Bulletin of Symbolic Logic 25 (2):196-207 (2019)

Abstract
By considering the new notion of the inverses of syllogisms such as Barbara and Celarent, we show how the rule of Indirect Proof, in the form used by Aristotle, may be dispensed with, in a system comprising four basic rules of subalternation or conversion and six basic syllogisms.
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DOI 10.1017/bsl.2018.59
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References found in this work BETA

Basic Proof Theory.Roy Dyckhoff - 2001 - Bulletin of Symbolic Logic 7 (2):280-280.
Stoic Syllogistic.Susanne Bobzien - 1996 - Oxford Studies in Ancient Philosophy 14:133-92.
What is a Syllogism?T. J. Smiley - 1973 - Journal of Philosophical Logic 2 (1):136 - 154.
Aristotle's Syllogistic and Core Logic.Neil Tennant - 2014 - History and Philosophy of Logic 35 (2):120-147.
Completeness of an Ancient Logic.John Corcoran - 1972 - Journal of Symbolic Logic 37 (4):696-702.

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