This chapter gives a critical exposition of Ockham’s innovations in the theory of the assertoric syllogism: his extensions of the class of syllogistic propositions to include singular propositions, propositions with a quantified predicate, tensed propositions and propositions in grammatical cases other than the nominative; his rules for the conversion of these propositions and for syllogisms containing them; his un-Aristotelian style of reducing syllogisms to the first figure by inference-to-inference transformations; and above all his use of supposition theory as a semantic base for this expanded syllogistic theory. His broadening of the scope of syllogistic theory resulted in abandoning several Aristotelian rules of the syllogism; it also resulted in a toleration of inferences containing redundant premises. The chapter provides proof-theoretic justifications for certain inferences that Ockham merely declared to be valid. It also argues that the originality of certain aspects of Ockham’s logic results from his philosophical nominalism.