Chapter 43
.'. A is not B.
(3). _Complex Constructive._
If A is B, C is D; and if E is F, G is H.
Either A is B or E is F.
.'. Either C is D or G is H.
(4). _Complex Destructive_.
If A is B, C is D; and if E is F, G is H.
Either C is not D or G is not H.
.'. Either A is not B or E is not F.
-- 783.
(1). _Simple Constructive_.
If she sinks or if she swims, there will be an end of her.
She must either sink or swim.
.'. There will be an end of her.
(2). _Simple Destructive_.
If I go to Town, I must pay for my ticket and pay my hotel bill.
Either I cannot pay for my ticket or I cannot pay my hotel bill.
.'. I cannot go to Town.
(3). _Complex Constructive_.
If I stay in this room, I shall be burnt to death, and if I jump out of the window, I shall break my neck.
I must either stay in the room or jump out of the window.
.'. I must either be burnt to death or break my neck.
(4). _Complex Destructive_.
If he were clever, he would see his mistake; and if he were candid, he would acknowledge it.
Either he does not see his mistake or he will not acknowledge it.
.'. Either he is not clever or he is not candid.
-- 784. It must be noticed that the simple destructive dilemma would not admit of a disjunctive consequent. If we said,
If A is B, either C is D or E is F, Either C is not D or E is not F,
we should not be denying the consequent. For 'E is not F' would make it true that C is D, and 'C is not D' would make it true that E is F; so that in either case we should have one of the alternatives true, which is just what the disjunctive form 'Either C is D or E is F'
insists upon.
-- 785. In the case of the complex constructive dilemma the several members, instead of being distributively a.s.signed to one another, may be connected together as a whole--thus--
If either A is B or E is F, either C is D or G is H.
Either A is B or E is F.
.'. Either C is D
In this shape the likeness of the dilemma to the partly conjunctive syllogism is more immediately recognisable. The major premiss in this shape is vaguer than in the former. For each antecedent has now a disjunctive choice of consequents, instead of being limited to one. This vagueness, however, does not affect the conclusion. For, so long as the conclusion is established, it does not matter from which members of the major its own members flow.
-- 786. It must be carefully noticed that we cannot treat the complex destructive dilemma in the same way.
If either A is B or E is F, either C is D or G is H.
Either C is not D or G is not H.
Since the consequents are no longer connected individually with the antecedents, a disjunctive denial of them leaves it still possible for the antecedent as a whole to be true. For 'C is not D' makes it true that G is H, and 'G is not H' makes it true that C is D. In either case then one is true, which is all that was demanded by the consequent of the major. Hence the consequent has not really been denied.
-- 787. For the sake of simplicity we have limited the examples to the case of two antecedents or consequents. But we may have as many of either as we please, so as to have a Trilemma, a Tetralemma, and so on.
TRILEMMA.
If A is B, C is D; and if E is F, G is H; and if K is L, M is N.
Either A is B or E is F or K is L.
.'. Either C is D or G is H or K is L.
-- 788. Having seen what the true dilemma is, we shall now examine some forms of reasoning which resemble dilemmas without being so.
-- 789. This, for instance, is not a dilemma--
If A is B or if E is F, C is D.
But A is B and E is F.
.'. C is D.
If he observes the sabbath or if he refuses to eat pork, he is a Jew.
But he both observes the sabbath and refuses to eat pork.
.'. He is a Jew.
What we have here is a combination of two partly conjunctive syllogisms with the same conclusion, which would have been established by either of them singly. The proof is redundant.
-- 790. Neither is the following a dilemma--
If A is B, C is D and E is F.
Neither C is D nor E is F.
.'. A is not B.
If this triangle is equilateral, its sides and its angles will be equal.
But neither its sides nor its angles are equal.
.'. It is not equilateral.
This is another combination of two conjunctive syllogisms, both pointing to the same conclusion. The proof is again redundant. In this case we have the consequent denied in both, whereas in the former we had the antecedent affirmed. It is only for convenience that such arguments as these are thrown into the form of a single syllogism. Their real distinctness may be seen from the fact that we here deny each proposition separately, thus making two independent statements--C is not D and E is not F. But in the true instance of the simple destructive dilemma, what we deny is not the truth of the two propositions contained in the consequent, but their compatibility; in other words we make a disjunctive denial.
-- 791. Nor yet is the following a dilemma--
If A is B, either C is D or E is F.
Neither C is D nor E is F.