may I believe, be safely inferred that every proposition of
type XIII., that of the premise in question, will have at
least two propositions neutral to it, of types II. and XII.
respectively.

Suppose it be required, as a second instance, to define the
precise points of agreement and difference of two disput-
ants, one of whom asserts that (1) 'Space = three-way
spread with points as elements' (Henrici); while his oppo-
nent holds that (2) 'Space = three-way spread,' and at
the same time (3) 'Space has points as elements,' but is not
known to be the only thing that has. The three assertions
are symbolised as below, the combinations excluded being
indicated by their Greek signs:

We see that the second disputant's assertions have a
logical force superior to that of the first by, namely,
which corresponds to assertion 5, or aBaBC. In addition,
then, to all that the first asserts, he affirms that 'a three-way
spread which is not space has points as elements.'

As a third instance of the power and flexibility of this
combinational logic, suppose it to be required to make an
exhaustive statement of all the inferences which can be
drawn from the theorem that 'Similar figures (A) consist
of all whose corresponding angles are equal (B), and whose
corresponding sides are proportional (C).' We proceed in
this way. The proposition is of the form ABC, of the
VIth type, and negatives ẞ, y, d, ε. Any proposition, then,
which negatives one, two, or three of these combinations

will be inferrible from the theorem, but not equivalent to it. All the possible inferences, therefore, are indicated in the following table of Index Numbers, which, taken in connection with the Logical Index, sufficiently explains itself:

These fourteen assertions, which are all the possible non-equivalent inferences, or the equivalents of these, were detected by the Logical Index in a few minutes; it would be doubtfully possible, and in any case a most laborious problem, to obtain an exhaustive statement of inferences by any other method, if indeed any other method exists.

The want of space alone prevents my giving more abundant illustrations of the multitudes of logical problems which may be solved infallibly and speedily by the use of the Logical Index. It may be safely said that in four pages of tables it gives the key to all possible logical questions, relations or problems involving three distinct logical terms. There is some possibility that the corresponding index for the relations of four terms may some day be worked out, as, when exhibited in like manner, it will occupy only one volume of 1024 pages of a rather larger size than those of this volume. There is no prospect whatever that the corresponding index for five terms will ever be exhaustively published, since it would fill a library of 65,536 volumes, each containing 1024 large pages. This fact will give some faint idea of the possible number and complexity of logical relations involving only a very moderate number of terms.

The Logical Stamp

In my previous logical books I described a Logical Slate with five series of the combinations of the Logical Alphabet engraved upon it. I first made such a slate in May 1863, and I have since frequently used it with much saving of labour. The recent extensive introduction of india-rubber printing stamps lately suggested to me the idea that the most convenient method of obtaining the logical combinations would be to stamp them on paper. Two stamps producing the combinations of three and of four terms as shown in columns IV. and V. of the Logical Alphabet (p. 181), were made for me at a cost of about eleven shillings.

They have been very successful, and leave nothing to be desired as regards the private study of logical problems. One great advantage of the stamps over the slate is evident, namely, that the work being done on paper can be preserved for reference without copying.

The ABCD stamp can readily be utilised for problems of five, six, or more terms. For six terms, for instance, it is requisite to make four impressions and distinguish them by writing EF, Ef, eF, ef, above the respective impressions.

India-rubber stamps of any design can now be easily ordered at all the principal stationers.

1 Pure Logic, 1864, p. 68; Substitution of Similars, 1869, p. 54; Elementary Lessons in Logic, 1870, p. 199; Principles of Science, 1874, Vol. i. p. 110; New Editions, p. 96.