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Such a breach of the fourth rule as is described above is exceedingly easy to detect, and is therefore very seldom committed.
But an illicit process or improper treatment of the major term is more common because it is not so transparently false. If we argued indeed that "because all Anglo-Saxons love liberty, and Frenchmen are not AngloSaxons, therefore they do not love liberty," the fallacy would be pretty apparent; but without a knowledge of logic it would not be easy to give a clear explanation of the fallacy. It is apparent that the major term loving liberty, is undistributed in the major premise, so that Anglo-Saxons must be assumed to be only a part of those who love liberty. Hence the exclusion of Frenchmen from the class Anglo-Saxons does not necessarily exclude them from the class who love liberty (see Fig. 8). The
conclusion of the false argument being negative distributes its predicate, the major term, and as this is undistributed in the major premise we have an illicit major as we may briefly call this fallacy. The following is an obscurer example of the same fallacy;-"Few students
are capable of excelling in many branches of knowledge, and such as can so excel are deserving of high commendation;" hence "few students are deserving of high commendation." The little word "few" has here the double meaning before explained (p. 67), and means that "a few are, &c., and the rest are not." The conclusion is thus really a negative proposition, and distributes the major term "deserving of high commendation." But this major term is clearly undistributed in the major premise, which merely asserts that those who can excel in many branches of knowledge are deserving, but says or implies nothing about other students.
The fifth rule is evidently founded on the principle noticed in the last lesson, that inference can only proceed where there is agreement, and that two differences or disagreements allow of no reasoning. Two terms, as the third Canon states, may both differ from a common term and yet may or may not differ from each other. Th if
we were to argue that Americans are not Europeans, and Virginians are not Europeans, we see that both terms disagree with the middle term Europeans, and yet they
agree between themselves. In other cases the two negative premises may be plainly true while it will be quite uncertain whether the major and minor terms agree or not. Thus it is true, for instance, that "Colonists are not Europeans, and Americans are not Europeans," but this gives us no right to infer that Colonists are or are not Americans. The two negative premises are represented in fig. 9, by excluding the circles of Colonists and Americans from that of Europeans; but this exclusion may still be effected whether Colonists and Americans coincide partially, or wholly, or not at all. A breach of this rule of the syllogism may be conveniently called the fallacy of negative premises. It must not however be supposed that the mere occurrence of a negative particle (not or no) in a proposition renders it negative in the manner contemplated by this rule. Thus the argument "What is not compound is an element. Gold is not compound;
Therefore Gold is an element."
contains negatives in both premises, but is nevertheless valid, because the negative in both cases affects the middle term, which is really the negative term not-compound.
The truth of the sixth rule depends upon that of the axiom, that if two terms agree with a common third term they agree with each other, whence, remembering that a negative proposition asserts disagreement, it is evident that a negative conclusion could not be drawn from really affirmative premises. The corresponding negative axiom prevents our drawing an affirmative conclusion if either premise should be really negative. Only practice however will enable the student to apply this and the preceding rules of the syllogism with certainty, since fallacy may be hidden and disguised by various forms of expression. Numerous examples are given at the end of
the book by which the student may acquire facility in the analysis of arguments.
The remaining rules of the syllogism, the 7th and 8th, are by no means of a self-evident character and are in fact corollaries of the first six rules, that is consequences which follow from them. We shall therefore have to shew that they are true consequences in a future Lesson. We may call a breach of the 7th rule a fallacy of particular premises, and that of the 8th rule the fallacy of a universal conclusion from a particular premise, but these fallacies may really be resolved into those of Illicit Process, or Undistributed Middle.
For many details concerning the Aristotelian and
THE MOODS AND FIGURES OF THE
WE are now in full possession of those principles of reasoning, and the rules founded upon them, by which a true syllogism may be known from one which only seems to be a true one, and our task in the present Lesson is to ascertain the various shapes or fashions in which a process of mediate inference or syllogism may be met with. We know that every syllogistic argument must contain three propositions and three distinct terms each occurring twice in those propositions. Each proposition
of the syllogism may, so far as we yet know, be either affirmative or negative, universal or particular, so that it is not difficult to calculate the utmost possible varieties of modes in which a syllogism might conceivably be constructed. Any one of the four propositions A, E, I, or O may in short be taken as a major premise, and joined with any one of the same form as a minor premise, and any one of the four again may be added as conclusion. We should thus obtain a series of the combinations or modes of joining the letters A, E, I, O, a few of which are here writ
It is obvious that there will be altogether 4×4×4 or 64 such combinations, of which 23 only are given above. The student can easily write out the remainder by carrying on the same systematic changes of the letters. Thus beginning with AAA we change the right-hand letter successively into E, I, and O, and then do the same beginning with AEA instead; after the middle letter has been carried through all its changes we begin to change the left-hand letter. With each change of this we have to repeat all the sixteen changes of the other letters, so that there will obviously be altogether 64 different conceivable modes of arranging propositions into syllogisms.
We call each of these triplets of propositions a mood or form of the syllogism (Latin modus, shape), and we have to consider how many of such forms can really be used in valid arguments, as distinguished from those which break one or more of the rules of the syllogism. Thus the mood AEA would break the 6th rule, that if one premise be negative the conclusion must be so too; AIE breaks the