This book examines the birth of the scientific understanding of motion. It investigates which logical tools and methodological principles had to be in place to give a consistent account of motion, and which mathematical notions were introduced to gain control over conceptual problems of motion. It shows how the idea of motion raised two fundamental problems in the 5th and 4th century BCE: bringing together being and non-being, and bringing together time and space. The first problem leads to the exclusion of motion from the realm of rational investigation in Parmenides, the second to Zeno's paradoxes of motion. Methodological and logical developments reacting to these puzzles are shown to be present implicitly in the atomists, and explicitly in Plato who also employs mathematical structures to make motion intelligible. With Aristotle we finally see the first outline of the fundamental framework with which we conceptualise motion today.

Logical paradoxes - like the Liar, Russell's, and the Sorites - are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses "dialetheic paraconsistency" - a formal framework where some contradictions can be true without absurdity - as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary.

Evidential Decision Theory is a radical theory of rational decision-making. It recommends that instead of thinking about what your decisions *cause*, you should think about what they *reveal*. This Element explains in simple terms why thinking in this way makes a big difference, and argues that doing so makes for *better* decisions. An appendix gives an intuitive explanation of the measure-theoretic foundations of Evidential Decision Theory.

Thomas Kuhn's The Structure of Scientific Revolutions offers an insightful and engaging theory of science that speaks to scholars across many disciplines. Though initially widely misunderstood, it had a profound impact on the way intellectuals and educated laypeople thought about science. K. Brad Wray traces the influences on Kuhn as he wrote Structure, including his 'Aristotle epiphany', his interactions, and his studies of the history of chemistry. Wray then considers the impact of Structure on the social sciences, on the history of science, and on the philosophy of science, where the problem of theory change has set the terms of contemporary realism/anti-realism debates. He examines Kuhn's frustrations with the Strong Programme sociologists' appropriations of his views, and debunks several popular claims about what influenced Kuhn as he wrote Structure. His book is a rich and comprehensive assessment of one of the most influential works in the modern sciences.

Your complete introduction to logic When you see black clouds looming and predict it's going to rain, you're using logic. When you decide that all swans are white because all the swans you've seen are white, that's logic too (even though the conclusion is false). As humans we want to understand how things are connected and why, so that we can derive things we don''t know yet from what we do. Logic studies the methods and principles to distinguish good and bad reasoning. This book introduces three basic logical systems: categorical logic, propositional logic and predicate logic. It identifies in each system essential methods to test arguments, and there are illustrations and exercises to enhance your mastery of these skills. By the time you finish you will understand what logicians were thinking when they designed logical systems and start to think like them. Whether you are preparing for an essay, studying for an exam or simply want to expand ypour knowledge, Logic: A Complete Introduction is your go-to guide.

Are psychopaths morally responsible? Should we argue with them? Remonstrate with them, blame them, sometimes even praise them? Is it worth trying to change them, or should we just try to prevent them from causing harm? In this book, Jim Baxter aims to find serious answers to these deep philosophical questions, drawing on contemporary insights from psychiatry, psychology, neuroscience and law. Moral Responsibility and the Psychopath is the first sustained, book-length philosophical work on this important and fascinating topic, and will be of deep interest and importance to researchers in these fields - not to mention anyone who has had to interact with a psychopath in their everyday life.

This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike.

The main aim of this Element is to introduce the topic of limited awareness, and changes in awareness, to those interested in the philosophy of decision-making and uncertain reasoning. While it has long been of interest to economists and computer scientists, this topic has only recently been subject to philosophical investigation. Indeed, at first sight limited awareness seems to evade any systematic treatment: it is beyond the uncertainty that can be managed. On the one hand, an agent has no control over what contingencies she is and is not aware of at a given time, and any awareness growth takes her by surprise. On the other hand, agents apparently learn to identify the situations in which they are more and less likely to experience limited awareness and subsequent awareness growth. How can these two sides be reconciled? That is the puzzle we confront in this Element.

This book argues that the meaning of negation, perhaps the most important logical constant, cannot be defined within the framework of the most comprehensive theory of proof-theoretic semantics, as formulated in the influential work of Michael Dummett and Dag Prawitz. Nils Kurbis examines three approaches that have attempted to solve the problem - defining negation in terms of metaphysical incompatibility; treating negation as an undefinable primitive; and defining negation in terms of a speech act of denial - and concludes that they cannot adequately do so. He argues that whereas proof-theoretic semantics usually only appeals to a notion of truth, it also needs to appeal to a notion of falsity, and proposes a system of natural deduction in which both are incorporated. Offering new perspectives on negation, denial and falsity, his book will be important for readers working on logic, metaphysics and the philosophy of language.

This Element presents a concise and accessible view of the central arguments of Wittgenstein's Tractatus Logico-Philosophicus. Starting from the difficulties found in historical and current debates, drawing on the background of Russell's philosophy, and grounded in the ladder structure expressed in the numbering system of the book, this Element presents the central arguments of the Tractatus in three lines of thought. The first concerns the role of the so-called 'ontology' and its relationship to the method of the Tractatus and its logical symbolism, which displays the formal essence of language and world. The second deals with the symbolic unity of language and its role in the 'ladder structure' and explains how and why the book is not self-defeating. The third elucidates Wittgenstein's claim to have solved in essentials all philosophical problems, whose very formulation, he says, rests on misunderstandings.

We are all captivated and puzzled by the infinite, in its many varied guises; by the endlessness of space and time; by the thought that between any two points in space, however close, there is always another; by the fact that numbers go on forever; and by the idea of an all-knowing, all-powerful God. In this acclaimed introduction to the infinite, A. W. Moore takes us on a journey back to early Greek thought about the infinite, from its inception to Aristotle. He then examines medieval and early modern conceptions of the infinite, including a brief history of the calculus, before turning to Kant and post-Kantian ideas. He also gives an account of Cantor's remarkable discovery that some infinities are bigger than others. In the second part of the book, Moore develops his own views, drawing on technical advances in the mathematics of the infinite, including the celebrated theorems of Skolem and Goedel, and deriving inspiration from Wittgenstein. He concludes this part with a discussion of death and human finitude. For this third edition Moore has added a new part, 'Infinity superseded', which contains two new chapters refining his own ideas through a re-examination of the ideas of Spinoza, Hegel, and Nietzsche. This new part is heavily influenced by the work of Deleuze. Also new for the third edition are: a technical appendix on still unresolved questions about different infinite sizes; an expanded glossary; and updated references and further reading. The Infinite, Third Edition is ideal reading for anyone interested in an engaging and historically informed account of this fascinating topic, whether from a philosophical point of view, a mathematical point of view, or a religious point of view.

Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naive and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics.

Our minds have physical effects. This happens, for instance, when we move our bodies when we act. How is this possible? Thomas Kroedel defends an account of mental causation in terms of difference-making: if our minds had been different, the physical world would have been different; therefore, the mind causes events in the physical world. His account not only explains how the mind has physical effects at all, but solves the exclusion problem - the problem of how those effects can have both mental and physical causes. It is also unprecedented in scope, because it is available to dualists about the mind as well as physicalists, drawing on traditional views of causation as well as on the latest developments in the field of causal modelling. It will be of interest to a range of readers in philosophy of mind and philosophy of science. This book is also available as Open Access.

The purpose of this book is to present unpublished papers at the cutting edge of research on dialetheism and to reflect recent work on the applications of the theory. It includes contributions from some of the most respected scholars in the field, as well as from young, up-and-coming philosophers working on dialetheism. Moving from the fringes of philosophy to become a main player in debates concerning truth and the logical paradoxes, dialetheism has thrived since the publication of Graham Priest's In Contradiction, and several of the papers find their roots in a conference on dialetheism held in Glasgow to mark the 25th anniversary of Priest's book. The content presented here demonstrates the considerable body of work produced in this field in recent years. With a broad focus, this book also addresses the applications of dialetheism outside the more familiar area of the logical paradoxes, and includes pieces discussing the application of dialetheism in metaphysics, philosophy of language, and philosophy of mind.

I have been thinking about the philosophical issue of truth for more than two decades. It is one of several fascinating philosophical issues that motivated me to change my primary re ective interest to philosophy after receiving BS in mathem- ics in 1982. Some serious academic work in this connection started around the late eighties when I translated into Chinese a dozen of Donald Davidson's representative essays on truth and meaning and when I assumed translator for Adam Morton who gave a series of lectures on the issue in Beijing (1988), which was co-sponsored by my then institution (Institute of Philosophy, Chinese Academy of Social Science). I have loved the issue both for its own sake (as one speci c major issue in the phil- ophy of language and metaphysics) and for the sake of its signi cant involvement in many philosophical issues in different subjects of philosophy. Having been attracted to the analytic approach, I was then interested in looking at the issue both from the points of view of classical Chinese philosophy and Marxist philosophy, two major styles or frameworks of doing philosophy during that time in China, and from the point of view of contemporary analytic philosophy, which was then less recognized in the Chinese philosophical circle.

The conditional, if...then, is probably the most important term in natural language and forms the core of systems of logic and mental representation. It occurs in all human languages and allows people to express their knowledge of the causal or law-like structure of the world and of others' behaviour, e.g., if you turn the key the car starts, if John walks the dog he stops for a pint of beer; to make promises, e.g., if you cook tonight, I'll wash up all week; to regulate behaviour, e.g., if you are drinking beer, you must be over 18 years of age; to suggest what would have happened had things been different, e.g., if the match had been dry it would have lit, among many other possible uses. The way in which the conditional is modelled also determines the core of most logical systems. Unsurprisingly, it is also the most researched expression in the psychology of human reasoning.
Cognition and Conditionals is the first volume for over 20 years (On Conditionals, 1986, CUP) that brings together recent developments in the cognitive science and psychology of conditional reasoning. Over the last 10 to 15 years, research on conditionals has come to dominate the psychology of reasoning providing a rich seam of results that have created new theoretical possibilities. This book shows how these developments have led researchers to view people's conditional reasoning behaviour more as succesful probabilistic reasoning rather than as errorful logical reasoning. It shows how the multifarious, and apparently competing, theoretical positions developed over the last 50 years in this area - mental logics, mental models, heuristic approaches, dual process theory, and probabilistic approaches-have responded to these insights. Its organisation reflects the view that an integrative approach is emerging that may need to exploit aspects of all these theoretical positions to explain the rich and complex phenomenon of reasoning with conditionals. It includes an introductory chapter relating the development of the psychology of reasoning to developments in the logic and semantics of the conditional. It also includes chapters by many of the leading figures in this field.
Cognition and Conditionals will be a valuable resource for cognitive scientists, psychologists and philosophers interested how people actually reason with conditionals.

Vagueness is the study of concepts that admit borderline cases: the property of being bald is vague because there are people who are neither definitely bald, nor definitely not bald. The epistemology of vagueness concerns the sorts of attitudes we ought to have towards propositions we know to be borderline. Is it possible to discover whether a borderline bald man is bald? Could two people with access to the same facts reasonably disagree about whether he is bald? Does it matter, when making practical decisions, whether he is bald? By drawing on such considerations, Andrew Bacon develops a novel theory of vagueness in which vagueness is fundamentally a property of propositions, and is explicated in terms of its role in thought. On this theory, language plays little role in explaining the central puzzles of vagueness. Part I of the book outlines some of the central questions regarding the logic and epistemology of vagueness, and criticizes some extant approaches to them. Part II concerns issues in the epistemology of vagueness, touching on the ramifications of vague thoughts on the study of evidence, ignorance, desire, probability theory, and decision theory. By examining the effects of vague information on one's beliefs about the precise, a positive theory of vagueness is proposed. Part III concerns the logic of vagueness, including the interaction between vagueness and modality, vague identity, and the paradoxes of higher-order vagueness. Bacon suggests that some familiar philosophical notions - including the concept of a fundamental proposition, a possible world and a precisification - need to be revised.

A long tradition, going back to Aristotle, conceives of logic in terms of necessity and possibility: a deductive argument is correct if it is not possible for the conclusion to be false when the premises are true. A relatively unknown feature of the analytic tradition in philosophy is that, at its very inception, this venerable conception of the relation between logic and necessity and possibility - the concepts of modality - was put into question. The founders of analytic philosophy, Gottlob Frege and Bertrand Russell, held that these concepts are empty: there are no genuine distinctions among the necessary, the possible, and the actual. In this book, the first of two volumes, Sanford Shieh investigates the grounds of this position and its consequences for Frege's and Russell's conceptions of logic. The grounds lie in doctrines on truth, thought, and knowledge, as well as on the relation between mind and reality, that are central to the philosophies of Frege and Russell, and are of enduring philosophical interest. The upshot of this opposition to modality is that logic is fundamental, and, to be coherent, modal concepts would have to be reconstructed in logical terms. This rejection of modality in early analytic philosophy remains of contemporary significance, though the coherence of modal concepts is rarely questioned nowadays because it is generally assumed that suspicion of modality derives from logical positivism, which has not survived philosophical scrutiny. The anti-modal arguments of Frege and Russell, however, have nothing to do with positivism and remain a challenge to the contemporary acceptance of modal notions.

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.

First published in 1986. Routledge is an imprint of Taylor & Francis, an informa company.

An agent often does not have precise probabilities or utilities to guide resolution of a decision problem. I advance a principle of rationality for making decisions in such cases. To begin, I represent the doxastic and conative state of an agent with a set of pairs of a probability assignment and a utility assignment. Then I support a decision principle that allows any act that maximizes expected utility according to some pair of assignments in the set. Assuming that computation of an option's expected utility uses comprehensive possible outcomes that include the option's risk, no consideration supports a stricter requirement.

This study of Plato's Phaedo promotes better understanding of its arguments for the soul's immortality by showing how Plato intended them, not as proofs, but as properly dialectical arguments functioning in accordance with the method of hypothesis. Unlike the argument for the soul's immortality in the Phaedrus, which does seem intended as a proof, the Phaedo arguments are proceeding toward the first principles that could serve as the basis for a proof - the most important being an account of the soul's own essential nature. This study attends to the substantial progress the Phaedo makes toward such an account. It also considers Socrates' epistemic situation in the dialogue and the problem of whether his confidence in the face of death is misplaced if his arguments have not been proofs before considering how the concluding myth draws together several of the dialogue's main themes.

Quine's set theory, New Foundations, has often been treated as an anomaly in the history and philosophy of set theory. In this book, Sean Morris shows that it is in fact well-motivated, emerging in a natural way from the early development of set theory. Morris introduces and explores the notion of set theory as explication: the view that there is no single correct axiomatization of set theory, but rather that the various axiomatizations all serve to explicate the notion of set and are judged largely according to pragmatic criteria. Morris also brings out the important interplay between New Foundations, Quine's philosophy of set theory, and his philosophy more generally. We see that his early technical work in logic foreshadows his later famed naturalism, with his philosophy of set theory playing a crucial role in his primary philosophical project of clarifying our conceptual scheme and specifically its logical and mathematical components.

Are there objects that are "thin" in the sense that not very much is required for their existence? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. Oystein Linnebo aims to do so by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed. The result is a distinctive approach to ontology. Abstract objects such as numbers and sets are demystified and allowed to exist alongside more familiar physical objects. And Linnebo also offers a novel approach to set theory which takes seriously the idea that sets are "formed" successively.

This book articulates and defends Fregean realism, a theory of properties based on Frege's insight that properties are not objects, but rather the satisfaction conditions of predicates. Robert Trueman argues that this approach is the key not only to dissolving a host of longstanding metaphysical puzzles, such as Bradley's Regress and the Problem of Universals, but also to understanding the relationship between states of affairs, propositions, and the truth conditions of sentences. Fregean realism, Trueman suggests, ultimately leads to a version of the identity theory of truth, the theory that true propositions are identical to obtaining states of affairs. In other words, the identity theory collapses the gap between mind and world. This book will be of interest to anyone working in logic, metaphysics, the philosophy of language or the philosophy of mind.