This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain point on, to wit: from the probabilities relative to that very B of certain atomic components and conjunctions of atomic components of A, but again to no further extent. These and other results are extended to the less studied case where A and B are compounded from atomic statements by means of “∀” as well as “~” and “&”. The absolute probability functions considered are those of Kolmogorov and Carnap, and the relative ones are those of Kolmogorov, Carnap, Rényi, and Popper.

Contemporary organismic and bio-social accounts of human behavior consider physical and psychological concepts as alternative or complementary linguistic descriptions of the same subject matter. The notion of complementarity is an important part of the organismic physicalistic synthesis which replaces the old duels between mechanism and vitalism, between physiology and psychology. The concept of complementarity comes from Bohr's solution for the difficulty of reconciling classical mechanics with quantum mechanics. He suggested that they are parallel and complementary rather than contradictory ways of looking at the universe. The continuity of the energy transfer according to the electromagnetic theory and the atomicity of light effects are “complementary aspects of one reality in the sense that each expresses an important feature of the phenomena of light, which, although irreconcilable from a mechanical point of view, can never be in direct contradiction, since a closer analysis of one or the other feature in mechanical terms would demand mutually exclusive experimental arrangements” (3).

C. S. Peirce argued that inductive reasoning and probability judgments are adequately secure only in the indefinitely long run, and that therefore it is illogical to employ these modes of inference unless one's chief devotion is to the interests of an ideal community of all rational beings, past, present, and future. He thought of this devotion as a “social sentiment”, involving self-sacrifice. An examination of his argument shows that the attitude presupposed by his conceptions of induction and probability is in fact not self-sacrificial and is social only in a very special sense. Furthermore, it seems doubtful that this attitude is characteristic of practicing scientists; and this is a reason for questioning Peirce's analysis of induction and probability.

It is helpful for any enterprise to stop occasionally and examine itself. Science has done this rather infrequently in its long and eventful history, and there has not been, in general, any continuity in these self-examinations. As a result the history of the philosophy of science has been a rather spotty affair. My belief is that the philosophy of science should also, at times, become self-critical. When a study is concerned primarily with methods of other disciplines it tends to underemphasize the role played by the methods which it itself uses. What is the method of the philosophy of science, and is it justifiable? Can there be a satisfactory study of science and can this be scientific?

Frank Jackson and Robert Pargetter (1977) propose a method for synchronizing clocks at rest at distant points of an inertial system in Euclidean space, which, they claim, (i) does not depend on Einstein's signalling method (Einstein 1923, pp. 27–29) and (ii) provides a basis for denying the conventionality of distant simultaneity. I am afraid, however, that the new method presupposes that the simultaneity of distant events relatively to the chosen inertial system has been already determined by Einstein's or some other method. Jackson and Pargetter describe their method as follows:

Nature exhibits a rich variety of adaptations. Cells contain complex biomolecular structures, such as proteins, that are exquisitely adapted to perform specific biological functions. Evolutionary biology explains how biomolecular structures evolve. Intelligent design creationists reject evolutionary explanations. They want to believe that all adaptations in nature are the handiwork of God. Their critics aver that “it ain't necessarily so.” The anthology under review is an excellent display of the issues between intelligent design creationists and their critics. I agree with the critics.

In this paper I shall argue that to a very significant extent mathematics is concept analysis, and that though the analysis of mathematical concepts is in a number of ways different from the analysis of philosophic concepts, the similarities between these two types of concept analyses are as important and far reaching as the differences. I shall argue that because mathematics and philosophy are each concerned with the analysis of concepts, they are much more like one another epistemologically than is often recognized. In insisting upon fundamental similarities between mathematics and philosophy, I shall be agreeing with the classical rationalists, but on a very different conception of both philosophy and mathematics from that held by the rationalists. The rationalists wished to assimilate philosophy to mathematics as understood in their time; viz. as a body of necessary propositions, which followed from self-evident axioms and postulates, revealed to the natural light of reason. As against this rationalistic position, I wish to make a comparison in the reverse direction, in which I shall presuppose a certain conception of philosophy as something given, and then insist that mathematics is in many important respects similar to philosophy as so understood. In particular, I wish to insist that there is a significant comparison between mathematics on the one hand, and philosophy as understood by probably a majority of philosophers today on the other—viz., philosophy understood as concept analysis—, where it is conceded that the analysis of philosophic concepts is inherently a tentative matter, wherein it is impossible—at least in the usual case—to offer any one analysis of a given philosophic concept as absolutely certain and beyond all revision. I shall argue that by virtue of the fact that mathematics, like philosophy, is concerned with the analysis of concepts, many at least of the propositions advanced within it are inherently revisable, and do not possess the kind of certainty the rationalists ascribed to them.

Relying on the universality of quantum mechanics and on recent results known as the “threshold theorems,” quantum information scientists deem the question of the feasibility of large-scale, fault-tolerant, and computationally superior quantum computers as purely technological. Reconstructing this question in statistical mechanical terms, this article suggests otherwise by questioning the physical significance of the threshold theorems. The skepticism it advances is neither too strong (hence is consistent with the universality of quantum mechanics) nor too weak (hence is independent of technological contingencies).

I should like to begin with a remark which, although personal, is not without relevance both to the special topic of this paper and the general purpose of an International Congress of Philosophy. Having had the opportunity of studying and teaching philosophy in various countries of the Western World I have come to realize strongly the enormous power of national traditions in philosophy. Much more so than scientific research and, I believe, even the arts, our philosophizing is tied down to and captivated by particular spiritual climates, particular national traditions. Notwithstanding the apparatus of international congresses, the various personal and professional contacts between philosophers of different nationality, and the translation of at least the most important works into other languages, there is, I have come to feel, a mutual contempt, a mutual misunderstanding, a mutual intolerance, that are frightening. You will remember Nietzsche's sarcasms concerning English philosophy. I have heard Heidegger say that the only language in which one can really philosophize, besides Greek, is German. The German contempt for French clarity and English factuality is equalled by the irony which French and English philosophers show towards German “depth.” The examples could be multiplied. Scrutinize your university programs and I am convinced you will be startled by the enormous emphasis upon philosophers of your own nationality. I know German Ph.D.s who have never read more than 200 pages of English and French philosophy. Courses in contemporary philosophy give usually 90 percent of their time to the philosophical scene in the particular country to which the teachers belongs and only the small remainder to the international scene.

It is natural to wonder what our multitude of successful physical theories tell us about the world—singly, and as a body. What are we to think when one theory tells us about a flat Newtonian spacetime, the next about a curved Lorentzian geometry, and we have hints of others, portraying discrete or higher-dimensional structures which look something like more familiar spacetimes in appropriate limits?

These considerations on Carl G. Hempel's and Paul Oppenheim's lucid article “Studies in the Logic of Explanation'”—LE for short—are an attempt to exploit some of the considerable advances made there. I wish to show how these advances may be incorporated into a technique for the investigation of problems of the philosophy of science, which has been called discourse-analytic. This discipline consists essentially in a technique of examining actual texts provided by specialists, such as physicists, psychologists, historians, etc., the task being to discover fundamental logical and semantical features in the discourses under investigation. I am preparing a book on historical explanation, which will contain such case studies of historical texts.

Epistemology in modern times has been much influenced by the growth of the mathematical theory of probability. The reverse influence is less obvious, but perhaps equally important. This interrelation affords, moreover, an excellent example of the way philosophic ideas, while directing attention in one fruitful line of inquiry, at the same time may inhibit other discoveries until long overdue.

In an essay recently published in this journal, Branden Fitelson argues that a variant of Miller's argument for the language dependence of the accuracy of predictions can be applied to Joyce's notion of accuracy of credences formulated in terms of scoring rules, resulting in a general potential problem for Joyce's argument for probabilism. We argue that no relevant problem of the sort Fitelson supposes arises since his main theorem and his supporting arguments presuppose the validity of nonlinear transformations of credence functions that Joyce's theory, charitably construed, would identify as invalid on the basis of the principle of simple dominance.