John Burgess

1 Choice conjecture In axiomatizing nonclassical extensions of classical sentential logic one tries to make do, if one can, with adding to classical sentential logic a finite number of axiom schemes of the simplest kind and a finite number of inference rules of the simplest kind. The simplest kind of axiom scheme in effect states of a particular formula P that for any substitution of formulas for atoms the result of its application to P is to count as an axiom. The simplest kind of onepremise inference rule in effect states of a particular pair of formulas P and Q that for any substitution of formulas for atoms, if the result of its application to P is a theorem, then the result of its application to Q is to count as a theorem; similarly for many-premise rules. Such are the schemes and rules of all the best-known modal and tense logics, for instance. Sometimes it is difficult to find such simple schemes and rules. In that case one may resort to less simple schemes or less simple rules. There is no generally recognized rigorous definition of "next simplest kind" of scheme. Neither is there any generally recognized definition of "next simplest kind" of rule, and hence there is no fully rigorous enunciation of the choice conjecture, the conjecture that schemes of the next simplest kind can always be avoided in favor of rules of the next simplest kind and vice versa. Nonetheless, there are cases where intuitively one does recognize that the schemes or rules in a given axiomatization are only slightly more complex than the simplest kind, including cases where one does have a choice between adopting slightly-more-complex-than-simplest schemes and adopting slightly-more-complex-than-simplest rules. In tense logic early examples of slightly more complex rules are found in [2] and [3] : there is one example of the embarrassed use of such rules in the former, and many examples of the enthusiastic use of such rules in the latter and its sequels. Accordingly the rules in question have come to be called "Gabbay-style" rules.

Recently it has become almost the received wisdom in certain quarters that Kripke models are appropriate only for something like metaphysical modalities, and not for logical modalities. Here the line of thought leading to Kripke models, and reasons why they are no less appropriate for logical than for other modalities, are explained. It is also indicated where the fallacy in the argument leading to the contrary conclusion lies. The lessons learned are then applied to the question of the status of the formula.

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.

Foundational work in mathematics by some of the other participants in the symposium helps towards answering the question whether a heterodox mathematics could in principle be used as successfully as is orthodox mathematics in scientific applications. This question is turn, it will be argued, is relevant to the question how far current science is the way it is because the world is the way it is, and how far because we are the way we are, which is a central question, if not the central question, of philosophy of science.

This long-awaited volume is a must-read for anyone with a serious interest in philosophy of mathematics. The book falls into two parts, with the primary focus of the first on ontology and structuralism, and the second on intuition and epistemology, though with many links between them. The style throughout involves unhurried examination from several points of view of each issue addressed, before reaching a guarded conclusion. A wealth of material is set before the reader along the way, but a reviewer wishing to summarize the author's views crisply will be frustrated. The chapter-by-chapter survey below conveys at best a very incomplete and imperfect impression of the work's virtues, and even of its contents, falling short even of supplying a full menu for the banquet of food for thought that Parsons serves up to his readers.

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.

Alan Weir’s new book is, like Darwin’s Origin of Species, ‘one long argument’. The author has devised a new kind of have-it-both-ways philosophy of mathematics, supposed to allow him to say out of one side of his mouth that the integer 1,000,000 exists and even that the cardinal ℵω exists, while saying out of the other side of his mouth that no numbers exist at all, and the whole book is devoted to an exposition and defense of this new view. The view is presented in the book in a way that can make it difficult for the reader to trace the main line of argument: with a great deal of apparatus, and with a great many digressions into subordinate issues. In what follows I will try to stick to what I take to be the essentials, even at the risk of oversimplifying some central but complicated issues, and at the cost of neglecting some interesting but peripheral ones.In chapter 1, the author introduces a distinction between what he calls ‘two aspects of meaning’ and dubs informational content and metaphysical content. Informational content is the aspect of meaning of primary interest to linguists, and the one of which speakers themselves are generally aware, at least upon reflection. Metaphysical content is supposed to be another aspect of meaning primarily of interest to philosophers. The basic idea is that if there are standards of correctness for assertions of a certain kind, then such an assertion may be called ‘true’ when those standards are met, even though the kind of correctness involved is not correctness in representing how the world is. What the world must be like in order for the utterance to be true is the metaphysical content of the assertion, but it need not be part of its …

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.

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.

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.