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AN INTENSIONAL LEIBNIZ SEMANTICS FOR ARISTOTELIAN LOGIC

Published online by Cambridge University Press:  17 March 2010

KLAUS GLASHOFF
KLAUS GLASHOFF*
Affiliation:
University of Hamburg
*
*DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HAMBURG, BUNDESSTRASSE 55, D-20146 HAMBURG, GERMANY E-mail:[email protected]
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Abstract

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” concepts—corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle’s theory via predicate logic.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 3 , Issue 2 , June 2010 , pp. 262 - 272
Copyright
Copyright © Association for Symbolic Logic 2010

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References

BIBIOGRAPHY

Arnauld, A. (1861). The Port Royal Logic. Translated from the French. Edinburgh, UK: James Gordon.Google Scholar
Boger, G. (1998). Completion, reduction and analysis: Three proof-theoretic processes in Aristotle’s Prior Analytics. History and Philosophy of Logic, 19, 187–226.CrossRefGoogle Scholar
Code, A. (1986). Aristotle: Essence and accident. In Grandy, R. E., and Warner, R., editors. Philosophical Grounds of Rationality. Oxford: Clarendon Press, pp. 411–440.Google Scholar
Corcoran, J., editor. (1972a). Ancient Logic and Its Modern Interpretation. Dordrecht, Holland/Boston, MA: D. Reidel Publishing Company.Google Scholar
Corcoran, J. (1972b). Completeness of an ancient logic. Journal of Symbolic Logic, 37(4), 696–702.CrossRefGoogle Scholar
Corcoran, J. (1974). Aristotle’s natural deduction system. In Corcoran, J., editor. Ancient Logic and Its Modern Interpretation. Dordrecht, Holland: Reidel, pp. 85–131.CrossRefGoogle Scholar
Ebbinghaus, K. (1964). Ein formales Modell der Syllogistik des Aristoteles. Number 9 in Hypomnemata. Göttingen, Germany: Vandenhoeck & Rupprecht.Google Scholar
Emilsson, E. (2009). Porphyry. In Zalta, E. N., editor. The Stanford Encyclopedia of Philosophy (Summer 2009 edition).Google Scholar
Frege, G. (1967). Concept script, a formal language of pure thought modelled upon that of arithmetic. In vanHeijenoort, J., editor, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, MA: Harvard University Press. Translation by S. Bauer-Mengelberg of ‘Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens’, Halle a. S.: Louis Nebert, 1879.Google Scholar
Frisch, J. C. (1969). Extension and Comprehension in Logic. New York: Philosophical Library.Google Scholar
Glashoff, K. (2002). On Leibniz’ characteristic numbers. Studia Leibnitiana, 34, 161.Google Scholar
Glashoff, K. (2005). Aristotelian syntax from a computational-combinatorial point of view. Journal of Logic and Computation, 15(6), 949–973.CrossRefGoogle Scholar
Kant, I. (1836). Metaphysical Works of the Celebrated Immanuel Kant. London: W. Simpkin and R. Marshall.Google Scholar
Kant, I. (1991). Schriften zur Metaphysik und Logik, Volume 2 of Werkausgabe Band VI. Frankfurt am Main, Germany: Suhrkamp.Google Scholar
Leibniz, G. W. (1999). Elementa calculi. In Sämtliche Schriften und Briefe IV, 1677-1690. Darmstadt/Leipzig/Berlin: Akademie-Verlag, pp. 195–205.Google Scholar
Lukasiewicz, J. (1957). Aristotle’s Syllogistic (second edition). Oxford: Clarendon Press.Google Scholar
Lyons, J. (1977). Semantics: Volume 1. Cambridge: Cambridge University Press.Google Scholar
Maldonado, A. M. (1998). Completitud de dos Cálculos Lógicos de Leibniz. Theso di grado, Departamento de Mathematicas, Universidad de Los Andes, Bogota.Google Scholar
Martin, J. N. (1997). Aristotle’s natural deduction reconsidered. History and Philosophy of Logic, 18, 1–15.CrossRefGoogle Scholar
Nedzynski, T. G. (1979). Quantification, domains of discourse, and existence. Notre Dame Journal of Formal Logic, XX(1), 130–140.Google Scholar
Porphyry. (1975). Isagoge. Trans. Warren, Edward. Toronto, Ontario, Canada: The Pontifical Institute of Medieaeval Studies.Google Scholar
Quine, W. v. O. (1951). Two dogmas of empiricism. The Philosophical Review, 60, 20–43.CrossRefGoogle Scholar
Smiley, T. (1973). What is a syllogism? Journal of Philosophical Logic, 2, 136–154.CrossRefGoogle Scholar
Smith, R. (1983). An ecthetic syllogistic. Notre Dame Journal of Formal Logic 24, 224–232.CrossRefGoogle Scholar
Tolley, C. (2007). Kant’s conception of logic (Immanuel Kant). PhD Thesis, The University of Chicago.Google Scholar