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Ancient Greek
Logic: Aristotle

Aristotle. Born: 384 BC in Stagirus, Macedonia, Greece; Died: 322 BC in Chalcis, Euboea, Greece. Aristotle's work on logic was the basis for any logical research until the mid of the 19th century. Aristotle, more than any other thinker, determined the orientation and the content of Western intellectual history. His concepts are still engraved in Western thinking.

Bild Aristoteles

  • An Intensional Leibniz Semantics for Aristotelian Logic [PDF] (The Review of Symbolic Logic, Published online by Cambridge University Press 17 Mar 2010 doi:10.1017/S1755020309990396)
  • 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.

  • Aristotelian Logic from a Computational-Combinatorical Point of View [PDF] (Journal of Logic and Computation 2005 15(6):949-973)
  • This paper translates Aristotle's syllogistic logic of the Analytica Priora into the sphere of computational - combinatorical research methods. The task is accomplished by formalising Aristotle's logical system in terms of rule-based reduction relations on a suitable basic set, which allow us to apply standard concepts of the theory of such structures (Newman lemma) to the ancient logical system.

  • Zur Übersetzung der Aristotelischen Logik in die Prädikatenlogik (in German) [PDF] (2005)
  • Seit den Anfängen der modernen symbolischen Logik hat es immer wieder Versuche gegeben, die Aristotelische assertorische Syllogistik in die Prädikatenlogik (mit monadischen, d.h. einstelligen Prädikaten) zu übersetzen. Aber alle diese Versuche modifizieren klassische Gesetze der Aristotelischen Logik oder beachten sie einfach nicht. Der Fehlschlag solcher Versuche liegt nun daran, dass es überhaupt keine vernünftige derartige Übersetzung gibt!

    Since the beginning of modern symbolic logic, there have been attempts to translate Aristotelian logic into monadic predicate logic. However, all these attempts do not reproduce classical laws of Aristotelian logic. In this paper we use a computer-based approach for the proof of the fact that there does not exist any satisfactory such translation of Aristotelian logic into predicate logic.

  • Algorithmic Aristotelian Logic:
    Online experiments with SYLLOgism
  • Aristotle's logic of the Analytica Priora has been formalized in many different ways since Leibniz' and, later but independently, Boole's invention of an Algebra of Logic. We present an online accessible site where Aristotelian logic has been implemented in the framework of a computer software utilizing "rule-based reasoning". Thus, our system SYLLOgism is a young relative of the "traditional" syllogistic theory as well as of the "systems of natural deduction" developed in the 1970s by Corcoran and Smiley. This site is meant not only as a tool for gaining elementary understanding of the basic syntactical and semantical concepts of Aristotelian logic but also as a research instrument for those who are convinced that the formal power of Aristotelian logic in the computer age has not yet come to light in an adequate manner.