The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

Al-Farahr’s commentarv and short treatise on Aristotle’s De interpretatione. Introduction and translation from Arabic by F. Zimmerman. Oxford: Published for the British Academy by Oxford University Press, 1987. clii + 287 pp. of English. £22.50 Johann Andreas Segner, Specimen logicae universaliter demonstrate. Appendices: Two dissertations De syllogismo. Edited by Mirella Capozzi. Bologna: Editrice CLUEB, 1990. clxxii + 281 pp. 85 000 Lire M. Borga, P. Fregugua And D. Palladino, I contribua fondazionali della scuola di Peano. Milano: Franco Angeli, 1985, 257 (...) pp. No price stated F. Cavaliere. La logica formale in Unione Sovietica. Gli anni del dibattito 1946-1965. Firenze: La Nuova Italia Editrice, 1990. ix + 140pp. Lire 42000 William S. Hatcher, Logic and logos. Essays on science, religion and philosophy. Oxford: George Ronald, 1990. x + 147 pp. £4.50/$9.50 E. J. Lowe, Kinds of being: a study of individuation, identity and the logic of sortal terms. Oxford and New York: Basil Blackwell, 1989. £25.00/$550.00 Reinhardt Grossmann, The fourth way. A theory of knowledge. Bloomington and Indianapolis: Indiana University Press, 1990. 312 pp.$35.00 J. Etchemendy. The concept of logical consequence. Cambridge, Mass, and London, England: Harvard University Press, 1990. vi 4- 174 pp. £19.95 Bernard Bolzano, Las paradojas del infinito. Translated from the German by L. Felipe Segura. Introduction by J. Sebestik. Mexico City: Faculty of Sciences, University of Mexico, 1991. 163 pp. No price stated W. Stantfy Jfvons, Pure logic and other minor works. Photoreprinted Bristol: Thoemmes, 1991. xxiii + 299 pp. No price stated Felix Kaufmann, L’infinito in matematica. Translated from the German, with an introduction, by L. Albertazzi. Gardolo di Trento: Luigi Reverdito Editore, 1990. 297 pp. L. 32.000 Massimo Libardi, Teorie de lie parti e delVintero. Mereologie estensionali. Trento: Centro Studi per la Filosofia Mitteleuropea, Quaderni 4-6, 302 pp. 22,000 lire M. S. Burcjin and V. I. Kuznetsoff, Aksio-logichcskiye aspekîi nauchnikh teorii. Kiev; Naukova Dymka, 1991. 184 pp. 3 roubles Hans Burkhardt and Barry Smith, Handbook of metaphysics and ontology. 2 vols. Munich: Philosophia, 1991. 1005 pp. 650 DM. (shrink)

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: ZFCAσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathsf {ZFCA}_{\sigma }$$\end{document} with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents.

The Geach‐Kaplan sentence is alleged to be an example of a non‐first‐orderizable sentence, and the proof of the alleged non‐first‐orderizability is credited to David Kaplan. However, there is also a widely shared intuition that the Geach‐Kaplan sentence is still first‐orderizable by invoking sets or other extra non‐logical resources. The plausibility of this intuition is particularly crucial for first‐orderism, namely, the thesis that all our scientific discourse and reasoning can be adequately formalized by first‐order logic. I first argue that the Geach‐Kaplan (...) sentence is, in fact, not first‐orderizable even by invoking extra non‐logical resources, in any sense that is acceptable for first‐orderism and adequately corresponds to the sense in which the Geach‐Kaplan sentence is deemed to be non‐first‐orderizable simpliciter via Kaplan's proof. To address this problem on behalf of first‐orderism, I then propose an alternative conception of first‐orderizability in the sense of which the Geach‐Kaplan sentence and any other second‐order sentences become first‐orderizable by invoking extra non‐logical resources; furthermore, in certain circumstances, they are first‐orderizable without incurring any extra ontological commitment. My analysis also turns out to yield (as a biproduct) a significant enhancement of the so‐called paradox of plurality. (shrink)

The purpose of the present paper is to prove the interpolation theorem for many-sorted languages which are, in terminology of Feferman neither restricted nor unrestricted. Such languages are often used in mathematical practice and have been investigated by several authors . The result is a generalization of the well-known Stern interpolation theorem for restricted m.s.l. and its proof depends heavily on that of Stern’s theorem. In place of the functions Rel + and Rel − our theorem treats the functions T (...) yp + and T yp − giving much more information about any formula. (shrink)