Deductive Logic Part 37

.'. If C is D, A is sometimes not not-D. (O)

-- 722.

(A) If a man is a smoker, he always drinks.

.'. If a man is a total abstainer, he never smokes. (E)

(E) If a man merely does his duty, no one ever thanks him.

.'. If people thank a man, he has always done more than his duty. (A)

(I) If a statesman is patriotic, he sometimes adheres to a party.

.'. If a statesman adheres to a party, he is sometimes not unpatriotic. (O)

(O) If a book has merit, it sometimes does not sell.

.'. If a book fails to sell, it sometimes has merit. (I)

(E per acc.) If the wind is high, rain never falls.

.'. If rain falls, the wind is sometimes high. (I)

(A per acc.) If a thing is common, it is always cheap.

.'. If a thing is cheap, it is sometimes not uncommon. (O)

-- 723. When applied to disjunctive propositions, the distinctive features of conversion by negation are still discernible. In each of the following forms of inference the converse differs in quality from the convertend and has the contradictory of one of the original terms (-- 515).

-- 724.

(A) Either A is B or C is always D.

.'. Either C is D or A is never not-B. (E)

(E) Either A is B or C is never D.

.'. Either C is not-D or A is always B. (A)

(I) Either A is B or C is sometimes D.

.'. Either C is not-D or A is sometimes not B. (O)

(O) Either A is B or C is sometimes not D.

.'. Either C is D or A is sometimes not-B. (I)

-- 725.

(A) Either miracles are possible or every ancient historian is untrustworthy.

.'. Either ancient historians are untrustworthy or miracles are not impossible. (E)

(E) Either the tide must turn or the vessel can not make the port.

.'. Either the vessel cannot make the port or the tide must turn. (A)

(1) Either he aims too high or the cartridges are sometimes bad.

.'. Either the cartridges are not bad or he sometimes does not aim too high. (0)

(O) Either care must be taken or telegrams will sometimes not be correct.

.'. Either telegrams are correct or carelessness is sometimes shown. (1)

-- 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 a.s.sumes 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.