diamagnetic substances are capable of polarity CCD, it follows by substitution that all substances are capable of polarity, or A = AD. We may divide the class substance again into the three subclasses, solid, liquid, and gas; and if we can show that in each of these forms it obeys Carnot's thermodynamic law, it follows that all substances obey that law. Similarly we may show that all vertebrate animals possess red blood, if we can show separately that fish, reptiles, birds, marsupials, and mammals possess red blood, there being, as far as is known, only five principal subclasses of vertebrata.

Our inductions will often be embarrassed by exceptions, real or apparent. We might affirm that all gems are incombustible were not diamond undoubtedly combustible. Nothing seems more evident than that all the metals are opaque until we examine them in fine films, when gold and silver are found to be transparent. All plants absorb carbonic acid except certain fungi; all the bodies of the planetary system have a progressive motion from west to east, except the satellites of Uranus and Neptune. Even some of the profoundest laws of matter are not quite universal; all solids expand by heat except india-rubber, and possibly a few other substances; all liquids which have been tested expand by heat except water below 4°C and fused bismuth; all gases have a coefficient of expansion increasing with the temperature except hydrogen. In a later chapter I shall consider how such anomalous cases may be regarded and classified; here we have only to express them in a consistent manner in our notation.

Let us take the case of the transparency of metals, and assign the terms thus

A = metal

B = gold
C = silver

D = iron

=

E, F &c. copper, lead, &c.
X = opaque.

and so on for the rest of the metals. Now evidently

Abc

=

(DEF...)/c,

and by substitution as before we shall obtain

Abc= AbcX,

or in words, 'All metals not gold nor silver are opaque ;' at the same time we have

=

A(B+ C) AB+ AC AB+ ACx A(B+ C)x, or 'Metals which are either gold or silver are not opaque.' In some cases the problem of induction assumes a much higher degee of complexity. If we examine the properties of crystallized substances we may find some properties. which are common to all, as cleavage or fracture in definite planes; but it would soon become requisite to break up the class into several minor ones. We should divide crystals according to the seven accepted systems-and we should then find that crystals of each system possess many common properties. Thus crystals of the Regular or Cubical system expand equally by heat, conduct heat and electricity with uniform rapidity, and are of like elasticity in all directions; they have but one index of refraction for light; and every facet is repeated in like relation to each of the three axes. Crystals of the system which possess one principal axis will be found to possess the various physical powers of conduction, refraction, elasticity, &c., uniformly in directions perpendicular to the principal axis, but in other directions their properties vary according to complicated laws. The remaining systems in which the crystals possess three unequal axes, or have inclined axes, exhibit still more complicated results, the

effects of the crystal upon light, heat, electricity, &c., varying in all directions. But when we pursue induction into the intricacies of its application to Nature we really enter upon the subject of classification which we must take up again in a later part of this work.

Complete Solution of the Inverse or Inductive Logical
Problem.

It is now plain that Induction consists in passing back from a series of combinations to the laws by which such combinations are governed. The natural law that all metals are conductors of electricity really means that in nature we find three classes of objects, namely

1. Metals, conductors;

2. Not-metals, conductors;

3. Not-metals, not-conductors.

It comes to the same thing if we say that it excludes the existence of the class, metals not-conductors.' In the same way every other law or group of laws will really mean the exclusion from existence of certain combinations of the things, circumstances or phenomena governed by those laws. Now in logic we treat not the phenomena and laws but, strictly speaking, the general forms of the laws; and a little consideration will show that for a finite number of things the possible number of forms or kinds of law governing them must also be finite. Using general terms we know that A and B can be present or absent in four ways and no more—thus

AB, Ab, aB, ab;

therefore every possible law which can exist concerning the relation of A and B must be marked by the exclusion of one or more of the above combinations. The number of possible laws then cannot exceed the number of selections which we can make from these four combinations,

and we arrive at this utmost number of cases by omitting any one or more of the four. The number of cases to be considered is therefore 2 x 2 x 2 x 2 or sixteen, since each may be present or absent; and these cases are all shown in the following table, in which the sign o indicates absence or non-existence of the combination shown at the left-hand column in the same line, and the mark 1 its presence :

Thus in column sixteen we find that all the conceivable combinations are present, which means that there are no special laws in existence in such a case, and that the combinations are governed only by the universal Laws of Identity and Difference. The example of metals and conductors of electricity would be represented by the twelfth column; and every other mode in which two things or qualities might present themselves is shown in one or other of the columns. More than half the cases may indeed be at once rejected, because they involve the entire absence of a term or its negative. It has been shown to be a necessary logical principle that every term must have its negative (p. 88), and where this is not the case some inconsistency between the laws or conditions of combinations must exist. Thus if we laid down the two following propositions, Graphite conducts electricity,' and Graphite does not conduct electricity,' it would amount to asserting the impossibility of graphite existing at all; or in general terms, A is B and A is not B result in destroying altogether the combinations containing A.

We therefore restrict our attention to those cases which may be represented in natural phenomena where at least two combinations are present, and which correspond to those columns of the table in which each of A, a, B, b appears. These cases are shown in the columns marked with an asterisk.

We find that seven cases remain for examination, thus characterised

Four cases exhibiting three combinations,
Two cases exhibiting two combinations,
One case exhibiting four combinations.

It has already been pointed out that a proposition of the form A = AB destroys one combination Ab, so that this is the form of law applying to the twelfth case. But by changing one or more of the terms in A = AB into its negative, or by interchanging A and B, a and b, we obtain no less than eight different varieties of the one form; thus

But the reader of the preceding sections will at once see that each proposition in the lower line is logically equivalent to, and is in fact the contrapositive of, that above it (p. 98). Thus the propositions A= Ab and B=aB both give the same combinations, shown in the eighth column of the table, and trial shows that the twelfth, eighth, fifteenth and fourteenth cases are thus fully accounted for. We come to this conclusion then-The general form of proposition A = AB admits of four logically distinct varieties, each capable of expression in two different modes.

In two columns of the table, namely the seventh and tenth, we observe that two combinations are missing. Now a simple identity A = B renders impossible both Ab and aB,