Deductive Logic Part 18

-- 484. The second rule is merely a conventional one. We may make a valid inference in defiance of it: but such an inference will be seen presently to involve something more than mere conversion.

-- 485. There are two kinds of conversion--

(1) Simple.

(2) Per Accidens or by Limitation.

-- 486. We are said to have simply converted a proposition when the quant.i.ty remains the same as before.

-- 487. We are said to have converted a proposition per accidens, or by limitation, when the rules for the distribution of terms necessitate a reduction in the original quant.i.ty of the proposition.

-- 488.

A can only be converted per accidens.

E and I can be converted simply.

O cannot be converted at all.

-- 489. The reason why A can only be converted per accidens is that, being affirmative, its predicate is undistributed (-- 293). Since 'All A is B' does not mean more than 'All A is some B,' its proper converse is 'Some B is A.' For, if we endeavoured to elicit the inference, 'All B is A,' we should be distributing the term B in the converse, which was not distributed in the convertend. Hence we should be involved in the fallacy of arguing from the part to the whole. Because 'All doctors are men' it by no means follows that 'All men are doctors.'

-- 499. E and I admit of simple conversion, because the quant.i.ty of the subject and predicate is alike in each, both subject and predicate being distributed in E and undistributed in I.

/ No A is B.

E < .'.="" no="" b="" is="">

/ Some A is B.

I < .'.="" some="" b="" is="">

-- 491. The reason why O cannot be converted at all is that its subject is undistributed and that the proposition is negative. Now, when the proposition is converted, what was the subject becomes the predicate, and, as the proposition must still be negative, the former subject would now be distributed, since every negative proposition distributes its predicate. Hence we should necessarily have a term distributed in the converse which was not distributed in the convertend. From 'Some men are not doctors,' it plainly does not follow that 'Some doctors are not men'; and, generally from 'Some A is not B' it cannot be inferred that 'Some B is not A,' since the proposition 'Some A is not B' admits of the interpretation that B is wholly contained in A.

[Ill.u.s.tration]

-- 492. It may often happen as a matter of fact that in some given matter a proposition of the form 'All B is A' is true simultaneously with 'All A is B.' Thus it is as true to say that 'All equiangular triangles are equilateral' as that 'All equilateral triangles are equiangular.' Nevertheless we are not logically warranted in inferring the one from the other. Each has to be established on its separate evidence.

-- 493. On the theory of the quantified predicate the difference between simple conversion and conversion by limitation disappears. For the quant.i.ty of a proposition is then no longer determined solely by reference to the quant.i.ty of its subject. 'All A is some B' is of no greater quant.i.ty than 'Some B is all A,' if both subject and predicate have an equal claim to be considered.

-- 494. Some propositions occur in ordinary language in which the quant.i.ty of the predicate is determined. This is especially the case when the subject is a singular term. Such propositions admit of conversion by a mere transposition of their subject and predicate, even though they fall under the form of the A proposition, e.g.

Virtue is the condition of happiness.

.'. The condition of happiness is virtue.

And again,

Virtue is a condition of happiness.

.'. A condition of happiness is virtue.

In the one case the quant.i.ty of the predicate is determined by the form of the expression as distributed, in the other as undistributed.

-- 495. Conversion offers a good ill.u.s.tration of the principle on which we have before insisted, namely, that in the ordinary form of proposition the subject is used in extension and the predicate in intension. For when by conversion we change the predicate into the subject, we are often obliged to attach a noun substantive to the predicate, in order that it may be taken in extension, instead of, as before, in intension, e.g.

Some mothers are unkind.

.'. Some unkind persons are mothers.

Again,

Virtue is conducive to happiness.

.'. One of the things which are conducive to happiness is virtue.

CHAPTER V.

_Of Permutation._

-- 496. Permutation [Footnote: Called by some writers Obversion.] is an immediate inference grounded on a change of quality in a proposition and a change of the predicate into its contradictory-term.

-- 497. In less technical language we may say that permutation is expressing negatively what was expressed affirmatively and vice versa.

-- 498. Permutation is equally applicable to all the four forms of proposition.

(A) All A is B.

.'. No A is not-B (E).

(E) No A is B.

.'. All A is not-B (A).

(I) Some A is B.

.'. Some A is not not-B (O).

(O) Some A is not B.

.'. Some A is not-B (I).

-- 499, Or, to take concrete examples--

(A) All men are fallible.

.'. No men are not-fallible (E).

(E) No men are perfect.

.'. All men are not-perfect (A).

(I) Some poets are logical.

.'. Some poets are not not-logical (O).

(O) Some islands are not inhabited.