Schrödinger logics are logical systems in which the principle of identity is not true in general. The intuitive motivation for these logics is both Erwin Schrödinger's thesis that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understand identity as meaning indistinguishability . Observing that these concepts are equivalent in classical logic and mathematics, which underly the usual physical theories, we present a higher-order logical system in which these concepts (...) are systematically separated. A 'classical' semantics for the system is presented and some philosophical related questions are mentioned. One of the main characteristics of our system is that Leibniz' Principle of the Identity of Indiscernibles cannot be derived. This fact is in accordance with some authors who maintain that quantum mechanics violates this principle. Furthermore, our system may be viewed as a way of making sense some of Schrödinger's logical intuitions about the nature of elementary particles. (shrink)

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)

Interpretability logic is a modal logic that can be used to describe relative interpretability between extensions of a given first-order arithmetical theory. Verbrugge semantics is a generalization of the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. The Van Benthem Correspondence Theorem establishes modal logic as the bisimulation invariant fragment of first-order logic. In this paper we show that a special type of bisimulations, the so-called w-bisimulations, enable an analogue of the Van Benthem (...) Theorem for interpretability logic w.r.t. Verbrugge semantics. (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)

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)