Deductive Logic Part 21

.'. Some unpleasant things are not unwholesome.

-- 525. The above admits of a.n.a.lysis in exactly the same way as the same process when applied to the A proposition.

Some A is not B.

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

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

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

The result, as in the case of the A proposition, is the converse by negation of the original proposition permuted.

-- 526. Contraposition may also be applied to the E proposition by the use of conversion per accidens in the place of simple conversion. But, owing to the limitation of quant.i.ty thus effected, the result arrived at is the same as in the case of the O proposition. Thus from 'No wholesome things are pleasant' we could draw the same inference as before. Here is the process in symbols, when expanded.

No A is B.

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

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

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

-- 527. In its una.n.a.lysed form conversion by contraposition may be defined generally as--A form of conversion in which both subject and predicate are replaced by their contradictories.

-- 528. Conversion by contraposition differs in several respects from conversion by negation.

(1) In conversion by negation the converse differs in quality from the convertend: whereas in conversion by contraposition the quality of the two is the same.

(2) In conversion by negation we employ the contradictory either of the subject or predicate, but in conversion by contraposition we employ the contradictory of both.

(3) Conversion by negation involves only two steps of immediate inference: conversion by contraposition three.

-- 529. Conversion by contraposition cannot be applied to the ordinary E proposition except by limitation (-- 526).

From 'No A is B' we cannot infer 'No not-B is not-A.' For, if we could, the contradictory of the latter, namely, 'Some not-B is not-A'

would be false. But it is manifest that this is not necessarily false. For when one term is excluded from another, there must be numerous individuals which fall under neither of them, unless it should so happen that one of the terms is the direct contradictory of the other, which is clearly not conveyed by the form of the expression 'No A is B. 'No A is not-A' stands alone among E propositions in admitting of full conversion by contraposition, and the form of that is the same after it as before.

-- 530. Nor can conversion by contraposition be applied at all to I.

[Ill.u.s.tration]

From 'Some A is B' we cannot infer that 'Some not-B is not-A.' For though the proposition holds true as a matter of fact, when A and B are in part mutually exclusive, yet this is not conveyed by the form of the expression. It may so happen that B is wholly contained under A, while A itself contains everything. In this case it will be true that 'No not-B is not-A,' which contradicts the attempted inference. Thus from the proposition 'Some things are substances' it cannot be inferred that 'Some not-substances are not-things,' for in this case the contradictory is true that 'No not-substances are not-things'; and unless an inference is valid in every case, it is not formally valid at all.

-- 531. It should be noticed that in the case of the [nu] proposition immediate inferences are possible by mere contraposition without conversion.

All A is all B.

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

For example, if all the equilateral triangles are all the equiangular, we know at once that all non-equilateral triangles are also non-equiangular.

-- 532. The principle upon which this last kind of inference rests is that when two terms are co-extensive, whatever is excluded from the one is excluded also from the other.

CHAPTER VII.

_Of other Forms of Immediate Inference._

-- 533. Having treated of the main forms of immediate inference, whether simple or compound, we will now close this subject with a brief allusion to some other forms which have been recognised by logicians.

-- 534. Every statement of a relation may furnish us with ail immediate inference in which the same fact is presented from the opposite side. Thus from 'John hit James' we infer 'James was. .h.i.t by John'; from 'd.i.c.k is the grandson of Tom' we infer 'Tom is the grandfather of d.i.c.k'; from 'Bicester is north-east of Oxford' we infer 'Oxford is south-west of Bicester'; from 'So and so visited the Academy the day after he arrived in London' we infer 'So and so arrived in London the day before he visited the Academy'; from 'A is greater than B' we infer 'B is less than A'; and so on without limit. Such inferences as these are material, not formal. No law can be laid down for them except the universal postulate, that

'Whatever is true in one form of words is true in every other form of words which conveys the same meaning.'

-- 535. There is a sort of inference which goes under the t.i.tle of Immediate Inference by Added Determinants, in which from some proposition already made another is inferred, in which the same attribute is attached both to the subject and the predicate, e.g.,

A horse is a quadruped.

.'. A white horse is a white quadruped.

-- 536. Such inferences are very deceptive. The attributes added must be definite qualities, like whiteness, and must in no way involve a comparison. From 'A horse is a quadruped' it may seem at first sight to follow that 'A swift horse is a swift quadruped.' But we need not go far to discover how little formal validity there is about such an inference. From 'A horse is a quadruped' it by no means follows that 'A slow horse is a slow quadruped'; for even a slow horse is swift compared with most quadrupeds. All that really follows here is that 'A slow horse is a quadruped which is slow for a horse.' Similarly, from 'A Bushman is a man' it does not follow that 'A tall Bushman is a tall man,' but only that 'A tall Bushman is a man who is tall for a Bushman'; and so on generally.

-- 537. Very similar to the preceding is the process known as Immediate Inference by Complex Conception, e.g.

A horse is a quadruped.

.'. The head of a horse is the head of a quadruped.

-- 538. This inference, like that by added determinants, from which it differs in name rather than in nature, may be explained on the principle of Subst.i.tution. Starting from the identical proposition, 'The head of a quadruped is the head of a quadruped,' and being given that 'A horse is a quadruped,' so that whatever is true of 'quadruped'

generally we know to be true of 'horse,' we are ent.i.tled to subst.i.tute the narrower for the wider term, and in this manner we arrive at the proposition,

The head of a horse is the head of a quadruped.

-- 539. Such an inference is valid enough, if the same caution be observed as in the case of added determinants, that is, if no difference be allowed to intervene in the relation of the fresh conception to the generic and the specific terms.

CHAPTER VIII.

_Of Mediate Inferences or Syllogisms._

-- 540. A Mediate Inference, or Syllogism, consists of two propositions, which are called the Premisses, and a third proposition known as the Conclusion, which flows from the two conjointly.

-- 541. In every syllogism two terms are compared with one another by means of a third, which is called the Middle Term. In the premisses each of the two terms is compared separately with the middle term; and in the conclusion they are compared with one another.

-- 542. Hence every syllogism consists of three terms, one of which occurs twice in the premisses and does not appear at all in the conclusion. This term is called the Middle Term. The predicate of the conclusion is called the Major Term and its subject the Minor Term.

-- 543. The major and minor terms are called the Extremes, as opposed to the Mean or Middle Term.

-- 544. The premiss in which the major term is compared with the middle is called the Major Premiss.