_Of Conversion._
§ 479. Conversion is an immediate inference grounded On the
transposition of the subject and predicate of a proposition.
§ 480. In this form of inference the antecedent is technically known
as the Convertend, i.e. the proposition to be converted, and the
consequent as the Converse, i.e. the proposition which has been
converted.
§ 481. In a loose sense of the term we may be said to have converted a
proposition when we have merely transposed the subject and predicate,
when, for instance, we turn the proposition 'All A is B' into 'All B
is A' or 'Some A is not B' into 'Some B is not A.' But these
propositions plainly do not follow from the former ones, and it is
only with conversion as a form of inference--with Illative Conversion
as it is called--that Logic is concerned.
§ 482. For conversion as a form of inference two rules have been laid
down--
(1) No term must be distributed in the converse which was not
distributed in the convertend.
(2) The quality of the converse must be the same as that of the
convertend.
§ 483. The first of these rules is founded on the nature of things. A
violation of it involves the fallacy of arguing from part of a term to
the whole.
§ 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
quantity 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 quantity 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 quantity 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 A.
/ Some A is B.
I <
\ .'. Some B is A.
§ 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.
[Illustration]
§ 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 quantity of a proposition is then no longer determined solely by
reference to the quantity of its subject. 'All A is some B' is of no
greater quantity 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
quantity 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 quantity of the predicate is determined by the
form of the expression as distributed, in the other as undistributed.
§ 495. Conversion offers a good illustration 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.