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 66 What is not compound is an element. 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 LESSON XVI. 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 0 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 written out: 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 0, 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 converse part of the same rule, that a negative conclusion can only be proved by a negative premise; while EEA, FEE &c., break the 5th rule, which prohibits our reasoning at all from two negative premises. Examples of any of these moods can easily be invented, and their falsity would be very apparent; thus for AEA we might take All Austrians are Europeans, No Australians are Europeans; Therefore, all Australians are Austrians. Many of the 64 conceivable moods are excluded by the 7th and 8th rules of the syllogism. Thus AIA and EIE break the rule, that if one premise be particular the conclusion must be so also, while IIA, 100, 010- and many others, break the rule against two particular premises. Some combinations of propositions may break more than one rule; thus 000 has both negative premises and particular premises, and OOA also violates as well the 6th rule. It is an admirable exercise in the use of the syllogistic rules to write out all the 64 combinations and then strike out such as break any rule; the task if pursued systematically will not be so long or tedious as might seem likely. It will be found that there are only twelve moods which escape exclusion, and may so far be considered good forms of reasoning, and these are Of these however IEO will have shortly to be rejected, because it will be found really to break the 4th rule, and involves Illicit process of the major term. There are, then, only eleven moods of the syllogism which are really valid; and we may thus account for the whole of the sixty-four moods. Number of moods. 16 12 12 8 6 4 4.. I We have by no means exhausted as yet all the possible varieties of the syllogism, for we have only determined the character, affirmative or negative, general or particular of the propositions, but have not decided the ways in which the terms may be disposed in them. The major term must be the predicate of the conclusion, but it may either be subject or predicate of the major premise, and similarly the minor term or subject of the conclusion, may be either the subject or predicate of the minor premise. There thus arise four different ways, or as they are called Figures, in which the terms can be disposed. These four figures of the syllogism are shewn in the following scheme, taking |