Deductive Logic Part 19

.'. Some islands are not-inhabited (I).

-- 500. The validity of permutation rests on the principle of excluded middle, namely--That one or other of a pair of contradictory terms must be applicable to a given subject, so that, when one may be predicated affirmatively, the other may be predicated negatively, and vice versa (-- 31).

-- 501. Merely to alter the quality of a proposition would of course affect its meaning; but when the predicate is at the same time changed into its contradictory term, the original meaning of the proposition is retained, whilst the form alone is altered. Hence we may lay down the following practical rule for permutation--

Change the quality of the proposition and change the predicate into its contradictory term.

-- 502. The law of excluded middle holds only with regard to contradictories. It is not true of a pair of positive and privative terms, that one or other of them must be applicable to any given subject. For the subject may happen to fall wholly outside the sphere to which such a pair of terms is limited. But since the fact of a term being applied is a sufficient indication of its applicability, and since within a given sphere positive and privative terms are as mutually destructive as contradictories, we may in all cases subst.i.tute the privative for the negative term in immediate inference by permutation, which will bring the inferred proposition more into conformity with the ordinary usage of language. Thus the concrete instances given above will appear as follows--

(A) All men are fallible.

.'. No men are infallible (E).

(E) No men are perfect.

.'. All men are imperfect (A).

(I) Some poets are logical.

.'. Some poets are not illogical (O).

(O) Some islands are not inhabited.

.'. Some islands are uninhabited (I).

CHAPTER VI.

_Of Compound Forms of Immediate Inference._

-- 503. Having now treated of the three simple forms of immediate inference, we go on to speak of the compound forms, and first of

_Conversion by Negation._

-- 504. When A and O have been permuted, they become respectively E and I, and, in this form, admit of simple conversion. We have here two steps of inference: but the process may be performed at a single stroke, and is then known as Conversion by Negation. Thus from 'All A is B' we may infer 'No not-B is A,' and again from 'Some A is not B'

we may infer 'Some not-B is A.' The nature of these inferences will be seen better in concrete examples.

-- 505.

(A) All poets are imaginative.

.'. No unimaginative persons are poets (E).

(O) Some parsons are not clerical.

.'. Some unclerical persons are parsons (I).

-- 506. The above inferences, when a.n.a.lysed, will be found to resolve themselves into two steps, namely,

(1) Permutation.

(2) Simple Conversion.

(A) All A is B.

.'. No A is not-B (by permutation).

.'. No not-B is A (by simple conversion).

(O) Some A is not B.

.'. Some A is not-B (by permutation).

.'. Some not-B is A (by simple conversion).

-- 507. The term conversion by negation has been arbitrarily limited to the exact inferential procedure of permutation followed by simple conversion. Hence it necessarily applies only to A and 0 propositions, since these when permuted become E and 1, which admit of simple conversion; whereas E and 1 themselves are permuted into A and 0, which do not. There seems to be no good reason, however, why the term 'conversion by negation' should be thus restricted in its meaning; instead of being extended to the combination of permutation with conversion, no matter in what order the two processes may be performed. If this is not done, inferences quite as legitimate as those which pa.s.s under the t.i.tle of conversion by negation are left without a name.

-- 508. From E and 1 inferences may be elicited as follows--

(E) No A is B.

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

(I) Some A is B.

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

(E) No good actions are unbecoming.

.'. All unbecoming actions are not-good (A).

(I) Some poetical persons are logicians.

.'. Some logicians are not unpoetical (O).

Or, taking a privative term for our subject,

Some unpractical persons are statesmen.

.'. Some statesmen are not practical.

-- 509. When the inferences just given are a.n.a.lysed, it will be found that the process of simple conversion precedes that of permutation.

-- 510. In the case of the E proposition a compound inference can be drawn even in the original order of the processes,

No A is B.

.'. Some not-B is A.

No one who employs bribery is honest.

.'. Some dishonest men employ bribery.

The inference here, it must be remembered, does not refer to matter of fact, but means that one of the possible forms of dishonesty among men is that of employing bribery.

-- 511. If we a.n.a.lyse the preceding, we find that the second step is conversion by limitation.

No A is B.

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

.'. Some not-B is A (by conversion per accidens).