According to a widespread view in medieval scholarship, theories of supposition are the medieval counterparts of theories of reference, and are thus essentially extensional theories. I propose an alternative interpretation: theories of supposition are theories of properties of terms, but whose aim is to allow for the interpretation of sentences. This holds especially of Ockham’s supposition theory, which is the main object of analysis in this paper. In particular, I argue for my intensional interpretation of his theory on the basis of two key-phrases in his Summa Logicae: ‘denotatur’ and ‘propositio est distinguenda’. Finally, I offer a reconstruction of his theory as a set of instructions to be carried out in order to generate the possible readings of (certain) sentences.