(3) De = DeBC, DE=DEbc. As five terms enter into these premises it is requisite to treat their thirty-two combinations, and it will be found that fourteen of them remain consistent with the premises, namely ABcdE a BCDe abCdE abCde abcDE abcdE abcde. If we examine the first four combinations, all of which contain A, we find that they none of them contain D; or again, if we select those which contain D, we have only two, thus Hence it is clear that no A is D, and vice versa no D is A. We might draw many other conclusions from the same premises; for instance DE = abcDE, or D and E never meet but in the absence of A, B, and C. Fallacies analysed by the Indirect Method. It has been sufficiently shown, perhaps, that we can by the Indirect Method of Inference extract the whole truth from a series of propositions, and exhibit it anew in any required form of conclusion. But it may also need to be shown by examples that so long as we follow correctly the almost mechanical rules of the method, we cannot fall into any of the fallacies or paralogisms which are often committed in ordinary discussion. Let us take the example of a fallacious argument, previously treated by the Method of Direct Inference (p. 62), Granite is not a sedimentary rock, Basalt is not a sedimentary rock, (1) (2) and let us ascertain whether any precise conclusion can be drawn concerning the relation of granite and basalt. Taking as before A = granite, B = sedimentary rock, Of the eight conceivable combinations of A, B, C, five agree with these conditions, namely Selecting the combinations which contain A, we find the description of granite to be A AbС Abc Ab(C + c), = = that is, granite is not a sedimentary rock, and is either basalt or not-basalt. If we want a description of basalt the answer is of like form C AbCabCbC (Aa), = that is basalt is not a sedimentary rock, and is either granite or not-granite. As it is already perfectly evident that basalt must be either granite or not, and vice versâ, the premises fail to give us any information on the point, that is to say the Method of Indirect Inference saves us from falling into any fallacious conclusions. This example sufficiently illustrates both the fallacy of Negative premises and that of Undistributed Middle of the old logic. The fallacy called the Illicit Process of the Major Term is also incapable of commission in following the rules of the method. Our example was (p. 65) All planets are subject to gravity, The false conclusion is that "fixed stars are not subject to gravity." The terms are A = B= fixed star The combinations which remain uncontradicted on com parison with these premises are that is, "a fixed star is not a planet, but is either subject or not, as the case may be, to gravity." Here we have no conclusion concerning the connection of fixed stars and gravity. The Logical Abacus. The Indirect Method of Inference has now been sufficiently described, and a careful examination of its powers will show that it is capable of giving a full analysis and solution of every question involving only logical relations. The chief difficulty of the method consists in the great number of combinations which may have to be examined; not only may the requisite labour become formidable, but a considerable chance of mistake arises. I have therefore given much attention to modes of facilitating the work, and have succeeded in reducing the method to an almost mechanical form. It soon appeared obvious that if the conceivable combinations of the Logical Alphabet, for any number of letters, instead of being printed in fixed order on a piece of paper or slate, were marked upon light movable pieces of wood, mechanical arrangements could readily be devised for selecting any required class of the combinations. The labour of comparison and rejection might thus be immensely reduced. This idea was first carried out in the Logical Abacus, which I have found useful in the lecture-room for exhibiting the complete solution of logical problems. A minute description of the construction and use of the Abacus, together with figures of the parts, has already been given in my essay called The Substitution of Similars, and I will here give only a general description. The Logical Abacus consists of a common school blackboard placed in a sloping position and furnished with four horizontal and equi-distant ledges. The combinations of the letters shown in the first four columns of the Logical Alphabet are printed in somewhat large type, so that each letter is about an inch from the neighbouring one, but the letters are placed one above the other instead of being in horizontal lines as in p. 94. Each combination of letters is separately fixed to the surface of 1 Pp. 55-59, 81-86. a thin slip of wood one inch broad and about one-eighth inch thick. Short steel pins are then driven in an inclined position into the wood. When a letter is a large capital representing a positive term, the pin is fixed in the upper part of its space; when the letter is a small italic representing a negative term, the pin is fixed in the lower part of the space. Now, if one of the series of combinations be ranged upon a ledge of the black-board, the sharp edge of a flat rule can be inserted beneath the pins belonging to any one letter-say A, so that all the combinations marked A can be lifted out and placed upon a separate ledge. Thus we have represented the act of thought which separates the class A from what is not-A. The operation can be repeated; out of the A's we can in like manner select those which are B's, obtaining the AB's; and in like manner we may select any other classes such as the aB's, the ab's, or the abc's. If now we take the series of eight combinations of the letters A, B, C, a, b, c, and wish to analyse the argument anciently called Barbara, having the premises A = B = BC, (1) (2) we proceed as follows-We raise the combinations marked a, leaving the A's behind; out of these A's we move to a lower ledge such as are b's, and to the remaining AB's we join the a's which have been raised. The result is that we have divided all the combinations into two classes, namely, the Ab's which are incapable of existing consistently with premise (1), and the combinations which are consistent with the premise. Turning now to the second premise, we raise out of those which agree with (1) the b's, then we lower the Be's; lastly we join the b's to the BC's. We now find our combinations arranged as below. The lower line contains all the combinations which are inconsistent with either premise; we have carried out in a mechanical manner that exclusion of self-contradictories which was formerly done upon the slate or upon paper. Accordingly, from the combinations remaining in the upper line we can draw any inference which the premises yield. If we raise the A's we find only one, and that is C, so that A must be C. If we select the c's we again find only one, which is a and also b; thus we prove that not-C is not-A and not-B. When a disjunctive proposition occurs among the premises the requisite movements become rather more complicated. Take the disjunctive argument A is either B or C or D, A is not C and not D, Therefore A is B. As there are four terms, we choose the series of sixteen combinations and place them on the highest ledge of the board but one. We raise the a's and out of the A's, which remain, we lower the b's. But we are not to reject all the Ab's as contradictory, because by the first premise A's may be either B's or C's or D's. Accordingly out of the Ab's we must select the c's, and out of these again the d's, so that only Abcd will remain to be rejected finally. Joining all the other fifteen combinations together again, and proceeding to premise (2), we raise the a's and lower the AC's, and thus reject the combinations inconsistent with (2); similarly we reject the AD's which are inconsistent with (3). It will be found that there remain, in addition to all the eight combinations containing a, only one containing A, namely ABcd, whence it is apparent that A must be B, the ordinary conclusion of the argument. In my "Substitution of Similars" (pp. 56-59) I have described the working upon the Abacus of two other logical problems, which it would be tedious to repeat in this place. |