1. IMPORTANCE OF THE SUBJECT In 1900, in A Critical Exposition of the Philosophy of Leihniz, Russell made the following assertion: "That all sound philosophy should begin with an analysis of propositions is a truth too evident, perhaps, to demand a proof". 1 Forty years later, the interest aroused by this notion had not decreased. C. J. Ducasse wrote in the Journal of Philosophy: "There is perhaps no question more basic for the theory of knowledge than that of the nature of 2 propositions and their relations to judgments, sentences, facts and inferences". Today, the great number of publications on the subject is proof that it is still of interest. One of the problems raised by propositions, the problem of deter mining whether propositions, statements or sentences are the primary bearers of truth and falsity, is even in the eyes of Bar-Hillel, "one of the major items that the future philosophy oflanguage will have to discuss". 3 gave a correct summary of the situation when he wrote in his Ph. Devaux Russell (1967): Since Peano and Schroder who, in fact, adhered more faithfully to Boole's logic of classes, the logical and epistemological status of the proposition together with its analysis have not ceased to be the object of productive philosophical controversies. And especially so since the establishment of contemporary symbolic logic, the foundations 4 of which have been laid out by Russell and Whitehead. * 2.

Though the subject of this work, "nominalism and contemporary nom inalism", is philosophical, it cannot be fully treated without relating it to data gathered from a great variety of domains, such as biology and more especially ethology, psychology, linguistics and neurobiology. The source of inspiration has been an academic work I wrote in order to obtain a postdoctoral degree, which is called in Belgium an "Aggregaat voor het Hoger Onderwijs" comparable to a "Habilitation" in Germany. I want to thank the National Fund of Scientific Research, which accorded me several grants and thereby enabled me to write the academic work in the first place and thereafter this book. I also want to thank Prof. SJ. Doorman (Technical University of Delft) and Prof. G. Nuchelmans (University of Leiden), who were members of the jury of the "Aggre gaatsthesis", presented to the Free University of Brussels in 1981 and who by their criticisms and suggestions encouraged me to write the present book, the core of which is constituted by the general ideas then formulated. I am further obliged to Mr. X, the referee who was asked by Jaakko Hintikka to read my work and who made a series of constructive remarks and recom mendations. My colleague Marc De Mey (University of Ghent) helped me greatly with the more formal aspects of my work and spent too much of his valuable time and energy to enable me to deliver a presentable copy. All remaining shortcomings are entirely my responsibility. I asked Prof.

This new volume on logic follows a recognizable format that deals in turn with the topics of mathematical logic, moving from concepts, via definitions and inferences, to theories and axioms. However, this fresh work offers a key innovation in its ‘pyramidal’ graph system for the logical formalization of all these items. The author has developed this new methodology on the basis of original research, traditional logical instruments such as Porphyrian trees, and modern concepts of classification, in which pyramids are the central organizing concept. The pyramidal schema enables both the content of concepts and the relations between the concept positions in the pyramid to be read off from the graph. Logical connectors are analyzed in terms of the direction in which they connect within the pyramid. Additionally, the author shows that logical connectors are of fundamentally different types: only one sort generates propositions with truth values, while the other yields conceptual expressions or complex concepts. On this basis, strong arguments are developed against adopting the non-discriminating connector definitions implicit in Wittgensteinian truth-value tables. Special consideration is given to mathematical connectors so as to illuminate the formation of concepts in the natural sciences. To show what the pyramidal method can contribute to science, a pyramid of the number concepts prevalent in mathematics is constructed. The book also counters the logical dogma of ‘false’ contradictory propositions and sheds new light on the logical characteristics of probable propositions, as well as on syllogistic and other inferences.

A collection of papers presented at the conference on Probability Theory - Philosophy, Recent History and Relations to Science, University of Roskilde, Denmark, September 16-18, 1998. Since the measure theoretical definition of probability was proposed by Kolmogorov, probability theory has developed into a mature mathematical theory. It is today a fruitful field of mathematics that has important applications in philosophy, science, engineering, and many other areas. The measure theoretical definition of probability and its axioms, however, are not without their problems; some of them even puzzled Kolmogorov. This book sheds light on some recent discussions of the problems in probability theory and their history, analysing their philosophical and mathematical significance, and the role pf mathematical probability theory in other sciences.

He shows direct and inverse links between foundations of science and new theories and empirical facts evolved from those, how among many potentially possible histories of science a culture selects just those directions which become a real history of science. The author analyses mechanisms of the generation of scientific theories and shows that those are changed in the process of historical development of science. He displays three historical types of scientific rationality (classical, non-classical and post-non-classical, which appears in modern science) and shows features of their coexistence and interplay. It is shown that along with the emerging of post-non-classical rationality science increases the sphere of its worldview applications. Science begins to correlate not only with the basic values of technogenic civilization but also with some values and patterns of traditional cultures.

Initially proposed as rivals of classical logic, alternative logics have become increasingly important in areas such as computer science and artificial intelligence. Fuzzy logic, in particular, has motivated major technological developments in recent years. Susan Haack's Deviant Logic provided the first extended examination of the philosophical consequences of alternative logics. In this new volume, Haack includes the complete text of Deviant Logic, as well as five additional papers that expand and update it. Two of these essays critique fuzzy logic, while three augment Deviant Logic's treatment of deduction and logical truth. Haack also provides an extensive new foreword, brief introductions to the new essays, and an updated bibliography of recent work in these areas. Deviant Logic, Fuzzy Logic will be indispensable to students of philosophy, philosophy of science, linguistics, mathematics, and computer science, and will also prove invaluable to experienced scholars working in these fields.

Since their appearance in the late 19th century, the Cantor--Dedekind theory of real numbers and philosophy of the continuum have emerged as pillars of standard mathematical philosophy. On the other hand, this period also witnessed the emergence of a variety of alternative theories of real numbers and corresponding theories of continua, as well as non-Archimedean geometry, non-standard analysis, and a number of important generalizations of the system of real numbers, some of which have been described as arithmetic continua of one type or another. With the exception of E.W. Hobson's essay, which is concerned with the ideas of Cantor and Dedekind and their reception at the turn of the century, the papers in the present collection are either concerned with or are contributions to, the latter groups of studies. All the contributors are outstanding authorities in their respective fields, and the essays, which are directed to historians and philosophers of mathematics as well as to mathematicians who are concerned with the foundations of their subject, are preceded by a lengthy historical introduction.