This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection - prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) - show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics - FilMat

This is the first textbook that approaches natural language semantics and logic from the perspective of Discourse Representation Theory, an approach which emphasizes the dynamic and incremental aspects of meaning and inference. The book has been carefully designed for the classroom. It is aimed at students with varying degrees of preparation, including those without prior exposure to semantics or formal logic. Moreover, it should make DRT easily accessible to those who want to learn about the theory on their own. Exercises are available to test understanding as well as to encourage independent theoretical thought. The book serves a double purpose. Besides a textbook, it is also the first comprehensive and fully explicit statement of DRT available in the form of a book. The first part of the book develops the basic principles of DRT for a small fragment of English (but which has nevertheless the power of standard predicate logic). The second part extends this fragment by adding plurals; it discusses a wide variety of problems connected with plural nouns and verbs. The third part applies the theory to the analysis of tense and aspect. Many of the problems raised in Parts Two and Three are novel, as are the solutions proposed. For undergraduate and graduate students interested in linguistics, theoretical linguistics, computational linguistics, artificial intelligence and cognitive science. Suitable for students with no previous exposure to formal semantics or logic.

This book provides a systematic analysis of many common argumentation schemes and a compendium of 96 schemes. The study of these schemes, or forms of argument that capture stereotypical patterns of human reasoning, is at the core of argumentation research. Surveying all aspects of argumentation schemes from the ground up, the book takes the reader from the elementary exposition in the first chapter to the latest state of the art in the research efforts to formalize and classify the schemes, outlined in the last chapter. It provides a systematic and comprehensive account, with notation suitable for computational applications that increasingly make use of argumentation schemes.

This collection presents the first sustained examination of the nature and status of the idea of principles in early modern thought. Principles are almost ubiquitous in the seventeenth and eighteenth centuries: the term appears in famous book titles, such as Newton's Principia; the notion plays a central role in the thought of many leading philosophers, such as Leibniz's Principle of Sufficient Reason; and many of the great discoveries of the period, such as the Law of Gravitational Attraction, were described as principles. Ranging from mathematics and law to chemistry, from natural and moral philosophy to natural theology, and covering some of the leading thinkers of the period, this volume presents ten compelling new essays that illustrate the centrality and importance of the idea of principles in early modern thought. It contains chapters by leading scholars in the field, including the Leibniz scholar Daniel Garber and the historian of chemistry William R. Newman, as well as exciting, emerging scholars, such as the Newton scholar Kirsten Walsh and a leading expert on experimental philosophy, Alberto Vanzo. The Idea of Principles in Early Modern Thought: Interdisciplinary Perspectives charts the terrain of one of the period's central concepts for the first time, and opens up new lines for further research.

Biologists, climate scientists, and economists all rely on models to move their work forward. In this book, Stephen M. Downes explores the use of models in these and other fields to introduce readers to the various philosophical issues that arise in scientific modeling. Readers learn that paying attention to models plays a crucial role in appraising scientific work. This book first presents a wide range of models from a number of different scientific disciplines. After assembling some illustrative examples, Downes demonstrates how models shed light on many perennial issues in philosophy of science and in philosophy in general. Reviewing the range of views on how models represent their targets introduces readers to the key issues in debates on representation, not only in science but in the arts as well. Also, standard epistemological questions are cast in new and interesting ways when readers confront the question, "What makes for a good (or bad) model?" All examples from the sciences and positions in the philosophy of science are presented in an accessible manner. The book is suitable for undergraduates with minimal experience in philosophy and an introductory undergraduate experience in science. Key features: The book serves as a highly accessible philosophical introduction to models and modeling in the sciences, presenting all philosophical and scientific issues in a nontechnical manner. Students and other readers learn to practice philosophy of science by starting with clear examples taken directly from the sciences. While not comprehensive, this book introduces the reader to a wide range of views on key issues in the philosophy of science.

The limit of language is one of the most pervasive notions found in Wittgenstein's work, both in his early Tractatus Logico-Philosophicus and his later writings. Moreover, the idea of a limit of language is intimately related to important scholarly debates on Wittgenstein's philosophy, such as the debate between the so-called traditional and resolute interpretations, Wittgenstein's stance on transcendental idealism, and the philosophical import of Wittgenstein's latest work On Certainty. This collection includes thirteen original essays that provide a comprehensive overview of the various ways in which Wittgenstein appeals to the limit of language at different stages of his philosophical development. The essays connect the idea of a limit of language to the most important themes discussed by Wittgenstein-his conception of logic and grammar, the method of philosophy, the nature of the subject, and the foundations of knowledge-as well as his views on ethics, aesthetics, and religion. The essays also relate Wittgenstein's thought to his contemporaries, including Carnap, Frege, Heidegger, Levinas, and Moore.

Originally published in 1973. This book presents a valid mode of reasoning that is different to mathematical probability. This inductive logic is investigated in terms of scientific investigation. The author presents his criteria of adequacy for analysing inductive support for hypotheses and discusses each of these criteria in depth. The chapters cover philosophical problems and paradoxes about experimental support, probability and justifiability, ending with a system of logical syntax of induction. Each section begins with a summary of its contents and there is a glossary of technical terms to aid the reader.

This book addresses the argument in the history of the philosophy of science between the positivists and the anti-positivists. The author starts from a point of firm conviction that all science and philosophy must start with the given... But that the range of the given is not definite. He begins with an examination of science from the outside and then the inside, explaining his position on metaphysics and attempts to formulate the character of operational acts before a general theory of symbolism is explored. The last five chapters constitute a treatise to show that the development from one stage of symbolismto the next is inevitable, consequently that explanatory science represents the culmination of knowledge.

Originally published in 1965. This is a textbook of modern deductive logic, designed for beginners but leading further into the heart of the subject than most other books of the kind. The fields covered are the Propositional Calculus, the more elementary parts of the Predicate Calculus, and Syllogistic Logic treated from a modern point of view. In each of the systems discussed the main emphases are on Decision Procedures and Axiomatisation, and the material is presented with as much formal rigour as is compatible with clarity of exposition. The techniques used are not only described but given a theoretical justification. Proofs of Consistency, Completeness and Independence are set out in detail. The fundamental characteristics of the various systems studies, and their relations to each other are established by meta-logical proofs, which are used freely in all sections of the book. Exercises are appended to most of the chapters, and answers are provided.

Originally published in 1967. The common aim of all logical enquiry is to discover and analyse correctly the forms of valid argument. In this book concise expositions of traditional, Aristotelian logic and of modern systems of propositional and predicative logic show how far that aim has been achieved.

This collection addresses metaphysical issues at the intersection between philosophy and science. A unique feature is the way in which it is guided both by history of philosophy, by interaction between philosophy and science, and by methodological awareness. In asking how metaphysics is possible in an age of science, the contributors draw on philosophical tools provided by three great thinkers who were fully conversant with and actively engaged with the sciences of their day: Kant, Husserl, and Frege. Part I sets out frameworks for scientifically informed metaphysics in accordance with the meta-metaphysics outlined by these three self-reflective philosophers. Part II explores the domain for co-existent metaphysics and science. Constraints on ambitious critical metaphysics are laid down in close consideration of logic, meta-theory, and specific conditions for science. Part III exemplifies the role of language and science in contemporary metaphysics. Quine's pursuit of truth is analysed; Cantor's absolute infinitude is reconstrued in modal terms; and sense is made of Weyl's take on the relationship between mathematics and empirical aspects of physics. With chapters by leading scholars, Metametaphysics and the Sciences is an in-depth resource for researchers and advanced students working within metaphysics, philosophy of science, and the history of philosophy.

Used in a variety of courses in various disciplines, Asking the Right Questions helps students bridge the gap between simply memorizing or blindly accepting information, and the greater challenge of critical analysis and synthesis. Specifically, this concise text teaches students to think critically by exploring the components of arguments--issues, conclusions, reasons, evidence, assumptions, language--and on how to spot fallacies and manipulations and obstacles to critical thinking in both written and visual communication. It teaches them to respond to alternative points of view and develop a solid foundation for making personal choices about what to accept and what to reject.

The volume consists of thirteen papers devoted to various problems of the philosophy of logic and mathematics. They can be divided into two groups. The first group contains papers devoted to some general problems of the philosophy of mathematics whereas the second group - papers devoted to the history of logic in Poland and to the work of Polish logicians and math-ematicians in the philosophy of mathematics and logic. Among considered problems are: meaning of reverse mathematics, proof in mathematics, the status of Church's Thesis, phenomenology in the philosophy of mathematics, mathematics vs. theology, the problem of truth, philosophy of logic and mathematics in the interwar Poland.

The primary objective of An Inquiry into the Nature of Aesthetic Theory in Its Relation to Theory of Knowledge in Kant's Critical Philosophy is to investigate Kant's aesthetic theory and its problematic relation to theory of knowledge in his transcendental philosophy. In the Critique of Judgment, Kant constructs his aesthetic theory by arguing that the aesthetic experience is based on a certain type of feeling, namely, the feeling of pleasure, rather than a concept. He grounds such a feeling on the aesthetic judgment of reflection. In spite of its nonconceptual and subjective characteristic, an aesthetic reflective judgment still has a claim to be universally valid. Here, the feeling of pleasure in beautiful is produced by the free harmonious relation between the imagination and the understanding. Judgment, in its reflective employment, does not determine its object but determines the feeling of pleasure in the judging subject. On the other hand, the categories, as pure concepts of the understanding, carry nearly all the weight in his theory of knowledge presented in the Critique of Pure Reason. The imagination, in this case, is strictly bound up by the rules that are imposed by the concepts of the understanding. By this way, judgment, as a cognitive faculty, determines its object and gains its objective validity. In this context, this book discusses the nature of Kant's aesthetic theory and the components that constitute a pure aesthetic judgment of reflection and attempts to clarify its proper place in critical philosophy regarding his theory of knowledge.

Originally published in 1966 On the Syllogism and Other Logical Writings assembles for the first time the five celebrated memoirs of Augustus De Morgan on the syllogism. These are collected together with the more condensed accounts of his researches given in his Syllabus of a Proposed System of Logic an article on Logic contributed to the English Cyclopaedia. De Morgan was among the most distinguished of nineteenth century British mathematicians but is chiefly remembered today as one of the founders of modern mathematical logic. His writings on this subject have been little read, however since apart from his Formal Logic, they lie buried for the most part in inaccessible periodicals. De Morgan's own later amendments are inserted in the text and the editorial introduction gives a summary of the whole and traces in some detail the course of the once-famous feud with Sir William Hamilton of Edinburgh.

This book provides an accessible and thorough analysis of "The Doctrine of Being," the first part of Hegel's Science of Logic. Though it received much scholarly attention in the past, interpreters of this text have generally refrained from examining it in a sufficiently detailed manner. Through a rigorous and critical reading of Hegel's speculative arguments, Mehmet Tabak illustrates that Hegel meant his logic to be both a presuppositionless analysis and development of the basic categories of thought, on the one hand, and a post-Kantian ontology on the other. However, the analysis of the text demonstrates that Hegel fails to deliver such logic. This volume promises to be an indispensable guide to those who wish to understand the first book of Science of Logic.

Lacan and the Formulae of Sexuation provides the first critical reading of Lacan's formulae of sexuation, examining both their logical consistency and clinical consequences. Are there two different entities named Man and Woman, separated by the gulf of sexual difference? Or is it better to conceive of this difference as something purely relative, each human being situated on a sort of continuum from more or less 'man' to more or less 'woman'? Sigmund Freud established the strange way through which sexuality determines being human: his concept of drive was no longer the heteronormative sexual instinct used by the psychiatrists of his time. With his provocative formula according to which 'there is no sexual relationship', Lacan has reinforced this perspective, combining logic and sexuality through the invention of a new operator, the concept 'not all', which points to a form of incompleteness at stake in his 'formulae of sexuation'. This book examines how these formulae have been constructed, and how we should read them in connection with, on one hand, their own logical consistency (a logical square different from Aristotelian tradition) and, on the other hand, a 'part object' in a very different sense to Melanie Klein's. The book also investigates the underlying logic of clinical vignettes, so much in favour in psychoanalytical literature today. The book represents essential reading for Lacanian psychoanalysts, as well as researchers at the cross-section of psychoanalysis, philosophy, and gender studies.

This volume contains twenty-four essays by the British/Australian analytic metaphysician, Brian Garrett. These essays are followed by four short dialogues that emphasize and summarize some of the main points of the essays and discuss new perspectives that have emerged since their original publication. The volume covers topics on the metaphysics of time, the nature of identity, and the nature and importance of persons and human beings. The chapters constitute the fruits of almost four decades of philosophical research, from Brian's two award-winning essays, published in Analysis in 1983 and The Philosophical Quarterly in 1992, to his latest ideas about Fatalism and the Grandfather Paradox. This book will be of interest to students and professional philosophers in the field of analytic philosophy.

This book offers a comprehensive account of logic that addresses fundamental issues concerning the nature and foundations of the discipline. The authors claim that these foundations can not only be established without the need for strong metaphysical assumptions, but also without hypostasizing logical forms as specific entities. They present a systematic argument that the primary subject matter of logic is our linguistic interaction rather than our private reasoning and it is thus misleading to see logic as revealing "the laws of thought". In this sense, fundamental logical laws are implicit to our "language games" and are thus more similar to social norms than to the laws of nature. Peregrin and Svoboda also show that logical theories, despite the fact that they rely on rules implicit to our actual linguistic practice, firm up these rules and make them explicit. By carefully scrutinizing the project of logical analysis, the authors demonstrate that logical rules can be best seen as products of the so called reflective equilibrium. They suggest that we can profit from viewing languages as "inferential landscapes" and logicians as "geographers" who map them and try to pave safe routes through them. This book is an essential resource for scholars and researchers engaged with the foundations of logical theories and the philosophy of language.

This book on proof theory centers around the legacy of Kurt Schutte and its current impact on the subject. Schutte was the last doctoral student of David Hilbert who was the first to see that proofs can be viewed as structured mathematical objects amenable to investigation by mathematical methods (metamathematics). Schutte inaugurated the important paradigm shift from finite proofs to infinite proofs and developed the mathematical tools for their analysis. Infinitary proof theory flourished in his hands in the 1960s, culminating in the famous bound 0 for the limit of predicative mathematics (a fame shared with Feferman). Later his interests shifted to developing infinite proof calculi for impredicative theories. Schutte had a keen interest in advancing ordinal analysis to ever stronger theories and was still working on some of the strongest systems in his eighties. The articles in this volume from leading experts close to his research, show the enduring influence of his work in modern proof theory. They range from eye witness accounts of his scientific life to developments at the current research frontier, including papers by Schutte himself that have never been published before.

Originally published in 1966. This is a self-instructional course intended for first-year university students who have not had previous acquaintance with Logic. The book deals with "propositional" logic by the truth-table method, briefly introducing axiomatic procedures, and proceeds to the theory of the syllogism, the logic of one-place predicates, and elementary parts of the logic of many-place predicates. Revision material is provided covering the main parts of the course. The course represents from eight to twenty hours work. depending on the student's speed of work and on whether optional chapters are taken.