I may here explain that the syllogism and the sorites can be expressed either in the order of extension or that of intension. In regard to the number of individual things the noble metals are part of the metals, and the metals are part of the elements; but in regard to intension, that is to say the qualities implied in the names, element is part of metal, and metal is part of noble metal. So again in extension the genus of plants Anemone is part of the order Ranunculaceæ, and this is part of the great class Exogens; but in intension the character of Exogen is part of the character of Ranunculaceæ, and this is part of the character of Anemone. Syllogistic reasoning is equally valid and evident in either case, and we might represent the two modes in ordinary language as follows: Extensive Syllogism. All Ranunculaceæ are Exogens; The Anemone is one of the Ranunculaceæ ; Intensive Syllogism. All the qualities of Ranunculaceæ are qualities of All the qualities of Exogen are qualities of Ranun- Therefore all the qualities of Exogen are qualities of Any sorites can be similarly represented either in extension or intension. Concerning the Aristotelian doctrine of the Enthymeme, see Mansel's Aldrich, App. Note F, and Hamilton's Lectures on Logic, Lecture XX. Port Royal Logic, translated by T. Spencer Baynes, 5th ed. Edinburgh, LESSON XIX. OF CONDITIONAL ARGUMENTS. It will be remembered that when treating of propositions we divided them into two distinct kinds, Categorical Propositions, and Conditional Propositions. The former kind alone has hitherto been considered, and we must now proceed to describe Conditional propositions and the arguments which may be composed of them. Logicians have commonly described Conditional propositions as composed of two or more Categorical propositions united by a conjunction. This union may happen in two ways, giving rise to two very different species of conditionals, which we shall call Hypothetical Propositions and Disjunctive Propositions. The way in which the several kinds of propositions are related will be seen in the following diagram : Categorical. Propositions are Conditional Hypothetical. A conditional proposition may be further described as one which makes a statement under a certain condition or qualification restricting its application. In the hypothetical form this condition is introduced by the conjunction if, or some other word equivalent to it. Thus "If iron is impure, it is brittle " is a hypothetical proposition consisting of two distinct categorical propositions, the first of which, "Iron is impure," is called the Antecedent; the second, "It is brittle," the Consequent. In this case "impurity" is the condition or qualification which limits the application of the predicate brittle to iron. It was asserted by Horne Tooke in his celebrated work The Diversions of Purley, that all conjunctions are the remains or corrupted forms of verbs. This is certainly true in the case of the hypothetical conjunction; for the word if in old English is written gif, or gyf, and is undoubtedly derived from the verb to give. We may actually substitute at present any verb of similar meaning, as for instance-grant, allow, suppose. Thus we may say "Grant that iron is impure, and it is brittle." The hypothetical proposition might be employed in arguments of various form, but only two of these are of sufficient importance to receive special names. The hypothetical syllogism consists of two premises, called the major and minor, as in the case of the ordinary syllogism. The major premise is hypothetical in form; the minor premise is categorical, and according as it is affirmative or negative the argument is said to be a Constructive or a Destructive hypothetical syllogism. Thus the form, If A is B, C is D; But A is B; Therefore C is D, is a constructive hypothetical syllogism, It must be carefully observed that the minor premise affirms the antecedent of the major premise, whence the argument is said to be of the modus ponens, or mood which posits or affirms. It is probably one of the most familiar and common kinds of argument. The form, If A is B, C is D; But C is not D; Therefore A is not B, represents the corresponding Destructive hypothetical syllogism, also called the modus tollens, or the mood which removes the consequent. It must be carefully observed again that it is the consequent, not the antecedent, which is denied. The only rule which is requisite for testing the validity of such syllogisms embodies what we have observed above; viz. that either the antecedent must be affirmed, or the consequent denied. If either part of this rule be broken, a serious fallacy will be committed. Thus the apparent argument, If A is B, C is D; is really a fallacy which we may call the fallacy of affirming the consequent, and its fallacious nature is readily understood by reflecting that "A being B" is not stated to be the only condition on which C is D. It may happen that when E is F, or G is H, or under a hundred other circumstances, C is D, so that the mere fact of C being D is no sufficient proof that A is B. Thus, if a man's character be avaricious he will refuse to give money for useful purposes; but it does not follow that every person who refuses to give money for such purposes is avaricious. There may be many proper reasons or motives leading him to refuse; he may have no money, or he may consider the purpose not a useful one, or he may have more useful purposes in view. A corresponding fallacy arises from denying the antecedent, as in the form If A is B, C is D; The error may be explained in the same way; for as "A being B" is not stated to be the only condition of C being D, we may deny this one condition to be true, but it is possible that the consequent may happen to be true for other reasons, of which we know nothing. Thus if a man is not avaricious we cannot conclude that he will be sure to give money whenever asked. Or take the following example: "If the study of Logic furnished the mind with a multitude of useful facts like the study of other sciences, it would deserve cultivation; but it does not furnish the mind with a multitude of useful facts; therefore it does not deserve cultivation." This is evidently a fallacious argument, because the acquiring of a multitude of useful facts is not the only ground on which the study of a science can be recommended. To correct and exercise the powers of judgment and reasoning is the object for which Logic deserves to be cultivated, and the existence of such other purpose is ignored in the above fallacious argument, which evidently involves the denial of the antecedent. Although it is usual in logical works to describe the hypothetical proposition and syllogism as if they were different in nature from the categorical proposition and syllogism, yet it has long been known that the hypotheticals can be reduced to the categorical form, and brought under the ordinary rules of the syllogism. As a general rule the hypothetical proposition can be readily converted into a universal affirmative proposition (A) of exactly the same meaning. Thus our instance, "If iron is impure, it is brittle," becomes simply "Impure iron is brittle." In making this alteration in a hypothetical syllogism it will be found necessary to supply a new minor term; thus in the case, |