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.

This book is dedicated to the life and work of the mathematician Joachim Lambek (1922-2014). The editors gather together noted experts to discuss the state of the art of various of Lambek's works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist (notably in ring theory). In the 1960s, he began to work in category theory, categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard's linear logic. In a surprising development (2000), he introduced a novel and deeper algebraic framework (which he called pregroup grammars) for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists.

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.

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.

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.

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.

'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

This book offers a comprehensive critical survey of issues of historical interpretation and evaluation in Bertrand Russell's 1918 logical atomism lectures and logical atomism itself. These lectures record the culmination of Russell's thought in response to discussions with Wittgenstein on the nature of judgement and philosophy of logic and with Moore and other philosophical realists about epistemology and ontological atomism, and to Whitehead and Russell's novel extension of revolutionary nineteenth-century work in mathematics and logic. Russell's logical atomism lectures have had a lasting impact on analytic philosophy and on Russell's contemporaries including Carnap, Ramsey, Stebbing, and Wittgenstein. Comprised of 14 original essays, this book will demonstrate how the direct and indirect influence of these lectures thus runs deep and wide.

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.

Current edition description Written by two of the leading experts in critical thinking, this book focuses on an integrated, universal concept of critical thinking that is both substantive and applicable to any and every situation in which human thinking is necessary. It provides readerse with the basic intellectual tools needed for life-long learning, helping them understand the mind and how its three functions - thinking, feeling, motivation - influence and are influenced by one another. This book fosters the development of fair-minded critical thinking. Features the intellectual standards: clarity, precision, accuracy, logicalness, significance, depth, breadth, and fairness; The importance of good questioning; and intellectual tools to read for deep and lasting comprehension, and to write in ways that show clarity of reasonability of thought. For all that want to improve their critical thinking skills to apply to their job or life. The text features: Think for Yourself activities - throughout each chapter. (Ex. pp 29, 127). ~Help students take ownership of basic concepts as they learn them. Practical and learnable format. ~Simplifies complex ideas to make learning easier for students. Focus on thinking across the disciplines. (Ex. pp 119-120). ~Helps students to think within the various disciplines, rather than memorizing facts. Students are taught to learn to think like an historian, like a scientist, like a psychologist, etc. Critical thinking focus - When students internalize intellectual standards - such as clarity, precision, accuracy, logicalness, significance, depth, breadth, and fairness - they use them on a daily basis to upgrade their thinking, and to assess the thinking of others. (Ex. 12, 152). ~Gives students intellectual standards they can use in every dimension of their thinking. Features intellectual tools to read for deep and lasting comprehension, and to write in ways that show clarity of reasonability of thought. (Ex. 133). ~Teaches students to read closely and write substantively. Good questions are the key to good thinking - Thinkers who know how to ask relevant questions in context are better able to think their way through complex issues. (Ex. pp 83, 87, 93). ~Teaches students to ask the questions the best thinkers ask. Website - www.criticalthinking.org. ~Links students to the world's largest and most prestigious critical thinking website and provides forums for student and faculty discussions. International approach - with translations into German, French, Spanish, Japanese, Chinese, Russian, and Malay. ~Provides students with the opportunity to read portions of the text in their native language. "This book is well-written, lucid and contains abundant examples and applications that not only enliven the subject matter but present relevant contexts for building understanding and advanced critical thinking. In addition, it is faithful to the complexity and work required to improve one's thinking. It does not soft-pedal the challenge but actually throws down the gauntlet to the worthy Reader to pick it up." --Stephen J. Knopp, Ph.D., Ohio University "This concise version is a more comprehensive and robust textbook. Many Critical Thinking books cover thinking from a narrow angle, but Paul and Elder offer a model of critical thinking that can be applied not only to academic disciplines but also to life in general." --Connie Wolfe, Surry Community College

Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters. The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes.

The volume is published on the occasion of the birth centennial of Eugenio Coseriu (1921-2002). It is the first collective volume to appear in English in which various scholars present a variety of perspectives on Coseriu's scholarly work and discuss its continuing relevance for the language sciences. Coseriu's international reputation has suffered from his commitment to publish in languages such as Spanish, German, French, Italian, Romanian and Portuguese, to the detriment of English. As a consequence, his work is less well-known outside Romance and German linguistics. The volume aims to raise the general awareness of Coseriu's work among linguists around the world, in accordance with Coseriu's own adage that it takes a constructive mindset (acknowledging "accomplishments and limitations") to do justice to all scholarly work in the humanities. The articles are organized into three major thematic clusters: 1) philosophy of language, 2) history of the language sciences and 3) theory and practice of "Integral Linguistics". The volume is essential reading for anyone working in these fields and for those seeking to gain deeper understanding of Coseriu's goal to develop a unitary approach to language which takes as its point of departure the "activity of speaking".

This book discusses how scientific and other types of cognition make use of models, abduction, and explanatory reasoning in order to produce important or creative changes in theories and concepts. It includes revised contributions presented during the international conference on Model-Based Reasoning (MBR'015), held on June 25-27 in Sestri Levante, Italy. The book is divided into three main parts, the first of which focuses on models, reasoning and representation. It highlights key theoretical concepts from an applied perspective, addressing issues concerning information visualization, experimental methods and design. The second part goes a step further, examining abduction, problem solving and reasoning. The respective contributions analyze different types of reasoning, discussing various concepts of inference and creativity and their relationship with experimental data. In turn, the third part reports on a number of historical, epistemological and technological issues. By analyzing possible contradictions in modern research and describing representative case studies in experimental research, this part aims at fostering new discussions and stimulating new ideas. All in all, the book provides researchers and graduate students in the field of applied philosophy, epistemology, cognitive science and artificial intelligence alike with an authoritative snapshot of current theories and applications of model-based reasoning.

Forms of Truth and the Unity of Knowledge addresses a philosophical subject-the nature of truth and knowledge-but treats it in a way that draws on insights beyond the usual confines of modern philosophy. This ambitious collection includes contributions from established scholars in philosophy, theology, mathematics, chemistry, biology, psychology, literary criticism, history, and architecture. It represents an attempt to integrate the insights of these disciplines and to help them probe their own basic presuppositions and methods. The essays in Forms of Truth and the Unity of Knowledge are collected into five parts, the first dealing with division of knowledge into multiple disciplines in Western intellectual history; the second with the foundational disciplines of epistemology, logic, and mathematics; the third with explanation in the natural sciences; the fourth with truth and understanding in disciplines of the humanities; and the fifth with art and theology. Contributors: Vittorio Hoesle, Keith Lehrer, Robert Hanna, Laurent Lafforgue, Thomas Nowak, Francisco J. Ayala, Zygmunt Pizlo, Osborne Wiggins, Allan Gibbard, Carsten Dutt, Aviezer Tucker, Nicola Di Cosmo, Michael Lykoudis, and Celia Deane-Drummond.

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.

A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

This new digital edition of The Trial and Death of Socrates: Euthyphro, Apology, Crito and Phaedo presents Benjamin Jowett's classic translations, as revised by Enhanced Media Publishing. A number of new or expanded annotations are also included.

This book introduces a new form of argumentative analysis: rhetorical heuremes. The method applies the concepts of heuristic thinking, probability, and contingency in order to develop a better understanding of complex arguments in classical oratory. A new theory is required because Greek and Roman rhetoric cannot provide detailed answers to problems of strategic argumentation in the analysis of speeches. Building on scholarship in Ciceronian oratory, this book moves beyond the extant terminology and employs a concept of heuristic reasoning derived from the psychology of decision making and mathematical problem solving. The author analyses selected passages from Cicero's forensic speeches where arguments of probability are deployed, and shows that the Sophistic concept of probability can link ancient rhetoric and modern theories of argumentation. Six groups of heuremes are identified, each of which represents a form of probabilistic reasoning by which the orator plays upon the perception of the jurors.

Saul Kripke's Naming and Necessity was one of the most influential philosophical works of the twentieth century. In this collection of essays leading specialists explore issues arising from this and other works of Kripke's.