or by the process of Ellipsis before described (p. 57)

A = Ac.

Third Example.

As a somewhat more complex example I take the argument thus stated, one which could not be thrown into the syllogistic form :—

"All metals except gold and silver are opaque; there

fore what is not opaque is either gold or silver or is not-metal."

There is more implied in this statement than is dis tinctly asserted, the full meaning being as follows:

All metals not gold or silver are opaque,
Gold is not opaque but is a metal,
Silver is not opaque but is a metal,
Gold is not silver.

Taking our letters thus

(1)

(2)

(3)

(4)

we may state the premises in the forms

To obtain a complete solution of the question we take the sixteen combinations of A, B, C, D, and striking out those which are inconsistent with the premises, there remain only

ABcd

AbCd

Abc D

abc D

abcd.

The expression for not-opaque things consists of the three combinations containing d, thus

or

d = ABcd AbCd | abcd, d = Ad (BcbC) abcd. In ordinary language, what is not-opaque is either metal which is gold, and then not-silver, or silver and then not gold, or else it is not-metal and neither gold nor silver.

Fourth Example.

A good example for the illustration of the Indirect Method is to be found in De Morgan's Formal Logic (p. 123), the premises being substantially as follows:

From A follows B, and from C follows D; but B and D are inconsistent with each other; therefore A and C are inconsistent.

The meaning no doubt is that where A is, B will be found, or that every A is a B, and similarly every C is a D; but B and D cannot occur together. The premises therefore appear to be of the forms

A = AB,

C = CD,

B = Bd.

(2)

(3)

On examining the series of sixteen combinations, only five are found to be consistent with the above conditions, namely,

ABcd

aBcd

abCD

abcD

abcd.

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

Fifth Example.

A more complex argument, also given by De Morgan,1 contains five terms, and is as stated below, except that the letters are altered.

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.

The meaning of the above premises is difficult to interpret, but seems to be capable of expression in the following symbolic forms

Formal Logic, p. 124. As Professor Croom Robertson has pointed out to me, the second and third premises may be thrown into a single proposition, D DeBC DEbc.

=

(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

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

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 AbC 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 (A + α),

=

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,

Fixed stars are not planets.

(1)

The false conclusion is that "fixed stars are not subject to

(2)

The combinations which remain uncontradicted on com

parison with these premises are

AbC

aBC

aBc

abC

abc.

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.