Deductive Logic Part 20

-- 512. From A again an inference can be drawn in the reverse order of conversion per accidens followed by permutation--

All A is B.

.'. Some B is not not-A.

All ingenuous persons are agreeable.

.'. Some agreeable persons are not disingenuous.

-- 513. The intermediate link between the above two propositions is the converse per accidens of the first--'Some B is A.' This inference, however, coincides with that from 1 (-- 508), as the similar inference from E (-- 510) coincides with that from 0 (-- 506).

-- 514. All these inferences agree in the essential feature of combining permutation with conversion, and should therefore be cla.s.sed under a common name.

-- 515. Adopting then this slight extension of the term, we define conversion by negation as--A form of conversion in which the converse differs in quality from the convertend, and has the contradictory of one of the original terms.

-- 516. A still more complex form of immediate inference is known as

_Conversion by Contraposition._

This mode of inference a.s.sumes the following form--

All A is B.

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

All human beings are fallible.

.'. All infallible beings are not-human.

-- 517. This will be found to resolve itself on a.n.a.lysis into three steps of inference in the following order--

(1) Permutation.

(2) Simple Conversion.

(3) Permutation.

-- 518. Let us verify this statement by performing the three steps.

All A is B.

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

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

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

All Englishmen are Aryans.

.'. No Englishmen are non-Aryans.

.'. No non-Aryans are Englishmen.

.'. All non-Aryans are non-Englishmen.

-- 519. Conversion by contraposition may be complicated in appearance by the occurrence of a negative term in the subject or predicate or both, e.g.

All not-A is B.

.'. All not-B is A.

Again,

All A is not-B.

.'. All B is not-A.

Lastly,

All not-A is not-B.

.'. All B is A.

-- 520. The following practical rule will be found of use for the right performing of the process--

Transpose the subject and predicate, and subst.i.tute for each its contradictory term.

-- 521. As concrete ill.u.s.trations of the above forms of inference we may take the following--

All the men on this board that are not white are red.

.'. All the men On this board that are not red are white.

Again,

All compulsory labour is inefficient.

.'. All efficient labour is free (=non-compulsory).

Lastly,

All inexpedient acts are unjust.

.'. All just acts are expedient.

-- 522. Conversion by contraposition may be said to rest on the following principle--

If one cla.s.s be wholly contained in another, whatever is external to the containing cla.s.s is external also to the cla.s.s contained.

[Ill.u.s.tration]

-- 523. The same principle may be expressed intensively as follows:--

If an attribute belongs to the whole of a subject, whatever fails to exhibit that attribute does not come under the subject.

-- 524. This statement contemplates conversion by contraposition only in reference to the A proposition, to which the process has. .h.i.therto been confined. Logicians seem to have overlooked the fact that conversion by contraposition is as applicable to the O as to the A proposition, though, when expressed in symbols, it presents a more clumsy appearance.

Some A is not B.

.'. Some not-B is not not-A.

Some wholesome things are not pleasant.