Deductive Logic Part 36

The original proposition

'If A is B, C is always D'

is equivalent to the simple proposition

'All cases of A being B are cases of C being D.'

This, when converted, becomes

'Some cases of C being D are cases of A being B,'

which, when thrown back into the conjunctive form, becomes

'If C is D, A is sometimes B.'

-- 712. This expression must not be misunderstood as though it contained any reference to actual existence. The meaning might be better conveyed by the form

'If C is D, A may be B.'

But it is perhaps as well to retain the other, as it serves to emphasize the fact that formal logic is concerned only with the connection of ideas.

-- 713. A concrete instance will render the point under discussion clearer. The example we took before of an A proposition in the conjunctive form--

'If kings are ambitious, their subjects always suffer'

may be converted into

'If subjects suffer, it may be that their kings are ambitious,'

i.e. among the possible causes of suffering on the part of subjects is to be found the ambition of their rulers, even if every actual case should be referred to some other cause. It is in this sense only that the inference is a necessary one. But then this is the only sense which formal logic is competent to recognise. To judge of conformity to fact is no part of its province. From 'Every AB is a CD' it follows that ' Some CD's are AB's' with exactly the same necessity as that with which 'Some B is A' follows from 'All A is B.' In the latter case also neither proposition may at all conform to fact. From 'All centaurs are animals' it follows necessarily that 'Some animals are centaurs': but as a matter of fact this is not true at all.

-- 714. The E and the I proposition may be converted simply, as above.

-- 715. O cannot be converted at all. From the proposition

'If a man runs a race, he sometimes does not win it,'

it certainly does not follow that

'If a man wins a race, he sometimes does not run it.'

-- 716. There is a common but erroneous notion that all conditional propositions are to be regarded as affirmative. Thus it has been a.s.serted that, even when we say that 'If the night becomes cloudy, there will be no dew,' the proposition is not to be regarded as negative, on the ground that what we affirm is a relation between the cloudiness of night and the absence of dew. This is a possible, but wholly unnecessary, mode of regarding the proposition. It is precisely on a par with Hobbes's theory of the copula in a simple proposition being always affirmative. It is true that it may always be so represented at the cost of employing a negative term; and the same is the case here.

-- 717. There is no way of converting a disjunctive proposition except by reducing it to the conjunctive form.

-- 718. _Permutation of Complex Propositions_.

(A) If A is B, C is always D.

.'. If A is B, C is never not-D. (E)

(E) If A is B, C is never D.

.'. If A is B, C is always not-D. (A)

(I) If A is B, C is sometimes D.

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

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

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

-- 719.

(A) If a mother loves her children, she is always kind to them.

.'. If a mother loves her children, she is never unkind to them. (E)

(E) If a man tells lies, his friends never trust him.

.'. If a man tells lies, his friends always distrust him. (A)

(I) If strangers are confident, savage dogs are sometimes friendly.

.'. If strangers are confident, savage dogs are sometimes not unfriendly. (O)

(O) If a measure is good, its author is sometimes not popular.

.'. If a measure is good, its author is sometimes unpopular. (I)

-- 720. The disjunctive proposition may be permuted as it stands without being reduced to the conjunctive form.

Either A is B or C is D.

.'. Either A is B or C is not not-D.

Either a sinner must repent or he will be d.a.m.ned.

.'. Either a sinner must repent or he will not be saved.

-- 721. _Conversion by Negation of Complex Propositions._

(A) If A is B, C is always D.

.'. If C is not-D, A is never B. (E)

(E) If A is B, C is never D.

.'. If C is D, A is always not-B. (A)

(I) If A is B, C is sometimes D.

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

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

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

(E per acc.) If A is B, C is never D.

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

(A per ace.) If A is B, C is always D.