 If some S is not P, and every D is P, then it isn’t true that every S is D. For if every S is D, and every D is P, then every S is P. But if some S is not P, then as we saw above, not every S is P. So given the premises of (10), adding ‘every S is D’ would lead to contradiction: every S is P, and not every S is P. So the premises imply the negation of ‘every S is D’. This reasoning shows how (10) can be reduced to inferential patterns that seem more basic—raising the question of how much reduction is possible. Euclid’s geometry had provided a model for how to present a body of knowledge as a network of propositions that follow from a few basic axioms. Aristotle himself indicated how to reduce all the valid syllogistic schemata to four basic patterns, given a few principles that govern how the basic patterns can be used to derive others; see Parsons (2014) for discussion. And further reduction is possible given insights from the medieval period.