315. Let X, Y, Z, P, Q, R be six propositions.
Given (1) If X is true, P is true ;
(2) If Y is true, Q is true ;
(3) If Z is true, R is true ;
(4) Of X, Y, Z one at least is true ;
(5) Of P, Q, R not more than one is true ;
prove syllogistically
(i) If P is true, X is true ;
(ii) If Q is true, Y is true ;
(iii) If R is true, Z is true ;
(iv) Of P, Q, R, one at least is true ;
(v) Of X, Y, Z, not more than one is true. [K.]
CHAPTER VI.
DISJUNCTIVE SYLLOGISMS.
316. The Disjunctive Syllogism.—A disjunctive (or alternative) syllogism may be defined as a formal reasoning in which a categorical premiss is combined with a disjunctive (alternative) premiss so as to yield a conclusion which is either categorical or else disjunctive (alternative) with fewer alternants than are contained in the disjunctive premiss.[389]
[389] Archbishop Thomson’s definition of the disjunctive syllogism—“An argument in which there is a disjunctive judgment” (Laws of Thought, p. 197)—must be regarded as too wide if, as is usually the case, an affirmative judgment with a disjunctive predicate is considered disjunctive. It would include such a syllogism as the following,—B is either C or D, A is B, therefore A is either C or D. The argument here in no way turns upon the alternation contained in the major premiss, and the reasoning may be regarded as an ordinary categorical syllogism in Barbara, the major term being complex.
Logicians have not, as a rule, given any distinctive recognition to arguments consisting of two disjunctive premisses and a disjunctive conclusion; and Mr Welton goes so far as to remark that “both premisses of a syllogism cannot be disjunctive since from two assertions as indefinite as disjunctive propositions necessarily are, nothing can be inferred” (Logic, p. 327). It is, however, clear that this is erroneous, if an argument consisting of two hypothetical premisses and a hypothetical conclusion is possible, and if a hypothetical can be reduced to the disjunctive form. As an example we may express in disjunctives the hypothetical syllogism given on page [348]: Either Q is not true or R is true, Either P is not true or Q is true, therefore, Either P is not true or R is true. Here questions of modality are left on one side. They would not, however, in any case materially affect the argument.
For example,
| A is either B or C, | |
| A is not B, | |
| therefore, | A is C ; |
| Either P or Q or R is true, | |
| P is not true, | |
| therefore, | Either Q or R is true. |
360 The categorical premiss in each of the above syllogisms denies one of the alternants of the alternative premiss, and the conclusion affirms the remaining alternant or alternants. Reasonings of this type are accordingly described as examples of the modus tollendo ponens.