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  1. Aristotle's demonstrative logic.John Corcoran - 2009 - History and Philosophy of Logic 30 (1):1-20.
    Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's two-volume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning showing (...)
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2015-01-22
INDIRECT REASONING IN PRIOR ANALYTICS
John Corcoran

University at Buffalo

REDUCTIO QUESTIONS

Corcoran’s 2009 ARISTOTLE’S DEMONSTRATIVE LOGIC deals decisively with several issues that had previously been handled by vague speculation and dogmatic pontification if at all. One possible example: Corcoran [2009, p. 13] proves conclusively that the imperfect syllogisms Baroco and Bocardo—which Aristotle completed indirectly [by reductio-ad-impossible]—cannot be completed directly. More generally, Corcoran shows that no valid premise-conclusion argument, regardless of the number of premises,  having an existential negative [“particular negative” or “O-proposition”] as a premise can be completed using a direct deduction—assuming of course that no premises are redundant and that the conclusion is not among the premises. To be clear this means that for no such argument is it possible to deduce the conclusion from the premises without using reductio.

This result, called the EXISTENTIAL-NEGATIVE EXCLUSION [ENE], was circulated informally by Corcoran much earlier but it seems not to have been printed before 2009.

Q1: Was ENE, or a stronger result, stated in print before 2009?

Q2: Was ENE, or a stronger result, proved in print before 2009?

Q3: Are there other categorical arguments besides those having existential negative premises that are not directly deducible?

Q4: What is a necessary and sufficient condition for a categorical argument to be directly deducible?

Q5: Does Aristotle say anything about incomplete syllogisms that cannot be completed directly?

Q6: Which later logicians say anything notable about premise-conclusion arguments that cannot be shown to be valid by direct deduction?

The above presupposes the system of deductions and the definitions given in Corcoran’s 2009 ARISTOTLE’S DEMONSTRATIVE LOGIC.

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