.. Either C is D or F is G... A is not B. (5) If either C is D or F is G, either X is Y or V is W. Either C is D or F is G; Neither X is Y nor V is W; .. Either X is Y or V is W... Neither C is D nor F is G. § 3. II. Disjunctive Syllogisms. A Disjunctive Syllogism is a syllogism of which the major premiss is a disjunctive, and the minor a simple proposition. We may indeed combine two disjunctive propositions, and draw conclusions from them, but we can only do so after reducing the disjunctive propositions to the conjunctive form. Thus from the two propositions Either A is B or C is D, Either A is B or E is F, we may draw four conclusions, viz. If C is D, E is F; If C is not D, E is not F; If E is F, C is D; If E is not F, C is not D. But as these conclusions are really drawn from conjunctive propositions which are involved in the two disjunctive propositions, we are not justified in calling the syllogisms disjunctive. Hence, as will be noticed, our definition of disjunctive is not so wide as that of conjunctive syllogisms. The disjunctive syllogism admits of four conclusions, which may be exhibited thus: Either A is B, or C is D, or E is F. (1) A is B; .. Neither C is D nor E is F. (5) Either A is B or C is D; .. E is not F. Note.-Mr. Mill (in his Examination of Sir W. Hamilton's Philosophy, ch. xxiii.) maintains that a disjunctive proposition merely implies that the two alternatives cannot both be false, but that it does not exclude the possibility of both of them being true. Thus, in the last example, he would maintain that there is nothing in the form of the assertion to exclude the supposition of the man being both a fool and a knave. In this opinion he is preceded by many other logicians, but it seems to us that in the expression 'either clude the possibility of both well as of both being false. wish to exclude the possibility of both being true, we add the words 'or both,' thus: He is either a fool or a or -'we distinctly exalternatives being true, as In fact, when we do not 6 knave, or both;' 'I shall come either to-day or to-morrow or perhaps both days.' I 4. The Dilemma. There remains the case in which one premiss of the complex syllogism is a conjunctive and the other a disjunctive proposition. This is called a Dilemma. The order of the premisses is indifferent, but it seems more natural that the conjunctive proposition should be the major. If we consider the case in which the major consists of one antecedent and several consequents, there is only one valid form of argument, and that is destructive. (1) If A is B, C is D and E is F; But either C is not D or E is not F; ... A is not B. If we asserted in the minor 'C is D and E is F' there would be no conclusion, and if we asserted 'Neither C is D nor E is F,' the minor would not be disjunctive. The assertionEither C is D or E is F' is, according to our view of the significance of a disjunctive proposition, equivalent to the assertion 'Either C is not D or E is not F,' and leads to the same conclusion. If the major consist of several antecedents and one consequent, there is only one valid form of argument, and that is constructive. (2) If A is B or if E is F, C is D; But either A is B or E is F; ... C is D. If we asserted in the minor 'C is not D,' it would not satisfy the requirements of the definition by being a disjunctive proposition. In the remaining case, where there are several antecedents and several consequents, there are two valid forms, one constructive and the other destructive. (3) If A is B, C is D; and if E is F, G is H; But either A is B, or E is F; (4) .. Either C is D, or G is H. If A is B, C is D; and if E is F, G is H; .. Either A is not B, or E is not F. It is evident that we may form a Trilemma, Tetralemma, &c., by increasing the number of antecedents or consequents or both, thus : If A is B, or if E is F, or if G is H, C is D; But either A is B, or E is F, or G is H; ... C is D. If A is B, C is D; and if E is F, G is H; and if I is J, K is L; But either A is B, or E is F, or I is J; ... Either C is D, or G is H, or K is L. It is not uncommon to mistake for a dilemma what is really only a conjunctive syllogism. Thus the two following syllogisms, when examined, will be found to be, the first a constructive, the second a destructive conjunctive. |