Philosophy
L. Mchedlishvili. Ecthesis in the Aristotelian Syllogistics. - Many-valued, Relevant and Paraconsistent Logics (Proceedings of the Scientific Seminar on Logic). - Moskow: Institute of Philosophy - 1984 - 119-129 pp. - Bibl. 9 - in Russian.
L. Mchedlishvili. A Reconstruction of the Method of Proof by Ecthesis and the Systems of Positive Syllogistics.- Aristotelian Logic. Proceedings of Symposium - Tbilisi: University press. - 1985 - 21-35 pp. - Bibl. 9 - in Russian.
L. Mchedlishvili. The Method by Ecthesis and a Reconstruction of the Aristotelian Apodictic Syllogistics. - Metalogical Investigations.-Tbilisi: Metsniereba - 1985 - 13-28 pp.- Bibl. 15 - in Russian.
L. Mchedlishvili. Proof by exposing in the Aristotelian Syllogistics.- Philosophical and Sociological Thought N 2- Kiev- 1989- 69-74 pp. - in Russian.
L. Mchedlishvili. Towards Semantics for the Aristotelian Apodictic Syllogistics. - Logical Investigations.- vol. 6.- Moskow: ROSSPEN - 1999 - 230-240 pp. - Bibl. 16 - in Russian, abstract in English
The Problem. At reduction of imperfect modes of syllogism to perfect modes of first figure in Prior Analytics the three methods are used by Aristotle - direct proof by conversion, reduction and impossible and proof by exposing (ecthesis) of term. The formal reconstruction of the first two methods in the syllogistic systems formed as independent theories on the basis of propositional calculus (in Lukasiewicz's mode) presents no difficulties, but in none of them - neither in traditional nor in formal presentation - among the means of inference nothing is found that could be interpreted as the reconstruction of the method by exposing (ecthesis). In reasoning by exposing of the term the crucial part is given to particular categorical propositions understood by the interpreters of the syllogistics since Alexander of Aphrodisias as sentences construction of which is completed by applying an existential quantifier. In the light of such interpretation as shown by Lukasiewicz proof by exposing turns out a particular instance of the correct method of reasoning using the rule of elimination of the existential quantifier. But the language of the independent formal syllogistics does not contain quantifiers and the problem of the possibility of reasoning by the method of exposing in the quantifier-free language of syllogistics remains unsolved.
Solution of the problem suggested by the author. Particular propositions "Some S are P" (SiP) and "Some S are not P" (SoP) really contain quantifiers, but for Aristotle quantifier in its present-day meaning is not an independent syntactical unit of a language and cannot be separated from the rest part of a particular proposition. Thus, sentences (MaS and MaP) and (MaS and MeP) have correspondingly the same relation to SiP and SoP as in the logic of predicates A(y) has to ExA(x) where "a" and "e" are syllogistical constants "all j are j" and "noj is j". The following explication is suggested: in reasoning by method of exposing rules of the type of subsidiary deduction analogues to the rule of elimination of an existential quantifier are used. These rules for Lukasiewicz's system of positive assertoric syllogistics are formed in the following way (in each of these rules it is understod that M does not occur either in G or in F):
Rule of o-elimination. If proved that from the premises G, MaS and MeP follows the conclusion F, then it is proved that from the premises G and SoP the same conclusion F follows.
Rule of i-elimination. If proved that from the premises G, MaS and MaP the conclusion F follows, then it is proved that from the premises G and SiP the same conclusion F follows.
Similar rules can be formulated for other systems of positive syllogistics different from the system of Lukasiewicz.
In Aristotelian apodictic syllogistics the method of exposing has some specificity determined by the modal character of particular apodictic propositions, but the analogy with the rule of elimination of an existential quantifier remains essential.
Rule of oL-elimination. If proved that from the premises G, MaS and MeLP the conclusion F follows, then it is proved that from the premises G and SoLP the same conclusion F follows.
Rule of iL-elimination. If proved that from the premises G, MaS and MaLP as well as from the premises G, MaLS and MaP the conclusion F follows, then it is proved that from the premises G and SiLP the same conclusion follows.('L' is a sign of the apodictic qualification of propositions).