  It is notorious that Aristotle's syllogistic logic cannot capture the logic of relations, and so is inadequate to the task of presenting the deductive structure of mathematics. This difficulty was handled in practice by placing the relational arguments outside the formal structure of syllogistic which formed the framework of a science. The principles of a demonstration, Aristotle thought, always had to be universal, reporting, as they do, necessary connections. Now he was accustomed to showing universal truths through one sort of ‘setting forth’ (ekthesis), working out the thing informally in a particular case and then using universal generalization to establish the universal truth. (Posterior Analytics I 4, 73b33-74a4 clearly sets out the procedure.) The ‘setting forth’ stood outside the formal structure of the syllogistic, and it allowed the establishment of universal premises with complex terms, for instance, a premise to the effect that “every triangle-with-its-base-extended-so-as-to-form-an-external-angle is a triangle-with-its-external-angle-equal-to-the-sum-of-the-two-angles-opposite-it-in-the-triangle.” Aristotle here uses the term “universal” to mean not only that the predicate belonged in every case to the subject, but also that it could not be proved of the subject simply because of its belonging to some wider universal to which that subject also belonged. So it would not do to show by ‘setting forth’ and assume as a first principle that an isosceles triangle has its angles equal to two right angles, for this is because it is a triangle, not because it is an isosceles triangle. A first principle must be not just universally quantified, but commensurately universal, so that the predicate belongs to the subject in every case, and belongs to whatever else it belongs to because it belongs to that subject, and because whatever else it belongs to, essentially or accidentally, falls under that subject.