Our principle of inference then includes the rule of negative premises whenever it is true, and discriminates correctly between the cases where it does and does not apply.

The paralogism, anciently called Undistributed Middle, is also easily exhibited and infallibly avoided by our system. Let the premises be

Hydrogen is an element,

All metals are elements.

(1)

(2)

According to the syllogistic rules the middle term element is here undistributed, and no conclusion can be obtained; we cannot tell then whether hydrogen is or is not a metal. Represent the terms as follows

A = hydrogen

B = element

C = metal.

The premises then become

which occurs in both with different letters. equivalent of AB.

The reader will here, as in a former page (p. 75), find it impossible to make any substitution. The only term premises is B, but it is combined For CB we cannot substitute the We have no right to decompose combinations; and if we adhere rigidly to the rule given, that if two terms are stated to be equivalent we may substitute one for the other, we cannot commit the fallacy. It is apparent that the form of premises given above is the same as that which we obtained by translating two negative premises into the affirmative form.

The old fallacy, technically called the Illicit Process of the Major Term, is more easy to commit and more difficult to detect than any other breach of the syllogistic rules. In our system it could hardly occur. From the premises All planets are subject to gravity, Fixed stars are not planets,

(1)

(2)

we might inadvertently but fallaciously infer that, 'Fixed stars are not subject to gravity.' To reduce the premises to symbolic form, let

The reader will try in vain to produce from these premises by legitimate substitution any relation between B and C; he could not then commit the fallacy of asserting that B is not C.

There remain two other kinds of paralogism, commonly known as the fallacy of Four Terms and the Illicit Process of the Minor term. They are so evidently impossible while we obey the rule of the substitution of equivalents, that it is not necessary to give any illustrations. When there are four distinct terms in two propositions there could be no opening for a substitution. As to the Illicit Process of the Minor it consists in a flagrant substitution for a term of another wider term which is not known to be equivalent to it, and which is therefore forbidden by our rule to be substituted for it.

CHAPTER V.

DISJUNCTIVE PROPOSITIONS.

IN the previous chapter I have exhibited various forms. of deductive reasoning by the process of substitution, so far as they can be treated without the use of disjunctive propositions; but we cannot long defer the consideration of this more complex class of identities. General terms arise, as we have seen (p. 29), from classifying or mentally uniting together all objects which agree in certain qualities, the value of this union consisting in the fact that the power of knowledge is multiplied thereby. In forming such classes or general notions, we overlook or abstract the points of difference which exist between the objects joined together, and fix our attention only on the points of agreement. But every process of thought may be said to have its inverse process, which consists in undoing the effects of the direct process. Just as division undoes multiplication, and evolution undoes involution, so we must have a process which undoes abstraction, or the operation of forming general notions. This inverse process will consist in distinguishing the separate objects or minor classes which are the constituent parts of any wider class. When we mentally unite together certain objects visible in the sky and call them planets, we shall afterwards need to distinguish the contents of this general notion, which we do in the disjunctive proposition

A planet is either Mercury or Venus or the Earth or or Neptune.

Having formed the very wide class 'vertebrate animal,'

we may specify its subordinate classes thus:-A vertebrate animal is either a mammalian, bird, reptile, or fish. Nor is there any limit to the number of possible alternatives, 'An exogenous plant is either a ranunculus, a poppy, a crucifer, a rose, or it belongs to some one of the other seventy natural orders of exogens at present recognised by botanists.' A cathedral church in England must be either that of London, Canterbury, Winchester, Salisbury, Manchester, or of one of about twenty-four cities possessing such churches. And if we were to attempt to specify the meaning of the term star, we should require to enumerate as alternatives, not only the many thousands of stars recorded in catalogies, but the many millions yet unnamed.

it homotor we thus distinguish the parts of a general mistion we employ a disjunctive proposition, in at least um side of which are several alternatives joined by the su called disjunctive conjunction or, a contracted form of wher. There must be some relation between the parts thue commented in one proposition; we may call it the sgustine or alternative relation, and we must carefully inquies into its nature and results. This relation is that of dould and ignorance, giving rise to choice or uncerLambs Whenever we classify and abstract we must open The way to such uncertainty. By fixing our attention on Clain attributes to the exclusion of others we necessarily leave it doubtful what those other attributes are. The term molar tooth' bears upon the face of it that it is a part of the wider term 'tooth.' But if we meet with the simple term 'tooth' there is nothing to indicate whether it is an incisor, a canine, or a molar tooth. This doubt, however, may be resolved by other information, and we have to consider what are the appropriate logical processes for treating disjunctive propositions in connection with other propositions disjunctive or otherwise.

Expression of the Alternative Relation.

In order to represent disjunctive propositions with convenience we require a sign of the alternative or disjunctive relation, equivalent to one meaning at least of the little conjunction or so frequently used in common language. I propose to use for this purpose the symbol . In my first logical Essay I followed the example of Dr. Boole and adopted the common sign + ; but this sign should not be employed unless there exists exact analogy between mathematical addition and logical alternation. We shall find that the analogy is of a very partial character, and that there is such profound difference between a logical and a mathematical term as should prevent our uniting them by the same symbol. Accordingly I have chosen a sign, which seems aptly to suggest whatever degree of analogy may exist without implying more. The exact meaning of the symbol we will now proceed to investigate and determine.

Nature of the Alternative Relation.

Before treating disjunctive propositions it is indispensable to decide whether the alternatives shall be considered exclusive or unexclusive. By exclusive alternatives we mean those which cannot contain the same things. Thus

Matter is solid, or liquid, or gaseous; but the same portion of matter cannot be at once solid and liquid, properly speaking; still less can we suppose it to be solid and gaseous, or solid, liquid and gaseous all at the same time. Many examples on the other hand can readily be suggested in which two or more alternatives may hold true of the same object. Thus

Luminous bodies are self-luminous or luminous by reflection.

G