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

A = planet

B fixed star

=

C = subject to gravity;

then we have the propositions

A = AC

B Ba.

=

(1)

(2) 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 as in A B and C = D, there could evidently be no opening for substitution. As to the Illicit Process of the Minor Term 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 not allowed by our rule to be substituted for it.

=

F

CHAPTER V.

DISJUNCTIVE PROPOSITIONS.

IN the previous chapter I have exhibited various cases of deductive reasoning by the process of substitution, avoiding the introduction 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. 24), 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 generalization, 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. If 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 verte

brate animal is either a mammal, bird, reptile, or fish." Nor is there any limit to the number of possible alterna tives. "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 recognized 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 catalogues, but the many millions unnamed.

Whenever we thus distinguish the parts of a general notion we employ a disjunctive proposition, in at least one side of which are several alternatives joined by the socalled disjunctive conjunction or, a contracted form of other. There must be some relation between the parts thus connected in one proposition; we may call it the disjunctive or alternative relation, and we must carefully inquire into its nature. This relation is that of ignorance and doubt, giving rise to choice. Whenever we classify and abstract we must open the way to such uncertainty. By fixing our attention on certain attributes to the exclusion of others, we necessarily leave it doubtful what those other attributes 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 further information, and we have to consider what are the appropriate logical processes for treating disjunctive propositions in connection with other propositions disjunctive or otherwise.

are.

Expression of the Alternative Relation.

In order to represent disjunctive propositions with convenience we require a sign of the alternative 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 practice of Boole and adopted

the 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 imperfect, and that there is such profound difference between logical and mathematical terms 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.

Nature of the Alternative Relation.

Before treating disjunctive propositions it is indispensable to decide whether the alternatives must be considered exclusive or unexclusive. By exclusive alternatives we mean those which cannot contain the same things. If we say "Arches are circular or pointed," it is certainly to be understood that the same arch cannot be described as both circular and pointed. 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.

It is undoubtedly possible, by the laws of optics, that the same surface may at one and the same moment give off light of its own and reflect light from other bodies. We speak familiarly of deaf or dumb persons, knowing that the majority of those who are deaf from birth are also dumb.

There can be no doubt that in a great many cases, perhaps the greater number of cases, alternatives are exclusive as a matter of fact. Any one number is incompatible with any other; one point of time or place is exclusive of all others. Roger Bacon died either in 1284 or 1292; it is certain that he could not die in both years. Henry Fielding was born either in Dublin or Somersetshire; he could not be born in both places. There is so much more precision and clearness in the use of exclusive alternatives that we ought doubtless to select them when possible. Old works on logic accordingly contained a rule directing that the Membra dividentia, the

parts of a division or the constituent species of a genus, should be exclusive of each other.

It is no doubt owing to the great prevalence and convenience of exclusive divisions that the majority of logicians have held it necessary to make every alternative in a disjunctive proposition exclusive of every other one. Aquinas considered that when this was not the case the proposition was actually false, and Kant adopted the same opinion. A multitude of statements to the same effect might readily be quoted, and if the question were to be determined by the weight of historical evidence, it would certainly go against my view. Among recent logicians Hamilton, as well as Boole, took the exclusive side. But there are authorities to the opposite effect. Whately, Mansel, and J. S. Mill have all pointed out that we may often treat alternatives as Compossible, or true at the same time. Whately gives us an example, "Virtue tends to procure us either the esteem of mankind, or the favour of God," and he adds-" Here both members are true, and consequently from one being affirmed we are not. authorized to deny the other. Of course we are left to conjecture in each case, from the context, whether it is meant to be implied that the members are or are not exclusive." Mansel says, "We may happen to know that two alternatives cannot be true together, so that the affirmation of the second necessitates the denial of the first; but this, as Boethius observes, is a material, not a formal consequence." Mill has also pointed out the absurdities which would arise from always interpreting alternatives as exclusive. If we assert," he says, "that a man who has acted in some particular way must be either a knave or a fool, we by no means assert, or intend to assert, that he cannot be both." Again, "to make an entirely unselfish use of despotic power, a man must be either a saint or a philosopher. Does the disjunctive premise necessarily imply, or must it be construed as supposing, that the same person cannot be both a

4

1 Mansel's Aldrich, p. 103, and Prolegomena Logica, p. 221.

2

Elements of Logic, Book II. chap. iv. sect. 4.

Aldrich, Artis Logica Rudimenta, p. 104.

Examination of Sir W. Hamilton's Philosophy, pp. 452-454.