_Of the Disjunctive Syllogism_.
§ 760. Roughly speaking, a Disjunctive Syllogism results from the
combination of a disjunctive with a simple premiss. As in the
preceding form, the complex proposition is regarded as the major
premiss, since it lays down a hypothesis, which is applied to fact in
the minor.
§ 761. The Disjunctive Syllogism may be exactly defined as follows--
A complex syllogism, which has for its major premiss a disjunctive
proposition, either the antecedent or consequent of which is in the
minor premiss simply affirmed or denied.
§ 762. Thus there are four types of disjunctive syllogism possible.
_Constructive Moods._
(1) Either A is B or C is D. (2) Either A is B or C is D.
A is not B. C is not D.
.'. C is D. .'. A is B.
Either death is annihilation or we are immortal.
Death is not annihilation.
.'. We are immortal.
Either the water is shallow or the boys will be drowned.
The boys are not drowned.
.'. The water is shallow.
_Destructive Moods_.
(3) Either A is B or C is D. (4) Either A is B or C is D.
A is B. C is D.
.'. C is not D. .'. A is not B.
§ 763. Of these four, however, it is only the constructive moods that
are formally conclusive. The validity of the two destructive moods is
contingent upon the kind of alternatives selected. If these are such
as necessarily to exclude one another, the conclusion will hold, but
not otherwise. They are of course mutually exclusive whenever they
embody the result of a correct logical division, as 'Triangles are
either equilateral, isosceles or scalene.' Here, if we affirm one of
the members, we are justified in denying the rest. When the major thus
contains the dividing members of a genus, it may more fitly be
symbolized under the formula, 'A is either B or C.' But as this admits
of being read in the shape, 'Either A is B or A is C,' we retain the
wider expression which includes it. Any knowledge, however, which we
may have of the fact that the alternatives selected in the major are
incompatible must come to us from material sources; unless indeed we
have confined ourselves to a pair of contradictory terms (A is either
B or not-B). There can be nothing in the form of the expression to
indicate the incompatibility of the alternatives, since the same form
is employed when the alternatives are palpably compatible. When, for
instance, we say, 'A successful student must be either talented or
industrious,' we do not at all mean to assert the positive
incompatibility of talent and industry in a successful student, but
only the incompatibility of their negatives--in other words, that, if
both are absent, no student can be successful. Similarly, when it is
said, 'Either your play is bad or your luck is abominable,' there is
nothing in the form of the expression to preclude our conceiving that
both may be the case.
§ 764. There is no limit to the number of members in the disjunctive
major. But if there are only two alternatives, the conclusion will be
a simple proposition; if there are more than two, the conclusion will
itself be a disjunctive. Thus--
Either A is B or C is D or E is F or G is H.
E is not F.
.'. Either A is B or C is D or G is H.
§ 765. The Canon of the Disjunctive Syllogism may be laid down as
follows--
To deny one member is to affirm the rest, either simply or
disjunctively; but from affirming any member nothing follows.