§ 726. In the above examples the converse of E looks as if it had undergone no change but the mere transposition of the alternative. This appearance arises from mentally reading the E as an A proposition: but, if it were so taken, the result would be its contrapositive, and not its converse by negation. § 727. The converse of I is a little difficult to grasp. It becomes easier if we reduce it to the equivalent conjunctive-- 'If the cartridges are bad, he sometimes does not aim too high.' Here, as elsewhere, 'sometimes' must not be taken to mean more than 'it may be that.' § 728. _Conversion by Contraposition of Complex Propositions._ As applied to conjunctive propositions conversion by contraposition assumes the following forms-- (A) If A is B, C is always D. .'. If C is not-D, A is always not-B. (O) If A is B, C is sometimes not D. .'. If C is not-D, A is sometimes not not-B. (A) If a man is honest, he is always truthful. .'. If a man is untruthful, he is always dishonest. (O) If a man is hasty, he is sometimes not malevolent. .'. If a man is benevolent, he is sometimes not unhasty. § 729. As applied to disjunctive propositions conversion by contraposition consists simply in transposing the two alternatives. (A) Either A is B or C is D. .'. Either C is D or A is B. For, when reduced to the conjunctive shape, the reasoning would run thus-- If A is not B, C is D. .'. If C is not D, A is B. which is the same in form as All not-A is B. .'. All not-B is A. Similarly in the case of the O proposition (O) Either A is B or C is sometimes not D. .'. Either C is D or A is sometimes not B. § 730. On comparing these results with the converse by negation of each of the same propositions, A and 0, the reader will see that they differ from them, as was to be expected, only in being permuted. The validity of the inference may be tested, both here and in the case of conversion by negation, by reducing the disjunctive proposition to the conjunctive, and so to the simple form, then performing the process as in simple propositions, and finally throwing the converse, when so obtained, back into the disjunctive form. We will show in this manner that the above is really the contrapositive of the 0 proposition. (O) Either A is B or C is sometimes not D. = If A is not B, C is sometimes not D. = Some cases of A not being B are not cases of C being D. (Some A is not B.) = Some cases of C not being D are not cases of A being B. (Some not-B is not not-A.) = If C is not D, A is sometimes not B. = Either C is D or A is sometimes not B. CHAPTER XX. _Of Complex Syllogisms_. § 731. A Complex Syllogism is one which is composed, in whole or part, of complex propositions. § 732. Though there are only two kinds of complex proposition, there are three varieties of complex syllogism. For we may have