Philosophy

Decision theory aims at a general account of rationality covering humans but to begin makes idealizations about decision problems and agents' resources and circumstances. It treats inerrant agents with unlimited cognitive power facing tractable decision problems. This book systematically rolls back idealizations and without loss of precision treats errant agents with limited cognitive abilities facing decision problems without a stable top option. It recommends choices that maximize utility using quantizations of beliefs and desires in cases where probabilities and utilities are indeterminate and using higher-order utility analysis in cases of limited access to probabilities and utilities. For agents burdened with mistakes, it advocates reasonable attempts to correct unacceptable mistakes before deciding. In decision problems without a stable top option, a topic of game theory, it proposes maximizing self-conditional utility among self-supporting options. In games of strategy, the new principles lead to solutions that are Pareto optimal among equilibria composed of jointly self-supporting strategies. Offering an account of bounded rationality, the bookmakes large strides toward realism in decision theory.

This book is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. It reassesses the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as the understanding of mathematics. The book argues that through the problem of arithmetic participates in the larger puzzle of the relationship between thought, language, experience, and the world, we can distinguish accounts that look to each of these to supply the content we require: those that involve the structure of our experience of the world; those that explicitly involve our grasp of a ‘third realm’ of abstract objects distinct from the concrete objects of the empirical world and the ideas of the author's private Gedankenwelt; those that appeal to something non-physical that is nevertheless an aspect of reality in harmony with which the physical aspect of the world is configured; and finally those that involve only our grasp of language.

This volume collects together 15 papers in which Hale and Wright develop their neo‐Fregean approach to the philosophy of mathematics. Two key aspects of Frege's philosophy are drawn out and defended: (1) the doctrine that mathematics is a body of knowledge about independently existing objects––Frege's platonism (2) the doctrine that this knowledge is a species of logical knowledge, broadly construed––Frege's logicism. In Wright and Hale's programme, a kind of contextual explanation––an Abstraction Principle––suffices to ground the fundamental concepts of a mathematical domain. A key result has become known as Frege's theorem: the basic axioms of arithmetic can be derived from a single ’abstraction principle’. Though Frege himself considers and rejects this principle as a foundation for arithmetic, Neo‐Fregeans take a more optimistic view. In the papers included in this volume, Wright and Hale combine elements drawn from Frege's own work with contemporary thinking to develop their distinctive approach to the philosophy of mathematics. The ground covered includes articles exploring the metaphysics, epistemology, and philosophy of language that forms the backdrop to the neo‐Fregean project; responses to critics of their programme; detailed exploration and defence of the case for neo‐Fregean logicism about arithmetic; and proposals for extending the neo‐Fregean programme to real analysis and set theory. A substantial introduction gives the framework for the neo‐Fregean programme and outlines how the papers fit together, while a postscript outlines a series of challenges for further research.

Although the study of reasons plays an important role in both epistemology and moral philosophy, little attention has been devoted to the question of how, exactly, reason interact to support the actions or conclusions they do. The goal of this book is to answer this question by providing a precise, concrete account of reasons and their interaction, based on the logic of default reasoning. The book begins with an intuitive, accessible introduction to default logic itself, and then argues that this logic can be adapted to serve as a foundation for a concrete theory of reasons. It then shows that the resulting theory helps to explain how the interplay among reasons can determine what we ought to do by developing two different deontic logics, capturing two different intuitions about moral conflicts. The central part of the book elaborates the basic theory to account for reasoning about the strength of our own reasons, and also about the related concepts of undercutting defeaters and exclusionary reasons. The theory is illustrated with an application to particularist arguments concerning the role of principles in moral theory. The book concludes by introducing a pair of issues new to the philosophical literature: the problem of determining the epistemic status of conclusions supported by separate but conflicting reasons, and the problem of drawing conclusions from sets of reasons that can vary aribtrarily in strength, or importance.

The book is an opinionated survey of philosophical work on paradoxes of truth and of related notions, such as property-instantiation, with occasional forays into related topics such as vagueness, the nature of validity, and the Gödel incompleteness theorems. It advocates a particular approach, according to which the paradoxes are to be resolved by the adoption of a non-classical logic: a logic in which excluded middle is restricted. (The logic is quite different from intuitionist logic, which doesn't avoid the paradoxes and also has many unnatural features; and it is much more powerful than the most familiar logic of the paradoxes, the strong Kleene logic, in that it contains a serious conditional.) The book also provides a systematic and detailed look at the main competing approaches. These include Tarski's theory, Kripke's theories, Lukasiewicz's theory, classical gap theories, classical glut theories, supervaluational theories, revision theories, stratified theories, contextual theories, and dialetheic theories. It attempts to compare the virtues of such theories on a range of issues. It also argues against the view that any solution to the paradoxes is inevitably faced with ‘revenge paradoxes’.

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. This book describes and practices a particularly austere form of naturalism called ‘Second Philosophy’. Without a definitive criterion for what counts as ‘science’ and what doesn't, Second Philosophy can't be specified directly — ‘trust only the methods of science!’ or some such thing — so the book proceeds instead by illustrating the behaviors of an idealized inquirer called here the ‘Second Philosopher’. This Second Philosopher begins from perceptual common sense and progresses from there to systematic observation, active experimentation, theory formation, and testing, working all the while to assess, correct, and improve methods along the way. ‘Second Philosophy’ is then the result of the Second Philosopher's investigations. This book delineates the Second Philosopher's approach by tracing reactions to various familiar sceptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. The book then undertakes to practice Second Philosophy in its reflections on the ground of logical truth, the methodology, ontology, and epistemology of mathematics, and the general prospects for metaphysics naturalized.

This book presents a philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. The book offers an account of cardinal and ordinal arithmetic, and the various axiom candidates. It 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. The book offers a simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. The book interweaves a presentation of the technical material with a philosophical critique. The book 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.

Among the various conceptions of truth is one according to which ‘is true’ is a transparent, entirely see-through device introduced for only practical (expressive) reasons. This device, when introduced into the language, brings about truth-theoretic paradoxes (particularly, the notorious Liar and Curry paradoxes). The options for dealing with the paradoxes while preserving the full transparency of ‘true’ are limited. This book presents and defends a modest, so-called dialetheic theory of transparent truth.

Numbers and other mathematical objects are exceptional in having no locations in space and time and no causes or effects in the physical world. This makes it difficult to account for the possibility of mathematical knowledge, leading many philosophers to embrace nominalism, the doctrine that there are no abstract entities. It has also led some of them to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects, eliminating so‐called ontological commitment to numbers, sets, and the like. These projects differ considerably in the apparatus they employ, and the spirit in which they are put forward. Some employ synthetic geometry, others modal logic. Some are put forward as revolutionary replacements for existing mathematics and science, others hermeneutic hypotheses about what they have meant all along. We attempt to cut through technicalities that have obscured previous discussions of these projects, and to present concise accounts with minimal prerequisites of a dozen strategies for nominalistic interpretation of mathematics. We also examine critically the aims and claims of such interpretations, suggesting that what they really achieve is something quite different from what the authors of such projects usually assume.

This book poses a broad challenge to the realist views of meaning and truth that have been prominent in recent philosophy. The book starts with a careful critical survey of the realism debate, guiding the reader through its complexities; it then presents a sustained defence of the anti-realist view that every truth is knowable in principle, and that grasp of meaning must be able to be made manifest. Sceptical arguments for the indeterminacy or non-factuality of meaning are countered; and the much-maligned notion of analyticity is reinvestigated and rehabilitated. The book goes on to show that an effective logical system can be based on an anti-realist view; the logical system that he advocates is justified as a body of analytic truths and inferential principles. Having laid the foundations for global semantic anti-realism, the book moves to the world of empirical understanding, and gives an account of the cognitive credentials of natural scientific discourse. The book shows that the same canon of constructive and relevant inference suffices both for intuitionistic mathematics and for empirical science.