I arranged the subject approached in this text so that the issues will generally fix up on the route from general/simple/undisputed to particular/complex/controversial. Our main task, however, is to make some sense out of the widely discussed notion of unexpressed premise(s), and not to work out theoretical solutions or solve decades-old disputes. As pointed in the chapeau of the first part, we aim for the “average” form of these concepts.
Although some might argue that an unusual haze covered the concept of enthymeme right from its debut in Aristotle’s Prior Analytics and then in Rhetoric, I think we can outgo the issue of Aristotle’s old (and somewhat incompatible) definitions. As it is obvious that over the centuries the definition from the Rhetoric prevailed and gathered adherents, and since linking the definitions from the two works cited above would not be that inconceivable, we will just stick with the mob, for which the main definition is this:
“The enthymeme is a sort of syllogism” (1355a)
“The enthymeme must consist of few propositions, fewer often than those which make up the normal syllogism” (1357a)
The most frequent example of an enthymeme is a crippled syllogism with a missing universal proposition:
Socrates is human,
Therefore, Socrates is mortal
A six year old will have no problem supplying the unexpressed premise here, namely the universal proposition: All men are mortal. The rules stay pretty much the same when the unexpressed proposition is the conclusion (consider stating both premises in the previous example; the six year old will nail it down without trouble), except that we might want to change “unexpressed premise” into “unexpressed conclusion” or “non-explicit assumption/conclusion” or others as such. We will pay little attention to unexpressed conclusions, though one should bear in mind the fact that they can be quite tricky.
Let us now take one step forward into our analysis with this question: how do we know that the premise we supplied is the good one? The spontaneous answer is “It makes the argument valid”. And it does, yet consider this example:
Dr. House is always making fun of patients.  He should be fired.
There is evidently a gap between  and , and yet no clear clues as to how to choose the (“best”? “correct”? “meant”?) unexpressed premise . The first problem is that one might not view this argument as an enthymeme at all, but instead take it as a non-sequitur, in which the conclusion simply does not follow. One suggested criterion is to look for topical relevance, in other words, search for “a common content expression, or the ability to produce a common content expression by making definitionally equivalent substitutions”. Since in our chosen example, the relevance is quite obvious, we should be able to make the next step. Still, several candidates are possible, all of which may render the argument valid, or at least strengthen it as opposed to the primary  therefore  form:
[2.1] If Dr. House is always making fun of patients, he should be fired
[2.2] Doctors that make fun of patients should be fired
[2.3] People violating their job contracts should be fired
[2.4.1] Doctors that make fun of patients are committing malpractice
[2.4.2] Doctors committing malpractice should be fired
[2.5] Doctors that make fun of patients should be fired or (V) France is in Italy
Things are getting a bit complicated. Let’s take a look at these sentences. Although fills the gap between the stated premise and conclusion, [2.1] functions only as a proposition which explicitly tells us that “it is permitted to infer the conclusion from the given premise”. It is not just the fact that [2.1] adds nothing new to the argument (being merely a “reiterative candidate”) or that it is superfluous, untestable, or not general enough; it’s just that by doing this we could get beaten up by epistemologists. By providing this sort of “if … then …” filling, every argument could be made deductively valid on subsequent analysis.
Propositions [2.2], [2.3] and [2.4] seem to be all improved versions of [2.1]. We may say that they are stronger – if we regard a certain definition of “propositional strength”, or more general or even testable. Some of them are even plausible – that is to say, it is quite ok to assume that someone had them in mind when putting forward this argument. All these characterizations might be true but, as one might have expected, they are not helping us to choose the (right) candidate – since we are trying to cut them down to just one. This leads us to our next step.
So are we allowed to add this content? And moreover, do we want to do that? Both the hearer and the analyst of this argument are always in danger of making two equally significant errors. First of all, they could be committing something that resembles a well known fallacy: “There is always the danger of the straw man fallacy, of attributing as a premise or conclusion of a speaker's argument a proposition that exaggerates or distorts the argument in order to make it easier to refute”. The remedy to this, continues Walton, is the principle of charity, which advises us to reconstruct the argument in such way to maximize its strength. Of course, one might ask “what kind of strength?” and while we’re at it “how come we always have to reconstruct them as deductively valid”? Whatever happened to induction or abduction?
Consider now the second (rather opposite) error: we are faced with an enthymeme which we transform into a deductively valid, maximally strengthen, fully generalized and testable argument: how does the hearer/analyst know this is the same argument from the enthymeme? How do we know that our charitable misinterpretation is not a degenerated one that takes into account … a different argument?
To my mind, the problem of unexpressed premises is nowadays fixed in this Golden Nasty “Lesser-of-two-evils” Rule that could be formulated thus: You are always susceptible of being closer to one of the two errors. Some authors describe it in the form of analyst vs. interpreter, used premises vs. needed premises, logical minimum vs. pragmatical optimum or argument’s assumption vs. arguer’s assumption. Either way, everyone feels that some distinction must be made between some sort of “action X” and “action Y” both of which are assumed to be performable when regarding the same argument but which differ in some respects.
- needed premises and used premises
In the early 1980s, two major works founded the needed/used distinction: “Identifying Implicit Assumptions” (1982), an article by Robert H. Ennis and “A Practical Study of Argument” (1985), a book written by Trudy Govier. The terminology would be later assumed by many theorists, the Canadian professor Douglas Walton being one of them. The distinction is at first easy to understand and useful. It is the theoretical back-up of such an approach that has received its criticism. The criterion for making such a differentiation is the goal of the analysis; it’s like pre-asking ourselves why we are searching for the “thing” that is missing. If we want to “evaluate” or “search the truth and validity” of an argument, we should fill in needed premises: ones that make the argument valid. It is after making this analytic step (with either one of the unexpressed premise from [2.1] to [2.5], in our example) and only after this that we are able “judge” the sustainability of what has been given. You might notice that this step only takes us from non-sequitur to enthymeme and nothing more. If we want to “interpret the meaning” and “attribute it” to the arguer, than we are in for a harder job, and that is finding used premises: the ones that are plausible to have been used by the arguer himself. This is highly problematic: how do we interpret arguers’ texts? How do we know what the speaker had in mind based on given (explicit) claims? Are we hoping for any degree of certainty when making such interpretation? This is sometimes referred to as “the attribution problem” and it has been responded to in many ways.
- logical minimum and pragmatical optimum
This distinction is mainly present among pragma-dialecticians and is sometimes offered as a solution to the “attribution problem”. The “logical minimum” is the sentence that has as its antecedent the explicit premise and as its consequent the conclusion of the explicit argument. In other words, the now well-known “if … then …” statement, as reiterative and useless as presented above. The “pragmatic optimum” is the generalized version of the logical minimum (in other words, “made as informative as required”) that does not ascribe unwarranted commitments to the speaker. A commitment is “warranted” if it can be inferred from the speech acts explicitly put forward. That is to say, denying the commitment and affirming the speech act would be a pragmatic inconsistency. In our example, explicitly putting forward  and  and denying [2.2] would be … pragmatically inconsistent.
I think this solution is strange in at least two ways. (1) It brings nothing new to analyzing enthymemes, and (2) It is useful in only some (happy) cases. The idea that we should use contextual information in order to make decisions about the right candidate (the used one) has always been taken into consideration. Nobody doubted it. Douglas Walton mentioned it as a “part solution” in 1983, based on Hamblin’s remark that speakers’ commitment stores should always be internally consistent (which might sound a bit familiar, now that we have mentioned Eemeren’s pragmatical consistency):
It is fairly clear how you can partly solve the problem of enthymemes […]. You can rule that a proposition is a fair assumption to make as a missing premiss in a player’s argument if that proposition is in that player’s commitment set.
Secondly, using this “rule for optimum” does not even apply to all cases of enthymematic argumentation. Gerritsen observes how “today’s theorists study larger samples of text and try to take the context into account” and that “the concept of an unexpressed premise is entirely context-dependent” . As much consensus as this closure might inspire, we cannot help noticing that if this is the pragmatic concept of enthymeme, it means that everything that has been done heretofore has been merely speculating and wishful thinking (“unnecessary guesswork”, see first note of part I) and that without any context, there’s no analysis. Which means that if we encounter a context-free, low-information, no-commitments-attached enthymeme, which we have reasons to regard as a “sequitur”, we will not be able to analyze it properly unless sudden details about context arise out of the … context.
This line of reasoning may be consistent, but it is nevertheless dismissive and restrictive, and this criticism might end up postulating the need to analyze only true (that is not hypothetical, nor made-up) examples. This again might be true, but one will have to explain decades of context-free analysis, and all the terminology that it has produced: charity, sufficiency, plausibility, preservation, strength, validity etc. Don S. Levi explained it thus:
I think of the absence of rhetorical context as the dirty little secret about the P[remise]-C[onclusion] sequence paraphrase, at least when understood in terms of the enthymeme approach to argument analysis
 GOULDING, Daniel J. 1965. Aristotle's Concept of the Enthymeme, Journal of the American
Forensic Society 2: 104-8., qtd. in: WALTON, Douglas, Enthymemes, Common Knowledge, and Plausible
Inference¸ Philosophy and Rhetoric, Vol. 34, No. 2, 2001 p. 98
 HITCHCOCK, David.. Enthymematic Arguments. Informal Logic, North America, Vol. 7, No. 2. jan. 1985, p. 87
 EEMEREN, Frans H. van, GROOTENDORST, Rob, 1992 p. 65
 BURKE, Michael, Unstated Premises, Informal Logic, Vol. 7, No. 2, Ian 1985, p. 108
 idem, p.110
 JOHNSON, Ralph H. Charity Begins at Home, Informal Logic Newsletter, 3:3, June 1981, p. 7 Proposition P is stronger than proposition Q if and only if P entails, and is not entailed by, Q, qtd. in BURKE, Michael, 1985, p. 109
 WALTON, Douglas, 2001, p. 95
 GERRITSEN, Susane, Unexpressed Premises, in Crucial Concepts in Argumentation Theory, Frans H. van Eemeren (ed.), Amsterdam University Press, 2001, Amsterdam, p. 51-56 for the deductivist – pluralist distinction among argumentation theorists see
 ENNIS, Robert H. Identifying Implicit Assumptions, in Synthese, 51, 1982 61-68, GOVIER, Trudy A Practical Study of Argument, Belmont, CA: Wadsworth, 1985, WALTON, Douglas Argument Structure: A Pragmatic Theory, Toronto/Buffalo/London: University of Toronto Press, 1996, WALTON, Douglas, 2001
 GERRITSEN, Susane, 2001, p. 67
 PAGLIERI, Fabio, 2007, p. 2
 EEMEREN, Frans H. van, GROOTENDORST, Rob, 1992 p. 64
 WALTON, Douglas, Enthymemes, in Logique et Analyse, No. 103-104, 1983, p. 400
 GERRITSEN, Susane, 2001, p.69
 LEVI, Don S., The Case of the Missing Premise, in “Informal Logic” Vol. 17, No.1, 1995, p.80