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Additional info for Aristotle's Modal Syllogisms
42 LAbb LAcc MAab Aad LAda LAcb LAbd MAdc * MAac. 51 1. The consequent is LIac, and the antecedents include neither LAaa nor LAcc, though they do include both Aac and Aca. If (a) any antecedent LAbia or LAbic (or both) is present, the expression is asserted. Otherwise (b) it is rejected. Proof. (a) Suppose we have LAbia among the antecedents. Then, since LAbia Aac Abic (Barbara X L X ) , we have Abic LAbia LIac (Darapti X L L ) , so that the expression is asserted. A similar argument applies if the antecedents include LAbic.
We find the convertibility of the LE-premiss in 25a29-31, and of the LAand LI-premisses in 25a32-34. In 25a37-25b2 Aristotle says that no matter in which sense the word “possible” is taken, affirmative possible premisses coiivert as affirmative assertorics do. Thus we have the convertibility of the MA- and MI-premisses. But in negative statements the case is different; only in the sense of “possible” in which we would be prepared to say that what is necessary is also possible (25b4) is it allowable to proceed from “Possibly no A is B” to “Possibly no B is A”.
CC MOaa M l a b C NMIab NMUaa (187, Df M , Df E , Df 0, RN) 188. CC MOaa MIab C LEab LAaa 188=C*7-*2 *7. C MOaa MIab VII=C*8-*7 *8. MIab, from which MAab, Iab, Aab, LIab, LAab may be rejected. This completes our rejection of all simple expressions of the L-X-M calculus which are not theorems. 21. A decision procedure for the L - X - M calculus Having seen how the process of rejection proceeds, we may now pass to the establishment of a decision procedure for the L-X-M calculus, in the course of which the remainder of the required axiomatic rejections will be given.