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An enthymeme is a categorical syllogism with one premise left unstated. We can use what we know about categorical syllogisms to reconstruct the missing premise.

Position and Terms

First, from the conclusion we will be able to identify the premise we have been given as either the major or the minor premise. If the subject term of the conclusion occurs in the premise, it is the minor premise, while if the the predicate term of the conclusion occurs in the premise it is the major premise. The other term given in the premise must be the middle term. Whichever premise we have been given the missing premise must be the other. Thus if we have been given the minor premise then the missing one must be the major premise and should be placed first and should have the major term as well as the middle term in it. On the other hand, if we have been given the major premise then the missing one must be the minor premise and should be placed second and should have the minor term as well as the middle term in it. In this way we can quickly identify the position of the premise in the argument and the terms which belong to the missing premise.

Type of Proposition and Order of Terms

Determining the type of proposition and the order for the terms is somewhat more difficult. At this point one should be guided by the principle of charity. In many cases one could easily construct an invalid argument, but this may not be the premise the originator had in mind. One ought instead to try to build a valid syllogism. Then if the enthymeme turns out to be invalid after your best efforts, the reason will be because it is unavoidably invalid rather than because you deliberately made it so. In other words, the fault will lie with the argument rather than with you. On the other hand, if the argument turns out to be valid, but you disagree with the conclusion, you will likely be able to attack one of the premises as untrue. Indeed, people will sometimes leave out an unreasonable premise in the hope that the audience will not bother to reconstruct the premise and so will not notice how unreasonable it is. Of course, there is a slight possibility that the given premise is the unreasonable one. Finally, do not overlook the possibility that you are wrong and the argument sound. (A sound argument is a valid argument with all true premises.)
In attempting to make the argument valid, one makes use of the rules from the rule method. One can quickly determine whether the premise must be affirmative or negative. If the conclusion is affirmative, the missing premise must also be so. If the conclusion is negative, the missing premise must be negative, unless the other premise is negative, in which case the missing premise must be affirmative. In this way one eliminates two proposition types. Whether the missing premise is particular or universal may sometimes be determined by the requirements of distribution. If the middle term is undistributed in the given premise, it will need to be distributed in the missing premise; so if one already knows that the missing premise is affirmative, this will show that it must be an A-form and the middle term must be in the subject position. On the other hand, if the missing premise must be negative, the distribution of the middle term alone will not determine the type. In such a case one would look at the term in the conclusion which is also in the missing premise. If the term is distributed in the conclusion it must also be so in its premise. Thus if both the middle term and the term form the conclusion required distributing, one would need an E-form proposition.
Note: Meeting the requirements of all the rules may be impossible (for example, one cannot distribute both terms and have an affirmative premise). Such an argument is simply invalid. One would choose to distribute as many of the terms as possible but which term you distribute is arbitrary (your choice).
Note: The order of the terms is already being determined as we apply the rules about distribution. The order of terms will not matter in the cases of E and I propositions (your choice).
Note: If the rules concerning distribution do not require a universal proposition rather than a particular, the principle of charity would favour a particular proposition over a universal. We have no reason to assume that the originator of the argument would have claimed more than he or she needed to claim.
Note: If the argument is about things which do not exist or you have been instructed to evaluate the argument using the Venn Diagram method, you will wish to also use a fifth rule in this reconstruction process, namely: If the conclusion is particular, one premise must also be particular. Do not use this rule unless you are reconstructing an enthymeme about things which do not exist or one to which you have been instructed to apply the Venn Diagram method.