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.
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.
