John Burgess

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

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.

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.

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.