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Completeness of an ancient logic

  • John Corcoran (a1)

In previous articles ([4], [5]) it has been shown that the deductive system developed by Aristotle in his “second logic” (cf. Bochenski [2, p. 43]) is a natural deduction system and not an axiomatic system as previously had been thought [6]. It was also pointed out that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument formable in the language of the system is demonstrable by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotélian system.

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[1] Aristotle, Prior analytics.
[2]Bochenski I. M., A history of formal logic (translated by Thomas Ivo), Chelsea, New York, 1970.
[3]Church Alonzo, Introduction to mathematical logic. Vol. I, Princeton University Press, Princeton, N.J., 1956.
[4]Corcoran John, Aristotle's natural deduction system, this Journal, vol. 37 (1972), p. 437. Abstract.
[5]Corcoran John, A mathematical model of Aristotle's syllogistic, Archiv für Geschichte der Philosophie (to appear).
[6]Łukasiewcz Jan, Aristotle's syllogistic from the standpoint of modern formal logic, Clarendon Press, Oxford, 1951.
[7]Rose Lynn, Aristotle's syllogistic, Springfield, Illinois, 1968.
[8]Ross W. D., Aristotle's prior and posterior analytics, Clarendon Press, Oxford, 1965.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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