While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental (...) mathematics on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed. (shrink)

This is a concise, advanced introduction to current philosophical debates about truth. A blend of philosophical and technical material, the book is organized around, but not limited to, the tendency known as deflationism, according to which there is not much to say about the nature of truth. In clear language, Burgess and Burgess cover a wide range of issues, including the nature of truth, the status of truth-value gaps, the relationship between truth and meaning, relativism and pluralism about (...) truth, and semantic paradoxes from Alfred Tarski to Saul Kripke and beyond. Following a brief introduction that reviews the most influential traditional and contemporary theories of truth, short chapters cover Tarski, deflationism, indeterminacy, realism, antirealism, Kripke, and the possible insolubility of semantic paradoxes. The book provides a rich picture of contemporary philosophical theorizing about truth, one that will be essential reading for philosophy students as well as philosophers specializing in other areas. (shrink)

Philosophical Logic is a clear and concise critical survey of nonclassical logics of philosophical interest written by one of the world's leading authorities on the subject. After giving an overview of classical logic, John Burgess introduces five central branches of nonclassical logic, focusing on the sometimes problematic relationship between formal apparatus and intuitive motivation. Requiring minimal background and arranged to make the more technical material optional, the book offers a choice between an overview and in-depth study, and it balances (...) the philosophical and technical aspects of the subject.The book emphasizes the relationship between models and the traditional goal of logic, the evaluation of arguments, and critically examines apparatus and assumptions that often are taken for granted. Philosophical Logic provides an unusually thorough treatment of conditional logic, unifying probabilistic and model-theoretic approaches. It underscores the variety of approaches that have been taken to relevantistic and related logics, and it stresses the problem of connecting formal systems to the motivating ideas behind intuitionistic mathematics. Each chapter ends with a brief guide to further reading.Philosophical Logic addresses students new to logic, philosophers working in other areas, and specialists in logic, providing both a sophisticated introduction and a new synthesis. (shrink)

John Burgess is the author of a rich and creative body of work which seeks to defend classical logic and mathematics through counter-criticism of their nominalist, intuitionist, relevantist, and other critics. This selection of his essays, which spans twenty-five years, addresses key topics including nominalism, neo-logicism, intuitionism, modal logic, analyticity, and translation. An introduction sets the essays in context and offers a retrospective appraisal of their aims. The volume will be of interest to a wide range of readers across (...) philosophy of mathematics, logic, and philosophy of language. (shrink)

Saul Kripke has been a major influence on analytic philosophy and allied fields for a half-century and more. His early masterpiece, _Naming and Necessity_, reversed the pattern of two centuries of philosophizing about the necessary and the contingent. Although much of his work remains unpublished, several major essays have now appeared in print, most recently in his long-awaited collection _Philosophical Troubles_. In this book Kripke’s long-time colleague, the logician and philosopher John P. Burgess, offers a thorough and self-contained guide (...) to all of Kripke’s published books and his most important philosophical papers, old and new. It also provides an authoritative but non-technical account of Kripke’s influential contributions to the study of modal logic and logical paradoxes. Although Kripke has been anything but a system-builder, Burgess expertly uncovers the connections between different parts of his oeuvre. Kripke is shown grappling, often in opposition to existing traditions, with mysteries surrounding the nature of necessity, rule-following, and the conscious mind, as well as with intricate and intriguing puzzles about identity, belief and self-reference. Clearly contextualizing the full range of Kripke’s work, Burgess outlines, summarizes and surveys the issues raised by each of the philosopher’s major publications. _Kripke_ will be essential reading for anyone interested in the work of one of analytic philosophy’s greatest living thinkers. (shrink)

Working from a realist Thomistic epistemology, Ashley asserts that we must begin our search for wisdom in the natural sciences; only then, he believes, can we ensure that our claims about immaterial and invisible things are rooted in reliable experience of the material. Any attempt to share wisdom, he insists, must derive from a context that is both interdisciplinary and intercultural. Ashley offers an ambitious analysis and synthesis of major historical contributions to the unification of knowledge, including non-Western (...) traditions. Beginning with the question "Metaphysics: Nonsense or Wisdom?" Ashley moves from a critical examination of the foundations of modern science to quantum physics and the Big Bang; from Aristotle's theory of being and change, through Aquinas's five ways, to a critical analysis of modern and postmodern thought. Ashley is able to interweave the approaches of the great philosophers by demonstrating their contributions to philosophical thought in a concrete, specific manner. In the process, he accounts for a contemporary culture overwhelmed by the fragmentation of data and thirsting for an utterly transcendent yet personal God. "This is an impressive, well-researched book, of great value. It offers the wider philosophical community a point of entrance, by a proponent of a certain type of Thomism, into a domain that all philosophers think they already understand. The result is the creation of a 'big picture' of human knowledge." -- _Mark Johnson, Marquette University_. (shrink)

Ethics and Educational Research: An Introduction Robert G. Burgess Ethical questions are the subject of interdisciplinary discussions and debates. ...

Fixing Frege is one of the most important investigations to date of Fregean approaches to the foundations of mathematics. In addition to providing an unrivalled survey of the technical program to which Frege’s writings have given rise, the book makes a large number of improvements and clarifications. Anyone with an interest in the philosophy of mathematics will enjoy and benefit from the careful and well informed overview provided by the first of its three chapters. Specialists will find the book an (...) indispensable reference and an invaluable source of insights and new results. Although Frege is widely regarded as the father of analytic philosophy, his work on the foundations of mathematics was for a long time rather peripheral to the ongoing research. The main reason for this is no doubt Russell’s discovery in 1901 that the paradox now bearing his name can be derived in Frege’s logical system. But recent decades have seen a huge surge of interest in Fregean approaches to the foundations of mathematics. A variety of consistent theories have been discovered that can be salvaged from Frege’s inconsistent system, and foundational and philosophical claims have been made on behalf of many of these theories. Burgess claims quite plausibly that the significance of any such modified Fregean theory will in large part depend on how much of ordinary mathematics it enables us to develop.1 His. (shrink)

“This is an impressive, well-researched book, of great value. It offers the wider philosophical community a point of entrance, by a proponent of a certain type of Thomism, into a domain that all philosophers think they already understand. The result is the creation of a ‘big picture’ of human knowledge.” —Mark Johnson, Marquette University Working from a realist Thomistic epistemology, noted scholar Benedict Ashley, O.P., asserts that we must begin our search for wisdom in the natural sciences; only then, (...) Ashley believes, can we ensure that our claims about immaterial and invisible things are rooted in reliable experience of the material. Any attempt to share wisdom, he insists, must derive from a context that is both interdisciplinary and intercultural. This capstone of a remarkable career will be welcomed by students in philosophy and theology. (shrink)

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. (shrink)

The first beginning logic text to employ the tree method--a complete formal system of first-order logic that is remarkably easy to understand and use--this text allows students to take control of the nuts and bolts of formal logic quickly, and to move on to more complex and abstract problems. The tree method is elaborated in manageable steps over five chapters, in each of which its adequacy is reviewed; soundness and completeness proofs are extended at each step, and the decidability proof (...) is extended at the step from truth functions to the logic of nonoverlapping quantifiers with a single variable, after which undecidability is demonstrated by example. The first three chapters are bilingual, with arguments presented twice, in logical notation and in English. The last three chapters consider the discoveries defining the scope and limits of formal methods that marked logic’s coming of age in the 20th century: Godel’s completeness and incompleteness theorems for first and second-order logic, and the Church-Turing theorem on the undecidability of first-order logic. This new edition provides additional problems, solutions to selected problems, and two new Supplements: Truth-Functional Equivalence reinstates material on that topic from the second edition that was omitted in the third, and Variant Methods, in which John Burgess provides a proof regarding the possibility of modifying the tree method so that it will always find a finite model when there is one, and another, which shows that a different modification—once contemplated by Jeffrey--can result in a dramatic speed--up of certain proofs. (shrink)

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental (...) mathematics on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed. (shrink)

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. Including a selection of exercises, adjusted for this edition, at the end of each chapter, it offers a new and simpler treatment of the representability of recursive functions, a (...) traditional stumbling block for students on the way to the Godel incompleteness theorems. (shrink)

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no such objects, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. This book cuts through a host of technicalities that have obscured previous (...) discussions of these projects, and presents clear, concise accounts of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed. (shrink)

Which concepts should we use to think and talk about the world and to do all of the other things that mental and linguistic representation facilitates? This is the guiding question of the field that we call ‘conceptual ethics’. Conceptual ethics is not often discussed as its own systematic branch of normative theory. A case can nevertheless be made that the field is already quite active, with contributions coming in from areas as diverse as fundamental metaphysics and social/political philosophy. In (...) this pair of papers, we try to unify the field, reflecting on its basic nature, structure, and methodology. (shrink)

Which concepts should we use to think and talk about the world, and to do all of the other things that mental and linguistic representation facilitates? This is the guiding question of the field that we call ‘conceptual ethics’. Conceptual ethics is not often discussed as its own systematic branch of normative theory. A case can nevertheless be made that the field is already quite active, with contributions coming in from areas as diverse as fundamental metaphysics and social/political philosophy. In (...) this pair of papers, we try to unify the field, reflecting on its basic nature, structure, and methodology. (shrink)

This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in ...

This volume reprints a dozen of the author’s papers, most with substantial postscripts, and adds one new one. The bulk of the material is on topics in philosophy of language, but there are also two papers on philosophy of mathematics written after the appearance of the author’s collected papers on that subject, and one on epistemology. As to the substance of Field’s contributions, limitations of space preclude doing much more below than indicating the range of issues addressed, and the general (...) orientation taken towards them. As to the style of his writing, it well exhibits the first of the two virtues, clarity and conciseness, that one looks for in philosophical prose. (shrink)

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

A new axiomatization of set theory, to be called Bernays-Boolos set theory, is introduced. Its background logic is the plural logic of Boolos, and its only positive set-theoretic existence axiom is a reflection principle of Bernays. It is a very simple system of axioms sufficient to obtain the usual axioms of ZFC, plus some large cardinals, and to reduce every question of plural logic to a question of set theory.

Appointment as a director of a company board often represents the pinnacle of a management career. Worldwide, it has been noted that very few women are appointed to the boards of directors of companies. Blame for the low numbers of women of company boards can be partly attributed to the widely publicized "glass ceiling". However, the very low representation of women on company boards requires further examination. This article reviews the current state of women's representation on boards of directors and (...) summarizes the reasons as to why women are needed on company boards. Given that more women on boards are desirable, the article then describes how more women could be appointed to boards, and the actions that organizations and women could take to help increase the representation of women. Finally, the characteristics of those women that have succeeded in becoming members of company boards are described from an international perspective. Unfortunately, answers to the vexing question of whether these women have gained board directorships in their own right as extremely competent managers, or whether they are mere token female appointments in a traditional male dominated culture, remains elusive. (shrink)

Work on a computer program called SMILE + IBP (SMart Index Learner Plus Issue-Based Prediction) bridges case-based reasoning and extracting information from texts. The program addresses a technologically challenging task that is also very relevant from a legal viewpoint: to extract information from textual descriptions of the facts of decided cases and apply that information to predict the outcomes of new cases. The program attempts to automatically classify textual descriptions of the facts of legal problems in terms of Factors, a (...) set of classification concepts that capture stereotypical fact patterns that effect the strength of a legal claim, here trade secret misappropriation. Using these classifications, the program can evaluate and explain predictions about a problem’s outcome given a database of previously classified cases. This paper provides an extended example illustrating both functions, prediction by IBP and text classification by SMILE, and reports empirical evaluations of each. While IBP’s results are quite strong, and SMILE’s much weaker, SMILE + IBP still has some success predicting and explaining the outcomes of case scenarios input as texts. It marks the first time to our knowledge that a program can reason automatically about legal case texts. (shrink)

The complexity of the set of truths of arithmetic is determined for various theories of truth deriving from Kripke and from Gupta and Herzberger.

The source, status, and significance of the derivation of the necessity of identity at the beginning of Kripke’s lecture “Identity and Necessity” is discussed from a logical, philosophical, and historical point of view.

This is the verbatim manuscript of a paper which has circulated underground for close to thirty years, reaching a metethical conclusion close to J. L. Mackie’s by a somewhat different route.

We provide a retrospective of 25 years of the International Conference on AI and Law, which was first held in 1987. Fifty papers have been selected from the thirteen conferences and each of them is described in a short subsection individually written by one of the 24 authors. These subsections attempt to place the paper discussed in the context of the development of AI and Law, while often offering some personal reactions and reflections. As a whole, the subsections build into (...) a history of the last quarter century of the field, and provide some insights into where it has come from, where it is now, and where it might go. (shrink)

Discussions of “indeterminacy” customarily distinguish two putative types: semantic indeterminacy (SI)—indeterminacy that’s somehow the product of the semantics of our words/concepts—and metaphysical indeterminacy (MI)—indeterminacy that exists as a mind/language-independent feature of reality itself. A popular and influential thought among philosophers is that all indeterminacy must be SI. In this paper we challenge this thought. Our challenge is guided by the question: What, exactly, does it take for a case of indeterminacy to count as SI? We argue that the only satisfactory (...) answer to this question must take SI to be grounded in a more basic type of MI. We conclude that SI cannot be made sense of without implicating MI. If there’s any indeterminacy, there must be indeterminacy in the world itself. (shrink)

The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Supecki and Bryll, is presented for the claim that the right demonstrability logic must be contained in S5, (...) and a more speculative argument for the claim that it does not include S4.2 is also presented. (shrink)

Quine correctly argues that Carnap's distinction between internal and external questions rests on a distinction between analytic and synthetic, which Quine rejects. I argue that Quine needs something like Carnap's distinction to enable him to explain the obviousness of elementary mathematics, while at the same time continuing to maintain as he does that the ultimate ground for holding mathematics to be a body of truths lies in the contribution that mathematics makes to our overall scientific theory of the world. Quine's (...) arguments against the analytic/synthetic distinction, even if fully accepted, still leave room for a notion of pragmatic analyticity sufficient for the indicated purpose. (shrink)