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.

This work, the first book-length study of its topic, is an important contribution to the literature of philosophical logic and philosophy of language, with implications for other branches of philosophy, including philosophy of mathematics. However, five of the book's ten chapters, including many of the author's most original contributions, are devoted to issues about natural language, and lie pretty well outside the scope of this journal, not to mention that of the reviewer's competence. For this reason I will here largely confine my attention to the other half of the book, and so will be far from doing full justice to the book as a whole; indeed, there is such a wealth of detail in the book that I will be unable to do full justice even to the five chapters selected for comment.Non-distributive predicates.To begin with some points that have often been remarked, formulas in classical first-order logic are built up from predicates by means of a limited range of logical operators. Natural language involves many other such operators, beginning with temporal and modal operators, that are of great philosophical interest, but these are ignored by classical first-order logic. The reasons why it ignores them are surely that classical first-order logic was developed primarily for analyzing mathematical arguments, and that such grammatical categories as tense and mood play no significant role in mathematics.It has been less often remarked that the range of predicates considered in classical first-order logic is also limited. From Frege onwards, predicates have been taken to be essentially sentences with one or more gaps or places suitable to be filled in by singular noun phrases. Natural language involves also plural predicates, as well as mixed predicates with some …

In this era when results of empirical scientific research are being appealed to all across philosophy, when we even find moral philosophers invoking the results of brain scans, many profess to practice "naturalized epistemology," or to be "epistemological naturalists." Such phrases derive from the title of a well-known essay by Quine,[1] but Paul Gregory's thesis in the work under review is that there is less connection than is usually assumed between Quine's variety of naturalized epistemology and what is today taken, by opponents and proponents alike, to constitute epistemological naturalism. To put it bluntly, as Gregory does in the opening sentence of his introduction, Quine "has not been well understood." If there is less connection between the Quinian and other epistemological naturalisms than there has often been taken to be, on Gregory's account there is also much more connection between Quine's position on epistemology and his positions on other contentious issues.

Philosophical Analysis in the Twentieth Century by Scott Soames reminds me of nothing so much as Lectures on Literature by Vladimir Nabokov. Both are works that arose immediately out of the needs of undergraduate teaching, yet each manages to say much of significance to knowledgeable professionals. Each indirectly provides an outline of the history of its field, through a presentation of selected major works, taken in chronological order and including items that are generally recognized as marking decisive turning points. Yet neither Soames’s work nor Nabokov’s is a history in any conventional sense, both being immediately disqualified from that category by the general absence of coverage of minor and middling works and writers. The emphasis is pedagogical rather than historiographical: the emphasis is on introducing the student to the field through very close examination of the limited number of key texts selected for inclusion. The author’s distinctive personality is also apparent in both works. Each writer has a favorite theme he repeatedly sounds: for Soames, the danger of conflating the analytic, the a priori, and the necessary; for Nabokov, the philistinism of expecting an uplifting “message” from works of literary art. Each also includes some quirky, individual selections: The Right and the Good, The Strange Case of Dr. Jekyll and Mr. Hyde. Few others would have taken R. L. Stevenson to be up there with Dickens, Flaubert, and Proust, or W. D. Ross with Russell, Wittgenstein, and Quine. Each also sets aside for separate treatment elsewhere a major body of work one might have expected to be covered. Nabokov reserves Russian literature for a companion volume, while Soames gives only slight coverage to what he describes as “work in logic, the foundations of logic, and the application of logical techniques to the study of language” — a category that in practice turns out to include the bulk of the relevant material (by such writers as Frege, Carnap, and Tarski) that published originally in German without simultaneous English translation..

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.