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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.
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.
If the major premise be suppressed, it is called an Enthymeme of the First Order; if the minor premise be wanting, it is said to be of the Second Order; if the conclusion be left to be understood, there is an Enthymeme of the Third Order.
Let the following be a complete Syllogism: All free nations are enterprising; The Dutch are a free nation: .’. Reduced to Enthymemes, this argument may be put thus: In the First Order: The Dutch are a free nation: .’. In the Second Order– All free nations are enterprising; .’. In the Third Order– All free nations are enterprising; And the Dutch are a free nation.
Against such occasions of error the logician can provide no safeguard, except the advice to be careful and discriminating in what you say or hear.
But as to any derangement of the elements of an argument, or the omission of them, Logic effectually aids the task of restoration; for it has shown what the elements are that enter into the explicit statement of most ratiocinations, namely, the four forms of propositions and what that connected order of propositions is which most easily and surely exposes the validity or invalidity of reasoning, namely, the premises and conclusion of the Syllogism.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.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.“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. 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.Two modes of abbreviating a Polysyllogism, are usually discussed, the Epicheirema and the Sorites. An Epicheirema is a syllogism for one or both of whose premises a reason is added; as– All men are mortal, for they are animals; Socrates is a man, for rational bipeds are men: .’. The Epicheirema is called Single or Double, says Hamilton, according as an “adscititious proposition” attaches to one or both of the premises. The Single Epicheirema is said to be of the First Order, if the adscititious proposition attach to the major premise; if to the minor, of the Second Order. The reader wonders what is the difference between these two forms.(Hamilton’s Logic: Lecture xix.) An Epicheirema, then, is an abbreviated chain of reasoning, or Polysyllogism, comprising an Episyllogism with one or two enthymematic Prosyllogisms. The Sorites is a Polysyllogism in which the Conclusions, and even some of the Premises, are suppressed until the arguments end. In the Aristotelian Sorites the minor term occurs in the first premise, and the major term in the last; whilst in the Goclenian the major term occurs in the first premise, and the minor in the last.The major premise in the above case, All men are mortal, for they are animals, is an Enthymeme of the First Order, suppressing its own major premise, and may be restored thus: All animals are mortal; All men are animals: .’. The minor premise, Socrates is a man, for rational bipeds are men, is an Enthymeme of the Second Order, suppressing its own minor premise, and may be restored thus: All rational bipeds are men; Socrates is a rational biped: .’. If the chain of arguments were freed of its enthymematic character, the suppressed conclusions would appear as premises of Episyllogisms. Bucephalus is a horse; An animal is a substance; A horse is a quadruped; A quadruped is an animal; A quadruped is an animal; A horse is a quadruped; An animal is a substance: Bucephalus is a horse: .'. But since the character of premises is fixed by their terms, not by the order in which they are written, there cannot be a better example of a distinction without a difference.