The briefer statement is less comprehensive, but it will cover any case that is likely to arise.* In this figure as in the second we must be careful not to be confused by negative relations. From the fact that all M's are P, and that no M's are S, we can infer that Negative relations. some P's are not S; but we cannot infer that some S's are not P.† From the Pope we may perhaps prove that there are infallible mortals, but not that there are fallible immortals. It takes the Devil for that. The sixth caution or its corollary--“ Evidence sufficient to prove that some S's are not P's may not be sufficient to prove that some P's are not S's"—is one which we tend to ignore or misunderstand continually. Altogether the best way to observe it without confusion, whether we are reasoning in the third figure or in one of the others, is to put our premises affirmatively, with the negative element, when there is one, in the predicate (i.e., to obvert negative premises and conclusions). When we say that there are infallible mortals or that there are fallible immortals, our meaning is much clearer and the distinction between the two statements is much more obvious I * This caution covers illicit minors in the third figure. Put in terms of causal relations the caution is this : A single coincidence proves the compatibility of relations, but no number of coincidences can prove their necessary connection. + This caution covers illicit majors in the third figure as well as in the second. # The statement in this form has moreover the advantage of directing attention to the fact that we are talking about real things. (See top of p. 173.) The diagrams in the text seem to me to accent the affirmative element which reasoning in the third figure particularly involves, as well as to guard M М. against the confusion referred to in the text P better than Euler's. Students always find it difficult to see why this figure does not s mean that some P's are not Sas well as that some S's are not P. But if we represent S by a vertical stroke and P by a horizontal the distinction between M which is S but not P $+ and M which is P but not S 0 is obvious, and with it the distinction between S not-P ++ and P not-S # than when we say that some mortals are not fallible or that some fallible beings are not mortal. By the coincidence of two relations we mean that they both belong to the same individual. Whether they do or not is primarily, of course, a matter of observation in Quantity each particular case; but when the coincidence of the premises. of the relations must be inferred by putting together statements about the existence of each we must remember one more CAUTION : 8) Two different relations can belong to individuals of the same class without belonging to the same individual, unless at least one of them belongs to every individual in the class. * If we know that this particular X is both Y and Z, we know of course that Y and Z coexist. If we know that every X' is Y and every X is Z, we can be sure that each X is both Y and Z; if we know that every X is Y and that some X is 2, we can be sure that some X or other is both Y and Z; but if we only know that some X's or other are Y and that some X's or other are Z, we cannot be sure that Y and Z ever belong to the same X. This is what is meant in this figure by the technical rule that from two particular premises no conclusion can be drawn. The technical rule should have added that from a particular premise and a singular premise in the third figure no conclusion can be drawn ; for it does no good to know which particular X's are Y so long as we do not know which are Z. The principle and all the cautions can be put together in such a general statement as this : The coincidence of relations—whether positive or negativeproves that they are compatible, but it does not prove that either of them involves the other, or that the absence of one is compatible either with the presence or with the absence of the other. Moreover the fact that two relations belong to objects of the same class will not prove that they belong to the same objects unless at least one of them belongs to all the objects in the class. * This covers undistributed middles in the third figure, CHAPTER XVI. THE ALLEGED FOURTH FIGURE. * of the So long as the various figures of the syllogism were distinguished by the mere arrangement of terms rather than by the relations involved in the reasoning, it seemed Origin reasonable that there should be a figure to reprefigure. sent every possible arrangement. Consequently to the three figures which we have discussed, and which were all that Aristotle recognized, Galen (131–200 A.D.) added a fourth. The arrangement of terms in each is as follows: First Figure. Second Figure. Third Figure. Fourth Figure. PM MS SP The four figures cover every possible permutation of the terms. Reasoning in the fourth figure outside of exercises in formal logic is extremely rare. Beyond mere questions of whether one class includes or excludes members of another, Three ways it has no significance; and though it is easy to of dealing with it. arrange problems in such a way that they will fall within the figure, they lose most of their meaning when so arranged and seem strained and unnatural. Nevertheless one ought to know how to deal with the problems when they arise. There are three ways of doing this. The first is to disregard their meaning and solve them by means of a set of purely mechanical rules. These rules are equally applicable to all four of the figures; but inasmuch as we have tried ( * This chapter is not essential, * to get along without them—or at least to interpret them-in the other figures, it seems a pity to take refuge in them now. The second way of dealing with syllogisms in the fourth figure is to assume that they are concerned merely with relations of inclusion or exclusion between a number of classes, all of which are assumed to exist, and then to test them by Euler's diagrams. For example, s! MIP All M's are S's; The conclusion follows; for we cannot possibly put the circle M within the circle S, and part of the circle P within the circle M, without some part of the area of S falling within the circle P. M S ס No M's are S's; * The rules, as stated by Jevons, are as follows : 1. Every syllogism has three and only three terms. These terms are called the major term, the minor term, and the middle term. 2. Every syllogism contains three and only three propositions. These propositions are called the major premise, the minor premise, and the conclusion. 3. The middle term must be distributed once at least, and must not be ambiguous. 4. No term must be distributed in the conclusion which was not distributed in one of the premises. 5. From negative premises nothing can be inferred. 6. If one premise be negative, the conclusion must be negative ; and vice versa, to prove a negative conclusion one of the premises must be negative. From the above rules may be deduced two subordinate rules, which it will nevertheless be convenient to state at once. 7. From two particular premises no conclusion can be drawn. These rules are not absolutely reliable unless we assume that the objects denoted by each term in the syllogism exist. The conclusion does not follow, for we can construct a diagram which represents the premises without representing the supposed conclusion. The third, and of course the best, way of dealing with syllogisms in the fourth figure which we are called upon to test is to try to give them a rational interpretation and thus work in the light. When we come to interpret such syllogisms we shall find that we must regard them as concerned either with the relations between classes which we have just discussed or with the relations peculiar to some one of the three other figures, to which the syllogism in question cai. be reduced' by converting the conclusion or one or both of the premises. If we turn back to the table which shows the arrangement of terms in each figure, it is easy to see that by converting the major premise of a syllogism in the fourth figure we Formal get the arrangement of terms found in the third ; reduction. by converting the minor we get that found in the second; and by converting the conclusion or by converting both premises we get that found in the first. It is thus formally possible to interpret the fourth figure by any one of the three others; and so long as we do not attempt to test a universal conclusion by the third figure (which is itself incapable of giving such a conclusion) or an affirmative conclusion by the second (which never gives it), that is to say, so long as we do not ask a figure to test a kind of conclusion that it is itself unable to draw, it does not make the slightest difference, as far as the formal process is concerned, which of the first three figures we use to test a syllogism in the fourth. If it is valid, there are always at least two of the first three figures in which the conclusion can be proved. But since the second and the third figures both have the limitations just referred to, any one who merely wants an easy formal test will save himself some thinking by making it a rule to test every argument in the fourth figure by reducing it to the first, In doing this he may assume that |