Enthymeme

The understanding of enthymenes as syllogisms with suppressed premises was first discussed in Arnauld's "The Art of Thinking: Port-Royal Logic", 1662. An example of an enthymene: All decontructions do not care if their statements are true or false; therefore she does not care if her statements are true or false." The suppressed premise is "she is a deconstructionist."
In ordinary language we frequently state arguments elliptically: we omitt a premiss or even a conclusion that we expect our listeners or readers to fill in. An argument that is stated ellipically or incompletely (part of it being implied or "understood") is called an enthymeme.  Testing the validity of an enthymematic argument requires supplying its missing premise or conclusion.
Enthymemes  have traditionally been divided into different orders, according to which part of the syllogism is left unexpressed:
        A first-order enthymeme is one in which the syllogism’s major premise is not stated.
        A second-order enthymeme is one in which the minor premise is not stated.
        A third-order enthymeme is one in which both premises are stated, but the conclusion is left unstated.

In testing an enthymeme for validity, two steps are involved: The first is to supply the missing premise or conclusion of the argument (if possibble the propostion added should be true and should make the syllogism valid); the second is to test the resulting syllogism.The difference between enthymemes and normal syllogisms is rhetorical, not logical: No new logical principle is need be to test whether aenthymeme is valid: they are tested by the same methods that apply to standard-form categorical syllogisms.

Steps in inventing or discovering the suppressed premise or conclusion:
1.) Determine the two terms that occur only once in the enthymeme. These two terms will occur in the proposition to be added to make it a syllogism.
2.) Determine the quality of the proposition to be added using rule 4 and  5.
3.) Determine the quantity of the proposition to be added using rule 6 (if conclusion is particular then one premise must be particular) and the rule that if the conclusion is universal then both premises must be universal.
4.) Use the rules for distribution of terms to decide the subject and predicate of the added proposition.