A comprehensive collection of historical readings in the philosophy of mathematics and a selection of influential contemporary work, this much-needed introduction reveals the rich history of the subject. An Historical Introduction to the Philosophy of Mathematics: A Reader brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates the development of the subject. Presenting historical background essential to understanding contemporary trends and a survey of recent work, An Historical Introduction to the Philosophy of Mathematics: A Reader is required reading for undergraduates and graduate students studying the philosophy of mathematics and an invaluable source book for working researchers.

According to Bayesian epistemology, rational learning from experience is consistent learning, that is learning should incorporate new information consistently into one's old system of beliefs. Simon M. Huttegger argues that this core idea can be transferred to situations where the learner's informational inputs are much more limited than Bayesianism assumes, thereby significantly expanding the reach of a Bayesian type of epistemology. What results from this is a unified account of probabilistic learning in the tradition of Richard Jeffrey's 'radical probabilism'. Along the way, Huttegger addresses a number of debates in epistemology and the philosophy of science, including the status of prior probabilities, whether Bayes' rule is the only legitimate form of learning from experience, and whether rational agents can have sustained disagreements. His book will be of interest to students and scholars of epistemology, of game and decision theory, and of cognitive, economic, and computer sciences.

This book analyses and defends the deflationist claim that there is nothing deep about our notion of truth. According to this view, truth is a 'light' and innocent concept, devoid of any essence which could be revealed by scientific inquiry. Cezary Cieslinski considers this claim in light of recent formal results on axiomatic truth theories, which are crucial for understanding and evaluating the philosophical thesis of the innocence of truth. Providing an up-to-date discussion and original perspectives on this central and controversial issue, his book will be important for those with a background in logic who are interested in formal truth theories and in current philosophical debates about the deflationary conception of truth.

Driven by an intercultural vision and comparative research, this is the first volume to bring together African and Western philosophical approaches to current issues in language, logic, metaphysics and ethics. Featuring contributions from an international line up of scholars across four continents including leading contemporary African philosophers, and new work from Saul Kripke in collaboration with Romina Padro, these chapters cover a wide range of topics. They advance our understanding of the linguistic, logical, metaphysical and ethical contributions of African philosophical thought in comparison with Western traditions, fostering discussion about how they relate, and are distinct, from one another. Reinvigorating the debate in cross-cultural and global philosophy, this is a groundbreaking resource for any scholar or student of each sub-discipline.

While probabilistic logics in principle might be applied to solve a range of problems, in practice they are rarely applied - perhaps because they seem disparate, complicated, and computationally intractable. This programmatic book argues that several approaches to probabilistic logic fit into a simple unifying framework in which logically complex evidence is used to associate probability intervals or probabilities with sentences. Specifically, Part I shows that there is a natural way to present a question posed in probabilistic logic, and that various inferential procedures provide semantics for that question, while Part II shows that there is the potential to develop computationally feasible methods to mesh with this framework. The book is intended for researchers in philosophy, logic, computer science and statistics. A familiarity with mathematical concepts and notation is presumed, but no advanced knowledge of logic or probability theory is required.

This book introduces the theory of graded consequence (GCT) and its mathematical formulation. It also compares the notion of graded consequence with other notions of consequence in fuzzy logics, and discusses possible applications of the theory in approximate reasoning and decision-support systems. One of the main points where this book emphasizes on is that GCT maintains the distinction between the three different levels of languages of a logic, namely object language, metalanguage and metametalanguage, and thus avoids the problem of violation of the principle of use and mention; it also shows, gathering evidences from existing fuzzy logics, that the problem of category mistake may arise as a result of not maintaining distinction between levels.

Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical logic, it also enjoys connections to and applications in diverse branches of mathematics, including algebra, number theory and analysis. Despite this, traditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics, paying particular attention to definable sets throughout. With numerous exercises of varying difficulty, this is an accessible introduction to model theory and its place in mathematics.

Metamathematics and the Philosophical Tradition is the first work to explore in such historical depth the relationship between fundamental philosophical quandaries regarding self-reference and meta-mathematical notions of consistency and incompleteness. Using the insights of twentieth-century logicians from Goedel through Hilbert and their successors, this volume revisits the writings of Aristotle, the ancient skeptics, Anselm, and enlightenment and seventeenth and eighteenth century philosophers Leibniz, Berkeley, Hume, Pascal, Descartes, and Kant to identify ways in which these both encode and evade problems of a priori definition and self-reference. The final chapters critique and extend more recent insights of late 20th-century logicians and quantum physicists, and offer new applications of the completeness theorem as a means of exploring "metatheoretical ascent" and the limitations of scientific certainty. Broadly syncretic in range, Metamathematics and the Philosophical Tradition addresses central and recurring problems within epistemology. The volume's elegant, condensed writing style renders accessible its wealth of citations and allusions from varied traditions and in several languages. Its arguments will be of special interest to historians and philosophers of science and mathematics, particularly scholars of classical skepticism, the Enlightenment, Kant, ethics, and mathematical logic.

Carnap, Quine, and Putnam held that in our pursuit of truth we can do no better than to start in the middle, relying on already-established beliefs and inferences and applying our best methods for re-evaluating particular beliefs and inferences and arriving at new ones. In this collection of essays, Gary Ebbs interprets these thinkers' methodological views in the light of their own philosophical commitments, and in the process refutes some widespread misunderstandings of their views, reveals the real strengths of their arguments, and exposes a number of problems that they face. To solve these problems, in many of the essays Ebbs also develops new philosophical approaches, including new theories of logical truth, language use, reference and truth, truth by convention, realism, trans-theoretical terms, agreement and disagreement, radical belief revision, and contextually a priori statements. His essays will be valuable for a wide range of readers in analytic philosophy.

Interest in the metaphysics and logic of possible worlds goes back at least as far as Aristotle, but few books address the history of these important concepts. This volume offers new essays on the theories about the logical modalities (necessity and possibility) held by leading philosophers from Aristotle in ancient Greece to Rudolf Carnap in the twentieth century. The story begins with an illuminating discussion of Aristotle's views on the connection between logic and metaphysics, continues through the Stoic and mediaeval (including Arabic) traditions, and then moves to the early modern period with particular attention to Locke and Leibniz. The views of Kant, Peirce, C. I. Lewis and Carnap complete the volume. Many of the essays illuminate the connection between the historical figures studied, and recent or current work in the philosophy of modality. The result is a rich and wide-ranging picture of the history of the logical modalities.

Computability and Logic has become a classic because of its accessibility to students without a mathematical background and because it covers not simply the staple topics of an intermediate logic course, such as Godel's incompleteness theorems, but also a large number of optional topics, from Turing's theory of computability to Ramsey's theorem. This 2007 fifth edition has been thoroughly revised by John Burgess. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a simpler treatment of the representability of recursive functions, a traditional stumbling block for students on the way to the Godel incompleteness theorems. This updated edition is also accompanied by a website as well as an instructor's manual.

'Don't hope that events will turn out the way you want, welcome events in whichever way they happen' How can we cope when life's events seem beyond our control? These words of consolation and inspiration from the three great Stoic philosophers - Epictetus, Seneca and Marcus Aurelius - offer ancient wisdom on how to face life's adversities and live well in the world. One of twenty new books in the bestselling Penguin Great Ideas series. This new selection showcases a diverse list of thinkers who have helped shape our world today, from anarchists to stoics, feminists to prophets, satirists to Zen Buddhists.

In The Open Future: Why Future Contingents are all False, Patrick Todd launches a sustained defense of a radical interpretation of the doctrine of the open future. He argues that all claims about undetermined aspects of the future are simply false. Todd argues that this theory is metaphysically more parsimonius than its rivals, and that objections to its logical and practical coherence are much overblown. Todd shows how proponents of this view can maintain classical logic, and argues that the view has substantial advantages over Ockhamist, supervaluationist, and relativist alternatives. Todd draws inspiration from theories of ''neg-raising'' in linguistics, from debates about omniscience within the philosophy of religion, and defends a crucial comparison between his account of future contingents and certain more familiar theories of counterfactuals. Further, Todd defends his theory of the open future from the charges that it cannot make sense of our practices of betting, makes our credences regarding future contingents unintelligible, and is at odds with proper norms of assertion. In the end, in Todd's classical open future, we have a compelling new solution to the longstanding "problem of future contingents".

This volume deals with the connection between thinking-and-speaking and our form(s) of life. All contributions engage with Wittgenstein's approach to this topic. As a whole, the volume takes a stance against both biological and ethnological interpretations of the notion "form of life" and seeks to promote a broadly logico-linguistic understanding instead. The structure of this book is threefold. Part one focuses on lines of thinking that lead from Wittgenstein's earlier thought to the concept of form of life in his later work. Contributions to part two examine the concrete philosophical function of this notion as well as the ways in which it differs from cognate concepts. Contributions to part three put Wittgenstein's notion of form of life in perspective by relating it to phenomenology, ordinary language philosophy and problems in contemporary analytic philosophy.

This book deals with an old conundrum: if God knows what we will choose tomorrow, how can we be free to choose otherwise? If all our choices are already written, is our freedom simply an illusion? This book provides a precise analysis of this dilemma using the tools of modern metaphysics and logic of time. With a focus on three intertwined concepts - God's nature, the formal structure of time, and the metaphysics time, including the relationship between temporal entities and a timeless God - the chapters analyse various solutions to the problem of foreknowledge and freedom, revealing the advantages and drawbacks of each. Building on this analysis, the authors advance constructive solutions, showing under what conditions an entity can be omniscient in the presence of free agents, and whether an eternal entity can know the tensed futures of the world. The metaphysics of time, its topology and the semantics of future tensed sentences are shown to be invaluable topics in dealing with this issue. Combining investigations into the metaphysics of time with the discipline of temporal logic this monograph brings about important advancements in the philosophical understanding of an ancient and fascinating problem. The answer, if any, is hidden in the folds of time, in the elusive nature of this feature of reality and in the infinite branching of our lives.

Many philosophy majors are shocked by the gap between the relative ease of lower-level philosophy courses and the difficulty of upper-division courses. This book serves as a necessary bridge to upper-level study in philosophy by offering rigorous but concise and accessible accounts of basic concepts and distinctions that are used throughout the discipline. It serves as a valuable advanced introduction to any undergraduate who is moving into upper-level courses in philosophy. While lower-level introductions to philosophy usually deal with popular topics accessible to the general student (such as contemporary moral issues, free will, and personal identity) in a piecemeal fashion, The Philosophy Major's Introduction to Philosophy offers coverage of important general philosophical concepts, tools, and devices that may be used for a long time to come in various philosophical areas. The volume is helpfully divided between a focus on the relation between language and the world in the first three chapters and coverage of mental content in the final two chapters, but builds a coherent narrative from start to finish. It also provides ample study questions and helpful signposts throughout, making it a must-have for any student attempting to engage fully with the problems and arguments in philosophy. Key Features Integrates topics from various areas of philosophy, such as philosophy of language, metaphysics, epistemology, ethics, and philosophical logic Provides descriptions of logico-mathematical tools necessary for philosophical studies, such as propositional logic, predicate logic, modal logic, set theory, mereology, and mathematical functions Makes connections with modern philosophy, including discussions of Descartes's skepticism and dualism, Locke's theory of personal identity, Hume's theory of causation, and Kant's synthetic a priori Includes well-known entertaining puzzles and thought experiments such as the Ship of Theseus, the Statue and the Clay, a Brain in a Vat, and Twin Earth Lists helpful Exercise Questions and Discussion Questions at the end of each chapter and answers selected questions at the back of the book

Many philosophy majors are shocked by the gap between the relative ease of lower-level philosophy courses and the difficulty of upper-division courses. This book serves as a necessary bridge to upper-level study in philosophy by offering rigorous but concise and accessible accounts of basic concepts and distinctions that are used throughout the discipline. It serves as a valuable advanced introduction to any undergraduate who is moving into upper-level courses in philosophy. While lower-level introductions to philosophy usually deal with popular topics accessible to the general student (such as contemporary moral issues, free will, and personal identity) in a piecemeal fashion, The Philosophy Major's Introduction to Philosophy offers coverage of important general philosophical concepts, tools, and devices that may be used for a long time to come in various philosophical areas. The volume is helpfully divided between a focus on the relation between language and the world in the first three chapters and coverage of mental content in the final two chapters, but builds a coherent narrative from start to finish. It also provides ample study questions and helpful signposts throughout, making it a must-have for any student attempting to engage fully with the problems and arguments in philosophy. Key Features Integrates topics from various areas of philosophy, such as philosophy of language, metaphysics, epistemology, ethics, and philosophical logic Provides descriptions of logico-mathematical tools necessary for philosophical studies, such as propositional logic, predicate logic, modal logic, set theory, mereology, and mathematical functions Makes connections with modern philosophy, including discussions of Descartes's skepticism and dualism, Locke's theory of personal identity, Hume's theory of causation, and Kant's synthetic a priori Includes well-known entertaining puzzles and thought experiments such as the Ship of Theseus, the Statue and the Clay, a Brain in a Vat, and Twin Earth Lists helpful Exercise Questions and Discussion Questions at the end of each chapter and answers selected questions at the back of the book

David E. Over is a leading cognitive scientist and, with his firm grounding in philosophical logic, he also exerts a powerful influence on the psychology of reasoning. He is responsible for not only a large body of empirical work and accompanying theory, but for advancing a major shift in thinking about reasoning, commonly known as the 'new paradigm' in the psychology of human reasoning. Over's signature mix of philosophical logic and experimental psychology has inspired generations of researchers, psychologists, and philosophers alike over more than a quarter of a century. The chapters in this volume, written by a leading group of contributors including a number who helped shape the psychology of reasoning as we know it today, each take their starting point from the key themes of Over's ground-breaking work. The essays in this collection explore a wide range of central topics-such as rationality, bias, dual processes, and dual systems-as well as contemporary psychological and philosophical theories of conditionals. It concludes with an engaging new chapter, authored by David E. Over himself, which details and analyses the new paradigm psychology of reasoning. This book is therefore important reading for scholars, researchers, and advanced students in psychology, philosophy, and the cognitive sciences, including those who are not familiar with Over's thought already.

This book is unique in presenting an interdisciplinary conversation between jurists and logicians. It brings together scholars from both law and philosophy, and looks at the application of 'the new logics' to law and legal ordering, in a number of legal systems. The first Part explores the ways in which the new logics shed light on the functioning of legal orders, including the structure of legal argumentation and the rules of evidence. The second addresses how non-classical logics can help us to understand the interactions between multiple legal orders, in a range of contexts including domestic and international law. The final Part examines particular issues in the applicability of non-classical logics to legal reasoning. This book will be of interest to jurisprudence and logic scholars and students who want to deepen their understanding of relationships between law and legal reasoning, and learn about recent developments in formal logic.

The Process of Argument: An Introduction is a necessary companion for anyone seeking to engage in successful persuasion: To organize, construct, and communicate arguments. It is both comprehensive and accessible: An authoritative guide to logical thinking and effective communication. The book begins with techniques to improve reading comprehension, including guides on navigating through fake news and internet trolls. Then, readers are taught how to reconstruct deductive, inductive, and abductive presentations so that the logical structure is explicit. And finally, there is a step-by-step guide for responding to these texts via the argumentative essay. Along the way are current examples from social media and elsewhere on the internet along with guides for assessing truth claims in an ever-complicated community worldview. Throughout, are carefully selected reading questions and exercises that will pace readers in order to ensure that the text is securely grasped and successfully applied. Key Features Offers guidance on how to read a text through self-analysis and social criticism Provides a step-by-step procedure for allowing the student to move from reading to reconstruction to being prepared to write an effective argumentative essay Presents truth theory and shows readers how they can helpfully acquaint themselves with a version of realistic, foundational epistemology Offers guidelines and helpful tools on how best to structure an argumentative, pro or con, essay Includes expansive coverage of inductive logic through the use and assessment of statistics Covers abductive logic as it applies to the analysis of narrative in argumentative writing Has up-to-date examples from the media, including from blogs, social media, and television Includes a helpful glossary of all important terms in the book

This book presents a systematic unifying-pluralist account-a "constructive-engagement" account-of how cross-tradition engagement in philosophy is possible. The goal of this "constructive-engagement" account is, by way of reflective criticism, argumentation, and methodological guiding principles, to inquire into how distinct approaches from different philosophical traditions can talk to and learn from each other for the sake of making joint contributions to the contemporary development of philosophy. In Part I of the book, Bo Mou explores a range of fundamental theoretic and methodological issues in cross-tradition philosophical engagement and philosophical interpretation. In Part II, he analyzes several representative case studies that demonstrate how relevant resources in the Western and Chinese philosophical traditions can constructively engage with each other. These studies cover issues in philosophical methodology, metaphysics, epistemology, philosophy of language and logic, and ethics. The book's theoretical and practical approaches expand the vision, coverage, and agenda of doing philosophy comparatively, and promote worldwide joint efforts of cross-tradition philosophical inquiries. Cross-Tradition Engagement in Philosophy will be of interest to graduate students and scholars interested in comparative philosophy and the intersection of Chinese and Western philosophy. It will also appeal to those who are interested in the ways in which cross-tradition philosophical engagement can enhance contemporary philosophical debates in metaphysics, epistemology, philosophy of language and logic, and ethics.

Traditionally, Aristotle is held to believe that philosophical contemplation is valuable for its own sake, but ultimately useless. In this volume, Matthew D. Walker offers a fresh, systematic account of Aristotle's views on contemplation's place in the human good. The book situates Aristotle's views against the background of his wider philosophy, and examines the complete range of available textual evidence (including neglected passages from Aristotle's Protrepticus). On this basis, Walker argues that contemplation also benefits humans as perishable living organisms by actively guiding human life activity, including human self-maintenance. Aristotle's views on contemplation's place in the human good thus cohere with his broader thinking about how living organisms live well. A novel exploration of Aristotle's views on theory and practice, this volume will interest scholars and students of both ancient Greek ethics and natural philosophy. It will also appeal to those working in other disciplines including classics, ethics, and political theory.

There can be no doubt that Kant thought we should be reflective: we ought to care to make up our own minds about how things are and what is worth doing. Philosophical objections to the Kantian reflective ideal have centred on concerns about the excessive control that the reflective person is supposed to exert over their own mental life, and Kantians who feel the force of these objections have recently drawn attention to Kant's conception of moral virtue as it is developed in his later work, chiefly the Metaphysics of Morals. Melissa Merritt's book is a distinctive contribution to this recent turn to virtue in Kant scholarship. Merritt argues that we need a clearer, and textually more comprehensive, account of what reflection is, in order not only to understand Kant's account of virtue, but also to appreciate how it effectively rebuts long-standing objections to the Kantian reflective ideal.

Recent work in argumentation theory has emphasized the nature of arguers and arguments along with various theoretical perspectives. Less attention has been given to the third feature of any argumentative situation - the audience. This book fills that gap by studying audience reception to argumentation and the problems that come to light as a result of this shift in focus. Christopher W. Tindale advances the tacit theories of several earlier thinkers by addressing the central problems connected with audience considerations in argumentation, problems that earlier philosophical theories overlook or inadequately accommodate. The main tools employed in exploring the central issues are drawn from contemporary philosophical research on meaning, testimony, emotion and agency. These are then combined with some of the major insights of recent rhetorical work in argumentation to advance our understanding of audiences and suggest avenues for further research.