Philosophy

Mathematics depends on proofs, and proofs have to begin somewhere, from some fundamental assumptions. Chapter I traces the historical rise of pure mathematics and the development of set theory, eventually axiomatic set theory, to play this foundational role for contemporary classical mathematics. Here the Euclidean ideal of postulates that are simply obvious or self-evident can't be the whole story, which raises two basic questions: what are the proper methods for defending set-theoretic axioms? And, why are these the proper methods? Chapter II introduces the meta-philosophical perspective, called Second Philosophy, from which the inquiry into these questions will take place, and identifies straightforward mathematical answers to the first question. Addressing the second requires engagement with the troublesome ontological and epistemological issues that have dogged the philosophy of mathematics from its beginnings. Chapters III and IV describe and explore two apparently conflicting stands on these issues—called Thin Realism and Arealism—not so much to recommend either one, but with an eye to suggesting that the question of which is correct has less bite than it might appear. In the end, the hope is to shift attention away from these elusive matters of truth and existence, and to direct it toward the distinctive type of mathematical objectivity emphasized in the opening section of Chapter V. The concluding sections of chapter V return, at last, to the question of set-theoretic method and draw some concrete morals for the project of defending the axioms.

What in our theoretical pronouncements commits us to objects? The Quinean standard for ontological commitment involves (nearly enough) commitments when we utter “there is” or “there are” statements without hope of eliminating these by paraphrase. Coupled with the indispensability of the truth of applied mathematical doctrine, the result is that the ontologically hard-nosed scientist is a Platonist—haplessly commited to abstracta. In this book Azzouni offers a way around the Quinean straitjacket: ontological commitment turns on how theories are (nearly enough) nailed to the world. The specifics of how theories are applied indicates which among the posits of a theory are mere mathematical garb and which are genuine connections to items out there. In the first part of the book Azzouni undercuts the arguments, both actual and possible, in support of Quine’s criterion. An alternative criterion for what exists—ontological independence—is offered, one in sturdy accord with ordinary folk views on the matter. In the second part of the book, a beginning is made of bringing this alternative to bear upon scientific theories with a rich mathematical component. Along the way, old philosophical issues about absolute space and time versus relative space and time, the status of mathematical posits, such as spatial and temporal points, and so on, are illuminated.

This book presents a history of modern logic from the Middle Ages through the end of the 20th century. In addition to a history of symbolic logic, the book also examines developments in the philosophy of logic and philosophical logic in modern times. The book begins with chapters on late medieval developments and logic and philosophy of logic from Humanism to Kant. The following chapters focus on the emergence of symbolic logic with special emphasis on the relations between logic and mathematics, on the one hand, and on logic and philosophy, on the other. This discussion is completed by a chapter on the themes of judgment and inference from 1837–1936. The book contains a section on the development of mathematical logic from 1900–1935, followed by a section on main trends in mathematical logic after the 1930s. The book goes on to discuss modal logic from Kant till the late 20th century, and logic and semantics in the 20th century; the philosophy of alternative logics; the philosophical aspects of inductive logic; the relations between logic and linguistics in the 20th century; the relationship between logic and artificial intelligence; and ends with a presentation of the main schools of Indian logic.

The Law of Non-Contradiction has been high orthodoxy in Western philosophy since Aristotle. The so-called Law has been the subject of radical challenge in recent years by dialetheism, the view that some contradictions are indeed true. Many philosophers have taken the Law to be central to many of our most important philosophical concepts. This book mounts the case against this view. Starting with an analysis of Aristotle on the Law, it discusses the nature of truth, rationality, negation, and logic itself, and argues that the Law is inessential to all of these things. The book develops Priest’s earlier ideas in In Contradiction.

This book is a collection of papers to celebrate the life and work of Jonathan Barnes. The papers cover a wide range of topics in ancient epistemology, ethics, metaphysics, and logic, with two papers on contemporary issues in logic.

This book attempts to show how rule-following can be understood as an essential element of rational action. The first step involves showing how rational choice theory can be modified to incorporate deontic constraint as a feature of rational deliberation. The second involves disarming the suspicion that there is something mysterious or irrational about the psychological states underlying rule-following. Human rationality is interpreted as a by-product of the so-called “language upgrade” that we receive as a consequence of the development of specific social practices. As a result, certain constitutive features of our social environment — such as the rule-governed structure of social life — migrate inwards, and become constitutive features of our psychological faculties. This in turn explains why there is an indissoluble bond between practical rationality and deontic constraint. In the end, the book offers a naturalistic, evolutionary argument in favor of the traditional Kantian view that there is an internal connection between being a rational agent and feeling the force of one's moral obligations.

A language is second‐order, or higher‐order, if it has bound variables that range over properties or sets of the items in the range of the ordinary, first‐order variables. This book presents a formal development of second‐ and higher‐order logic and an extended argument that higher‐order systems have an important role to play in the philosophy and foundations of mathematics. The development includes the languages, deductive systems, and model‐theoretic semantics for higher‐order languages, and the basic and advanced results in its meta‐theory: completeness, compactness, and the Löwenheim–Skolem theorems for Henkin semantics, and the failure of those results for standard semantics. Argues that second‐order theories and formalizations, with standard semantics, provide better models of important aspects of mathematics than their first‐order counterparts. Despite the fact that Quine is the main opponent of second‐order logic (arguing that second‐order logic is set‐theory in disguise), the present argument is broadly Quinean, proposing that there is no sharp line dividing mathematics from logic, especially the logic of mathematics. Also surveys the historical development in logic, tracing the emergence of first‐order logic as the de facto standard among logicians and philosophers. The connection between formal deduction and reasoning is related to Wittgensteinian issues concerning rule‐following. The book closes with an examination of several alternatives to second‐order logic: first‐order set theory, infinitary languages, and systems that are, in a sense, intermediate between first order and second order.

Frege’s Coneption of Logic explores the relationship between Frege’s understanding of conceptual analysis and his understanding of logic. It is argued that the fruitfulness of Frege’s conception of logic, and the illuminating differences between that conception and those more modern views that have largely supplanted it, are best understood against the backdrop of a clear account of the role of conceptual analysis in logical investigation. The first part of the book locates the role of conceptual analysis in Frege’s logicist project. It is argued that, despite a number of difficulties, Frege’s use of analysis in the service of logicism is a powerful and coherent tool. As a result of coming to grips with his use of that tool, we can see that there is, despite appearances, no conflict between Frege’s intention to demonstrate the grounds of ordinary arithmetic and the fact that the numerals of his derived sentences fail to co-refer with ordinary numerals. The second part of the book explores the resulting conception of logic itself, and some of the straightforward ways in which Frege’s conception differs from its now-familiar descendants. In particular, it’s argued that consistency, as Frege understands it, differs significantly from the kind of consistency demonstrable via the construction of models. To appreciate this difference is to appreciate the extent to which Frege was right in his debate with Hilbert over consistency- and independence-proofs in geometry. For similar reasons, modern results such as the completeness of formal systems and the categoricity of theories do not have for Frege the same importance they are commonly taken to have by his post-Tarskian descendants. These differences, together with the coherence of Frege’s position, provide reason for caution with respect to the appeal to formal systems and their properties in the treatment of fundamental logical properties and relations.