I first learned about enthymeme’s from Frederick Norris’s excellent commentary on Gregory of Nazianzus’s Theological Orations, Faith Gives Fullness to Reasoning, which needs to be reprinted in paperback for the general betterment of humankind. Norris defines an enthymeme as a rhetorical syllogism: a syllogism that has one of its premises unstated as it is assumed to be supplied by the audience. But this “missing premise” definition is, at least according to James Allen in his book Inference from Signs: Ancient Debates about the Nature of Evidence, not really what an enthymeme is about.
It would be a mistake to picture the orator trimming premisses from full-blown categorical syllogisms that he has first framed before his mind’s eye in order to present them in the form suitable to the rhetorical occasion (p. 24).
So then what is an enthymeme? And why is one of the premises so often missing?
Enthymemes are the rhetorical counterpart to dialectic’s syllogisms (likewise, paradigms are the rhetorical counterpart to induction) (p. 19). However, they are not concerned with valid deduction, but reputable (ἔνδοξος) arguments (p. 20). An enthymeme is further divided between 1) enthymemes from likelihoods and 2) enthymemes from signs (Rhet. 1.2 [70a9–11; 1357a32–3]), the former being a more reputable form of enthymematic argument (p. 23). What is a likelihood? That, Aristotle says (Rhet. 1.2, 1357a34), is “something people know comes to be or not for the most part” (p. 23).
An enthymeme from likelihoods is an argument bringing a particular case under an acknowledged general rule permitting exceptions (1357a34–b1) (p. 23).
What is a sign? For Aristotle (An. pr. 2.27, 70a7), a sign is “a premiss that is or is such as to be reputable (ἔνδοξος)” (p. 23). In an enthymeme from signs,
the sign—a particular fact or alleged fact—is put forward as a ground for the conclusion of which the orator wishes to convince his audience (p. 23).
How does this affect the shape of the enthymeme? Allen says (p. 25) that
In an enthymeme from likelihoods the crucial element on which the argument turns is the generalization under which the particular item in question is being brought. This must in all cases be stated, while the minor premiss, which states that the subject term of the major premiss belongs to the item under discussion, can and often will go without saying. And it is the generalization stated in the major premiss that is also the potential object of controversy. An orator who needs to oppose an argument from likelihood will try to show that his opponent’s conclusion is not likely because it is based on a generalization that is false or in some way not appropriate to the case at hand . . . By contrast, he will treat the syllogistic structure of this argument and the truth of its minor premiss as unproblematic. This is what it is to treat an argument as an argument from likelihood, and to evaluate its merits and faults as such.
In the case of an argument from signs, on the other hand, the new element on which the argument is seen to hinge will be the new piece of evidence to which the orator wishes to direct his auditors’ attention and which counts as evidence against a background of uncontroversial assumptions. If we concentrate for the moment on the relatively simple case of the valid first-figure sign-inference [i.e., Barbara], it is clear that the sign functions as a ground in this way in virtue of what we should call a covering generalization (cf. An. pr. 1.32, 47a16–27). And it is this covering generalization, e.g. that the feverish are ill, formulated in the major premiss, that is typically treated as part of the background of uncontentious assumptions in virtue of which the sign is able to serve as evidence for the conclusion at issue. For this reason, it can and typically will be omitted in the presentation of the sign-inference.
So, Allen is arguing that a missing premise is not the definition of an enthymeme, but a characteristic of it. What premise is assumed (and so missing), depends on the type of enthymematic argument being pursued. If one is arguing from likelihood, the minor premise may be assumed (and so not explicitly stated), whereas if one argues from a sign, then the major premise may be assumed (and so not explicitly stated).
Interestingly, only one type of sign-inference is valid, which Aristotle calls a token (τεκμήριον). This takes the Barbara form. Aristotle’s example is:
P1: All those with fever are ill.
P2: This man has fever (the sign)
C: Therefore he is ill
The two other sign-inferences (not given a technical name but associated with Cesare and Darapti respectively) Aristotle says are invalid, but according to Allen, they are still enthymemes (Rhet. 1357a27–32; b5, 14, 22; cf. 1403a11) (pp. 28, 30). This adds to his argument that it is wrong to define an enthymeme merely as a syllogism with one of the premises because some enthymemes are not valid syllogisms but are still, in Allen’s argument, reputable (ἔνδοξος, though Aristotle does not specifically state this) (p. 30).