Most seventeenth-century textbooks of logic follow the usual pattern according to which a discussion of the constituents of propositions and of propositions themselves leads up to a part dealing with those combinations of propositions which exhibit a valid form of deductive argument. This chapter first builds up arguments against and in favour of syllogism. In spite of the manifold attempts to discredit the syllogism, several seventeenth-century writers succeeded in presenting the Aristotelian and scholastic treatment of that form of reasoning in more or less original guises. Of the many patterns of reasoning that are not syllogisms in a strict sense, such immediate inferences as the laws of the so-called square of opposition and the laws of conversion were usually assumed in the process of proving the validity of syllogisms. Gottfried Wilhelm Leibniz was exceptional in reversing that order by using syllogisms in his derivation of the laws of subalternation and conversion. The chapter also discusses Leibniz's conception of abstract calculus.