qualities is wanting, the combined term will not apply. Hence the negative of the whole term is

Not-malleable or not-dense or not-metallic. In the above the conjunction or must clearly be interpreted as unexclusive; for there may readily be objects which are both not-malleable, and not-dense, and perhaps not-metallic at the same time. If in fact we were required to use or in a strictly exclusive manner, it would be requisite to specify seven distinct alternatives in order to describe the negative of a combination of three single terms. The negatives of four or five terms would consist of fifteen or thirty-one alternatives. This consideration alone is sufficient to prove that the meaning of or cannot be always exclusive in common language.

Every disjunctive term, then, is the negative of a combined term, and vice versa.

Apply this result to the combined term AAA, and its negative is

a | a | a.

Now since AAA is by the Law of Simplicity equivalent to A, so a ↑ a † a must be by the Law of Unity equivalent Each law thus necessarily presupposes the other.

to a.

We

Symbolic expression of the Law of Duality.

may now employ our symbol of alternation to express in a clear and formal manner the third Fundamental Law

of Thought, which I have called the Law of Duality. Taking A to represent any class or object or quality, and B any other class, object or quality, we may always assert that A either agrees with B, or does not agree. Thus we may say

A AB+ Ab.

This is a formula which will henceforth be constantly employed, and it lies at the basis of reasoning.

The reader may perhaps wish to know why A is inserted. in both alternatives of the second member of the identity, and why the law is not stated in the form

A =B+b.

But if he will consider the contents of the last section (p. 87), he will see that the latter expression cannot be correct, otherwise no term would have any negative. For the negative of Bb is bB, or a self-contradictory term; so that if A were identical with Bb, its negative a would be non-existent. This result would generally be an absurd one, and I see much reason to think that in a strictly logical point of view it would always be absurd. In all probability we ought to assume as a fundamental logical axiom that every term has its negative in thought. We cannot think at all without separating what we think about from other different things, and these things necessarily form the negative notionf. If so, it follows that any term of the form Bb is just as self-contradictory as one of the form Bb.

It will be convenient to recapitulate in this place the three great Laws of Thought in their symbolic form, thus

f 'Pure Logic, p. 65. See also the criticism of this point by De Morgan in the Athenæum,' No. 1892, 30th January, 1864; p. 155.

Various Forms of the Disjunctive Proposition.

Disjunctive propositions may occur in a great variety of forms, of which the old logicians took very insufficient notice. There may be any number of alternatives each of which may be a combination of any number of simple terms. A proposition, again, may be disjunctive in one or both members. The proposition

Solids or liquids or gases are electrics or conductors of

electricity

is an example of the doubly disjunctive form. The meaning of any such proposition is that whatever falls under any one or more alternatives on one side must fall under one or more alternatives on the other side. From what has been said before, it is apparent that the proposition A + B C + D

will correspond to

ab= cd,

each member of the latter being the negative of a member of the former proposition.

As an instance of a complex disjunctive proposition. I may give Senior's definition of wealth, namely 'Wealth is what is transferable, limited in supply, and either productive of pleasure or preventive of pain g.'

but if we develop the alternatives by a method to be afterwards more fully considered, it becomes

A =BCDE+ BCDe + BCdE.

* Boole's 'Laws of Thought,' p. 106. Jevons' 'Pure Logic,' p. 69.

An example of a still more complex proposition may be found in De Morgan's writingsh, and is as follows:'He must have been rich, and if not absolutely mad was weakness itself, subjected either to bad advice or to most unfavourable circumstances.'

If we assign the letters of the alphabet in succession, thus,

E =

F

=

subjected to bad advice

subjected to most unfavourable circumstances,

the proposition will take the form

A = AB{C + D (E + F)},

and if we develop the alternatives, expressing some of the different cases which may happen, we obtain A = ABC + ABCDEF+ ABCDEƒ+ ABcDeF.

Inference by Disjunctive Propositions.

Before we can make a free use of disjunctive propositions in the processes of inference we must consider how disjunctive terms can be combined together or with simple terms. In the first place, to combine a simple term with a disjunctive one, we must combine it with every alternative of the disjunctive term. A vegetable, for instance, is either a herb, a shrub, or a tree. Hence an exogenous vegetable is either an exogenous herb, or an exogenous shrub, or an exogenous tree. Symbolically stated this process of combination is as follows

Secondly, to combine two disjunctive terms with each other, combine each alternative of one separately with each

h On the Syllogism,' No. iii. p. 12. Camb. Phil. Trans., vol. x.

part i.

alternative of the other. Since flowering plants are either exogens or endogens, and are at the same time either herbs, shrubs or trees, it follows that there are altogether six alternatives-namely, exogenous herbs, exogenous shrubs, exogenous trees, endogenous herbs, endogenous shrubs, endogenous trees. This process of combination is shown in the general form

(A + B) (C + D) = AC+ AD + BC + BD.

It is hardly necessary to point out that, however numerous the terms combined, or the alternatives in those terms, we may effect the combination provided each alternative is combined with each alternative of the other terms, as in the algebraic process of multiplication.

Some processes of deduction may at once be exhibited. We may always, for instance, unite the same qualifying term to each side of an identity even though one or both members of the identity be disjunctive. Thus let

A=B+ C.

Now it is self-evident that

AD=AD,

and in one side of this identity we may for A substitute its equivalent B + C obtaining

Since 'a gaseous element is either hydrogen, or oxygen, or nitrogen, or chlorine, or fluorine,' it follows that a free gaseous element is either free hydrogen, or free oxygen, or free nitrogen, or free chlorine, or free fluorine.'

This process of combination will lead to most useful inferences when the qualifying adjective combined with both sides of the proposition is a negative of one or more alternatives. Since chlorine is a coloured gas, we may infer that a colourless gaseous element is either (colourless) hydrogen, oxygen, nitrogen, or fluorine.' The alternative chlorine disappears because colourless chlorine does not exist. Again, since a tooth is either an incisor,