Aristotle - The Organon - index for ANALYTICA PRIORIA Book 1 Part 25

Every demonstration proceeds through three terms

Paragraph 1 It is clear too that every demonstration will proceed through three terms and no more, unless the same conclusion is established by different pairs of propositions;
Paragraph 2 (1) If it is E the syllogism will have A and B for its sole premisses.
Paragraph 3 (2) But if from the propositions A and B there follows not E but some other conclusion, and if from C and D either A or B follows or something else, then there are several syllogisms, and they do not establish the conclusion proposed:
Paragraph 4 So it is clear that every demonstration and every syllogism will proceed through three terms only.
Paragraph 5 This being evident, it is clear that a syllogistic conclusion follows from two premisses and not from more than two.
Paragraph 6 If then syllogisms are taken with respect to their main premisses, every syllogism will consist of an even number of premisses and an odd number of terms (for the terms exceed the premisses by one), and the conclusions will be half the number of the premisses.

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