Quintilian, and J.S. Watson. Institutes of Oratory.Edited by Lee Honeycutt, 2010. Kindle Electronic Edition. Book V Chapter 14.
In this chapter Quintilian discusses the enthymeme, the epicheirema, and the syllogism. An enthymeme, as used by a rhetorician, is the actual enunciation of an argument (Quintilian V.14.1). It may or may not have the complete logical conclusion.
The epicheirema consists of, at least, a proposition, an assumption, and a conclusion. The proposition and assertions made may have proofs attached (Quintilian V.14.6). Quintilian describes several slight variations in form.
A syllogism is essentially an epicheirema, “except that the syllogism has a greater number of forms and infers truth from truth, while the epicheirema is generally employed about probabilities” (Quintilian V.14.14).
Quintilian points out the need to bring solid conclusions throughout arguments, often varying the form of the arguments (Quintilian V.14.17).
“The enthymeme is called by some an oratorical syllogism, by others a part of a syllogism, because the syllogism has always its regular proposition and conclusion and establishes by means of all its parts that which it has proposed, while the enthymeme is satisfied if merely what is stated in it be understood” (Quintilian V.14.24). The purpose may be analogous but the forms meet different criteria. The orator does not need to prove each step of an argument. He simply needs to gain acceptance of his overall goal (Quintilian V.14.32).