts to develop non-Aristotelian logics or ideas that present affinities with imaginary logic. It then looks at the contribution of traditional logic in elaborating non-classical ideas. This logic allows the author to deal with incomplete objects just as imaginary logic does with contradictory ones. Both logics are objects of interesting analysis by modern researchers. This volume will appeal to graduate students and scholars interested not only in Vasil'ev's work, but also in the history of non-classical logics.

The aim of this volume is to collect original contributions by the best specialists from the area of proof theory, constructivity, and computation and discuss recent trends and results in these areas. Some emphasis will be put on ordinal analysis, reductive proof theory, explicit mathematics and type-theoretic formalisms, and abstract computations. The volume is dedicated to the 60th birthday of Professor Gerhard Jager, who has been instrumental in shaping and promoting logic in Switzerland for the last 25 years. It comprises contributions from the symposium "Advances in Proof Theory", which was held in Bern in December 2013. Proof theory came into being in the twenties of the last century, when it was inaugurated by David Hilbert in order to secure the foundations of mathematics. It was substantially influenced by Goedel's famous incompleteness theorems of 1930 and Gentzen's new consistency proof for the axiom system of first order number theory in 1936. Today, proof theory is a well-established branch of mathematical and philosophical logic and one of the pillars of the foundations of mathematics. Proof theory explores constructive and computational aspects of mathematical reasoning; it is particularly suitable for dealing with various questions in computer science.

This is a concise and readable study of five intertwined themes at the heart of Wittgenstein's thought, written by one of his most eminent interpreters. David Pears offers penetrating investigations and lucid explications of some of the most influential and yet puzzling writings of twentieth-century philosophy. He focuses on the idea of language as a picture of the world; the phenomenon of linguistic regularity; the famous "private language argument"; logical necessity; and ego and the self.

This book presents the state of the art in the fields of formal logic pioneered by Graham Priest. It includes advanced technical work on the model and proof theories of paraconsistent logic, in contributions from top scholars in the field. Graham Priest's research has had a considerable influence on the field of philosophical logic, especially with respect to the themes of dialetheism-the thesis that there exist true but inconsistent sentences-and paraconsistency-an account of deduction in which contradictory premises do not entail the truth of arbitrary sentences. Priest's work has regularly challenged researchers to reappraise many assumptions about rationality, ontology, and truth. This book collects original research by some of the most esteemed scholars working in philosophical logic, whose contributions explore and appraise Priest's work on logical approaches to problems in philosophy, linguistics, computation, and mathematics. They provide fresh analyses, critiques, and applications of Priest's work and attest to its continued relevance and topicality. The book also includes Priest's responses to the contributors, providing a further layer to the development of these themes .

This volume collects the most important articles on the metaphysics of modality by noted philosopher Alvin Plantinga. The book chronicles Plantinga's thought from the late 1960's to the present. Plantinga is here concerned with fundamental issues in metaphysics: what is the nature of abstract objects like possible worlds,properties, propositions, and such phenomena? Are there possible but non-actual objects? Can objects that do not exist exemplify properties? In this thorough and searching book, Plantinga addresses these and many other questions that continue to preoccupy philosophers in the field. This volume contains some of the best work in metaphysics from the past 30 years, and will remain a source of critical contention and keen interest among philosophers of metaphysics and philosophical logic for years to come.

Barry Taylor's book mounts an argument against one of the fundamental tenets of much contemporary philosophy, the idea that we can make sense of reality as existing objectively, independently of our capacities to come to know it. Part One sets the scene by arguings that traditional realism can be explicated as a doctrine about truth - that truth is objective, that is, public, bivalent, and epistemically independent. Part Two, the centrepiece of the book, shows how a form of Hilary Putnam's model-theoretic argument demonstrates that no such notion of truth can be founded on the idea of correspondence, as explained in model-theoretic terms (more traditional accounts of correspondence having been already disposed of in Part One). Part Three argues that non-correspondence accounts of truth - truth as superassertibility or idealized rational acceptability, formal conceptions of truth, Tarskian truth - also fail to meet the criteria for objectivity; along the way, it also dismisses the claims of the latterday views of Putnam, and of similar views articulated by John McDowell, to constitute a new, less traditional form of realism. In the Coda, Taylor bolsters some of the considerations advanced in Part Three in evaluating formal conceptions of truth, by assessing and rejecting the claims of Robert Brandom to have combined such an account of truth with a satisfactory account of semantic structure. He concludes that there is no defensible notion of truth which preserves the theses of traditional realism, nor any extant position sufficiently true to the ideals of that doctrine to inherit its title. So the only question remaining is which form of antirealism to adopt.

Published in honor of Sergio Galvan, this collection concentrates on the application of logical and mathematical methods for the study of central issues in formal philosophy. The volume is subdivided into four sections, dedicated to logic and philosophy of logic, philosophy of mathematics, philosophy of science, metaphysics and philosophy of religion. The contributions adress, from a logical point of view, some of the main topics in these areas. The first two sections include formal treatments of: truth and paradoxes; definitions by abstraction; the status of abstract objects, such as mathematical objects and universal concepts; and the structure of explicit knowledge. The last two sections include papers on classical problems in philosophy of science, such as the status of subjective probability, the notion of verisimilitude, the notion of approximation, and the theory of mind and mental causation, and specific issues in metaphysics and philosophy of religion, such as the ontology of species, actions, and intelligible worlds, and the logic of religious belonging.

Kit Fine has since the 1970s been one of the leading contributors to work at the intersection of logic and metaphysics. This is his eagerly-awaited first book in the area. It draws together a series of essays, three of them previously unpublished, on possibility, necessity, and tense. These puzzling aspects of the way the world is have been the focus of considerable philosophical attention in recent decades. Fine gives here the definitive exposition and defence of certain positions for which he is well known: the intelligibility of modality de re; the primitiveness of the modal; and the primacy of the actual over the possible. But the book also argues for several positions that are not so familiar: the existence of distinctive forms of natural and normative necessity, not reducible to any form of metaphysical necessity; the need to make a distinction between the worldly and the unworldly, analogous to the distinction between the tensed and the tenseless; and the viability of a non-standard form of realism about tense, which recognizes the tensed character of reality without conceding that there is any privileged standpoint from which it is to be viewed. Modality and Tense covers a wide range of topics from many different areas: the possible-worlds analysis of counterfactuals; the compatibility of special relativity with presentism; the implications of ethical naturalism; and the nature of first-personal experience. A helpful introduction orients the reader and offers a way into some of the most original work in contemporary philosophy.

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.

F. H. Bradley was the greatest of the British Idealists, but for much of this century his views have been neglected, primarily as a result of the severe criticism to which they were subjected by Russell and Moore. In recent years, however, there has been a resurgence of interest in and a widespread reappraisal of his work. W. J. Mander offers a general introduction to Bradley's metaphysics and its logical foundations, and shows that much of his philosophy has been seriously misunderstood. Dr Mander argues that any adequate treatment of Bradley's thought must take full account of his unique dual inheritance from the traditions of British empiricism and Hegelian rationalism. The scholarship of recent years is assessed, and new interpretations are offered of Bradley's views about truth, predication, and relations, and of his arguments for idealism. This book is a clear and helpful guide for those new to this difficult but fascinating thinker, and at the same time an original and stimulating contribution to the re-evaluation of his work.

Alfred Tarski was one of the two giants of the twentieth-century development of logic, along with Kurt Goedel. The four volumes of this collection contain all of Tarski's papers and abstracts published during his lifetime, as well as a comprehensive bibliography. Here will be found many of the works, spanning the period 1921 through 1979, which are the bedrock of contemporary areas of logic, whether in mathematics or philosophy. These areas include the theory of truth in formalized languages, decision methods and undecidable theories, foundations of geometry, set theory, and model theory, algebraic logic, and universal algebra.

Critical thinking is becoming increasingly prominent as an academic discipline taught and examined in schools and universities, as well as a crucial skill for everyday life. To be a successful critical thinker it is vital to understand how the different concepts and terms are defined and used. The terminology often presents a stumbling block for the beginner, since much of it is used imprecisely in everyday language. This definitive A to Z guide provides precise definitions for over 130 terms and concepts used in critical thinking. Each entry presents a short definition followed by a more detailed explanation and authoritative clarification. Armed with the tools and knowledge provided in these pages, the reader will be able to distinguish an assertion from an argument, a flaw from a fallacy, a correlation from a cause and a fact from an opinion. The book is an invaluable resource for teachers and students of critical thinking, providing all the tools necessary to effectively analyse, evaluate, question and reason for yourself.

Isaac Levi's new book develops further his pioneering work in formal epistemology, focusing on the problem of belief contraction, or how rationally to relinquish old beliefs. Levi offers the most penetrating analysis to date of this key question in epistemology, offering a completely new solution and explaining its relation to his earlier proposals. He mounts an argument in favour of the thesis that contracting a state of belief by giving up specific beliefs is to be evaluated in terms of the value of the information lost by doing so. The rationale aims to be thoroughly decision theoretic. Levi spells out his goals and shows that certain types of recommendations are obtained if one seeks to promote these goals. He compares his approach to his earlier account of inductive expansion. The recommendations are for 'mild contractions'. These are formally the same as the 'severe withdrawals' considered by Pagnucco and Rott. The rationale, however, is different. A critical part of the book concerns the elaboration of these differences. The results are relevant to accounts of the conditions under which it is legitimate to cease believing and to accounts of conditionals. Mild Contraction will be of great interest to all specialists in belief revision theory and to many students of formal epistemology, philosophy of science, and pragmatism.

Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics are covered.

This book presents reflections on the relationship between narratives and argumentative discourse. It focuses on their functional and structural similarities or dissimilarities, and offers diverse perspectives and conceptual tools for analyzing the narratives' potential power for justification, explanation and persuasion. Divided into two sections, the first Part, under the title "Narratives as Sources of Knowledge and Argument", includes five chapters addressing rather general, theoretical and characteristically philosophical issues related to the argumentative analysis and understanding of narratives. We may perceive here how scholars in Argumentation Theory have recently approached certain topics that have a close connection with mainstream discussions in epistemology and the cognitive sciences about the justificatory potential of narratives. The second Part, entitled "Argumentative Narratives in Context", brings us six more chapters that concentrate on either particular functions played by argumentatively-oriented narratives or particular practices that may benefit from the use of special kinds of narratives. Here the focus is either on the detailed analysis of contextualized examples of narratives with argumentative qualities or on the careful understanding of the particular demands of certain well-defined situated activities, as diverse as scientific theorizing or war policing, that may be satisfied by certain uses of narrative discourse.

In three comprehensive volumes, Logic of the Future presents a full panorama of Charles S. Peirce's important late writings. Among the most influential American thinkers, Peirce took his existential graphs to be his greatest contribution to human thought. The manuscripts from 1895-1913, most of which are published here for the first time, testify the richness and open-endedness of his theory of logic and its applications. They also invite us to reconsider our ordinary conceptions of reasoning as well as the conventional stories told about the evolution of modern logic. This second volume collects Peirce's writings on existential graphs related to his Lowell Lectures of 1903, the annus mirabilis of his that became decisive in the development of the mature theory of the graphical method of logic.

'David Pole, in his The Later Philosophy of Wittgenstein, makes an admirable attempt to clarify the central points of Wittgenstein's philosophy in a straightforward manner. He approaches it from the outside with sympathy and good sense. And since he combines a clear head with a fluent style of writing - a combination that is rare among the initiated - his book will prove an excellent introduction for those who need a succinct account of Wittgenstein's later philosophy without any mystical overtones.' - The Economist 'There is now a real need for a commentary on what must in frankness be admitted to be an exceedingly difficult corpus of philosophy. Mr Pole's little book is a response to that need; if small in bulk, it is rich in ideas... and all students of Wittgenstein will turn to it with gratitude.' - Sunday Times

Aristotle's Topics is a handbook for dialectic, which can be understood as a philosophical debate between a questioner and a respondent. In book 2, Aristotle mainly develops strategies for making deductions about 'accidents', which are properties that might or might not belong to a subject (for instance, Socrates has five fingers, but might have had six), and about properties that simply belong to a subject without further specification. In the present commentary, here translated into English for the first time, Alexander develops a careful study of Aristotle's text. He preserves objections and replies from other philosophers whose work is now lost, such as the Stoics. He also offers an invaluable picture of the tradition of Aristotelian logic down to his time, including innovative attempts to unify Aristotle's guidance for dialectic with his general theory of deductive argument (the syllogism), found in the Analytics. The work will be of interest not only for its perspective on ancient logic, rhetoric, and debate, but also for its continuing influence on argument in the Middle Ages and later.

This volume comprises nine lively and insightful essays by leading scholars on the philosophy of Ludwig Wittgenstein, focusing mainly on his early work. The essays are written from a range of perspectives and do not belong to any one exegetical school; they approach Wittgenstein's work directly, seeking to understand it in its own terms and by reference to the context in which it was produced. The contributors cover a wide range of aspects of Wittgenstein's early philosophy, but three central themes emerge: the relationship between Wittgenstein's account of representation and Russell's theories of judgment; the role of objects in the tractarian system; and Wittgenstein's philosophical method. Collectively, the essays demonstrate how progress in the understanding of Wittgenstein's work is not to be made by focusing on overarching, ideological issues, but by paying close attention to his engagement with specific philosophical problems.

Alain Badiou's Being and Event continues to impact philosophical investigations into the question of Being. By exploring the central role set theory plays in this influential work, Burhanuddin Baki presents the first extended study of Badiou's use of mathematics in Being and Event. Adopting a clear, straightforward approach, Baki gathers together and explains the technical details of the relevant high-level mathematics in Being and Event. He examines Badiou's philosophical framework in close detail, showing exactly how it is 'conditioned' by the technical mathematics. Clarifying the relevant details of Badiou's mathematics, Baki looks at the four core topics Badiou employs from set theory: the formal axiomatic system of ZFC; cardinal and ordinal numbers; Kurt Goedel's concept of constructability; and Cohen's technique of forcing. Baki then rebuilds Badiou's philosophical meditations in relation to their conditioning by the mathematics, paying particular attention to Cohen's forcing, which informs Badiou's analysis of the event. Providing valuable insights into Badiou's philosophy of mathematics, Badiou's Being and Event and the Mathematics of Set Theory offers an excellent commentary and a new reading of Badiou's most complex and important work.