The process of finding the missing premise in an Enthymeme of either the First or the Second Order, so as to constitute a syllogism, is sometimes called Reduction; and for this a simple rule may be given: Take that term of the given premise which does not occur in the conclusion (and which must therefore be the Middle), and combine it with that term of the conclusion which does not occur in the given premise; the proposition thus formed is the premise which was requisite to complete the Syllogism.If the premise thus constituted contain the predicate of the conclusion, the Enthymeme was of the First Order; if it contain the subject of the conclusion, the Enthymeme was of the Second Order.In any Polysyllogism, again, a syllogism whose conclusion is used as the premise of another, is called in relation to that other a Prosyllogism; whilst a syllogism one of whose premises is the conclusion of another syllogism, is in relation to that other an Episyllogism.
“The Three Bases for the Enthymeme: A Dialogical Theory,” Douglas Walton develops a theory of enthymematic arguments in accordance with dialogical theory, illustrated with several examples first published in 1.
“Enthymeme” is not used in contemporary logic in the Aristotelian sense of the word.
These practices give a great advantage to sophists; who would find it very inconvenient to state explicitly in Mood and Figure the pretentious antilogies which they foist upon the public; and, indeed, such licences of composition often prevent honest men from detecting errors into which they themselves have unwittingly fallen, and which, with the best intentions, they strive to communicate to others: but we put up with these drawbacks to avoid the inelegance and the tedium of a long discourse in accurate syllogisms.
Many departures from the strictly logical statement of reasonings consist in the use of vague or figurative language, or in the substitution for one another of expressions supposed to be equivalent, though, in fact, dangerously discrepant.
That a statement in the form of a Hypothetical Proposition may really be an Enthymeme (as observed in chap. Section 4) can easily be shown by recasting one of the above Enthymemes thus: If all free nations are enterprising, the Dutch are enterprising.
Such statements should be treated according to their true nature.It is certainly very common to meet with arguments whose statement may be represented by one or other of these three forms; indeed, the Enthymeme is the natural substitute for a full syllogism in oratory: whence the transition from Aristotle’s to the modern meaning of the term.The most unschooled of men readily apprehend its force; and a student of Logic can easily supply the proposition that may be wanted in any case to complete a syllogism, and thereby test the argument’s formal validity.Each Enthymeme may be isolated and expanded into a syllogism.And we may then inquire: (1) whether the syllogisms are formally correct according to Barbara (or whatever the appropriate Mood); (2) whether the premises, or the ultimate premises, are true in fact. A Monosyllogism is a syllogism considered as standing alone or without relation to other arguments.CHAPTER XI ABBREVIATED AND COMPOUND ARGUMENTS Section 1.In ordinary discussion, whether oral or written, it is but rarely that the forms of Logic are closely adhered to.In any Enthymeme of the Third Order, especially, to supply the conclusion cannot present any difficulty at all; and hence it is a favourite vehicle of innuendo, as in Hamilton’s example: Every liar is a coward; And Caius is a liar.The frankness of this statement and its reticence, together, make it a biting sarcasm upon Caius. Rather than classifying this argument in accordance with Aristotle's concept of “enthymeme” contemporary usage would label the argument as an example of the informal fallacy of converse accident. Aristotle says this argument is refutable because it does not form a syllogism.