In these combinations the only A which appears is joined to c, and similarly C is joined to a, or A is inconsistent with C.

A more complex argument, also given by De Morgan f, contains five terms, and is as stated below, except that I have altered the letters.

'Every A is one only of the two B or C; D is both B

and C, except when B is E, and then it is neither; therefore no A is D.'

A little reflection will show that these premises are capable of expression in the following symbolic forms

As five letters, A, B, C, D, E, 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

Now 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—

DaBCDe + abcDE.

Hence it is clear that no A is D, and vice versa no D is A. We might also draw many other conclusions from the premises; for instance

or D and E never meet but in the absence of A, B, and C.

f Formal Logic,' p. 124.

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 any 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 common fallacies or paralogisms which are not seldom committed in ordinary discussion. Let us take the example of a fallacious argument, previously treated by the Method of Direct Inference (p. 75),

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

(2)

the premises become

C = Cb.

Of the eight conceivable combinations of A, B, C, five agree

with these conditions, namely

that is, granite is not a sedimentary rock but is either basalt or not-basalt. If we want a description of basalt the answer is of like form

Basalt is a sedimentary rock, and either granite or notgranite. As it is already perfectly evident that basalt

must be either granite or not, and vice versa, 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 (pp. 75-77).

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. 77)

All planets are subject to gravity,

Fixed stars are not planets.

(1)

(2)

The false conclusion is that fixed stars are not subject to gravity.' The terms are

The combinations which remain uncontradicted on comparison with these premises are

B = aBC + a Bc,

that is, a fixed star is not a planet, but is either subject or not, as the case may be, to 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 any simply 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 may arise. 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 abecedarium, for any number of letters, instead of being printed in fixed order on a piece of paper or slate, were marked upon light moveable pieces of wood, mechanical arrangements could readily be devised for selecting the combinations in any required order. 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 lectureroom 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 Similarss, and I will here give only a general description.

The abacus consists of a common school black-board 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 abecedarium (see p. 109), 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. 109. Each combination of letters is separately fixed to the surface of a thin slip of wood one inch broad and about oneeighth 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

Pp. 55-59, 81-86.

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, obtaining the AB's; and in like manner we might select any other class 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

we proceed as follows:-Firstly 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 not-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 should now find our combinations arranged as below.