John Burgess

Juliette Kennedy’s new book reminds us that some have claimed that the existence of so-called deviant encodings or numeral systems raises doubts about the wide acceptance of “Church’s thesis” understood as the identification of computability with recursiveness for numerical functions. It is argued that if responses to such doubts, by Stewart Shapiro and others, in the literature do not already suffice to dispel such doubts, attention to careful discussions, by Charles Parsons and others, of ontological issues in philosophy of mathematics should be more than enough to do so.

In justifying Russian aggression against Ukraine, President Vladimir Putin asserts that Ukraine is neither a distinct nation nor a viable state. In response, this essay will establish a Christian account of Ukraine’s right to self-defense not only via just war criteria but also in relation to its purpose theologically as a nation-state. This essay, after reviewing Christian ethical positions that either reject or embrace the nation-state, draws on the Niebuhr brothers and Karl Barth to develop key theological criteria for judging whether particular nation-states are true to their divine purpose. While affirming the role of the nation-state in protecting a people’s distinct identity, Christians will call on the nation-state to direct its people outward, toward their shared humanity with the world’s other nations. The essay concludes that the Ukrainian nation-state is fulfilling these criteria, whereas the Russian nation-state is not.

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.

There is no prospect of discovering measurable cardinals by radio astronomy, but this does not mean that higher set theory is entirely irrelevant to applied mathematics broadly construed. By way of example, the bearing of some celebrated descriptive-set-theoretic consequences of large cardinals on measurable-selection theory, a body of results originating with a key lemma in von Neumann’s work on the mathematical foundations of quantum theory, and further developed in connection with problems of mathematical economics, will be considered from a philosophical point of view.

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.

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.

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.

Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.

What is the simplest and most natural axiomatic replacement for the set-theoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? Several axiomatic systems, old and new, are examined and evaluated as candidate answers to these questions, with results of Harvey Friedman playing a significant role in the examination.