  A demonstrative science then, requires for its being understood (1) an understanding of the terms entering into it (later commentators suggest that “nominal definitions” provide this), (2) knowledge of certain axioms applicable to many sciences (for instance, the law of the excluded middle), (3) knowledge of first principles, that is,indemonstrable truths proper to the science under consideration (PA I 1, 71a11-17; 2, 72a14-22; 10), (4) knowledge of real definitions of the subject of the science, and its various species and predicates (PA I 33, 89a18; II 3, 90b24; 13, 96b23; 17, 99b22). The principles of a demonstration must be true, indemonstrable, and such as to provide the reason for the truth of the conclusion, but they must also be necessary and per se (PA I 2). These last two requirements were variously interpreted by medieval thinkers, but we can note that Aristotle allows a statement can be per se in two ways: (1) if its subject term somehow includes its predicate within its definition, so that “A human being is rational” is per se, and (2) if its predicate term includes its subject implicitly in its definition as the proper subject for that sort of accident, so that “A human being is able to laugh” is per se (PA I 4, 73a35-b24. (There are two other senses of “per se” given here, and although it is reasonably supposed they have no bearing on demonstration, some medieval commentators tried to make something of them.) The middle term of a demonstration must express the cause why the predicate of the conclusion belongs to its subject. The predicate of the conclusion, the “attribute,” will not be part of the essence of the subject, but will somehow follow on its essence. It was disputed in the Middle Ages whether the middle term in the highest sort of demonstration would be the real definition of the subject, or the real definition of the attribute, or somehow include both, but it was generally agreed that it would be a real definition, and that one of the premisses of a demonstration would express a necessary truth not derivable by a simple analysis of either the nominal or the real definition of its subject. (The central text here, and that a very difficult one suggesting a number of different possible lines of analysis, is PA II 8-10.)