CHAPTER V.

CONVERSION OF PROPOSITIONS, AND IMMEDIATE INFERENCE.

1. THE student is referred to the Elementary Lessons in Logic, or to other elementary text-books, for the common rules of conversion and immediate inference, but, for the sake of easy reference, the ancient square of opposition is given below.

All the relations of propositions and the methods of inference applying to a single proposition will be found fully exemplified and described in the following questions and

answers.

2. It appears to be indispensable, however, to endeavour to introduce some fixed nomenclature for the relations of propositions involving two terms. Professor Alexander Bain has already made an innovation by using the name obverse, and Professor Hirst, Professor Henrici and other reformers of the teaching of geometry have begun to use the terms converse and obverse in meanings inconsistent with those attached to them in logical science (Mind, 1876, p. 147). It seems needful, therefore, to state in the most explicit way the nomenclature here proposed to be adopted with the concurrence of Professor Robertson.

Taking as the original proposition all A are B,' the following are what we may call the related propositions—

Reciprocal. All not A are not B.

It must be observed that the converse, obverse, and contrapositive are all true if the original proposition is true. The same is not necessarily the case with the inverse and reciprocal. These latter two names are adopted from the excellent work of Delbœuf, Prolegomènes Philosophiques de la Géométrie, pp. 88-91, at the suggestion of Professor Croom Robertson. (Mind, 1876, p. 425.)

QUESTIONS AND ANSWERS.

3. Give all the logical opposites of the proposition, 'All metals are conductors.'

This is a universal affirmative proposition, having the symbol A. By its logical opposites we mean the corresponding propositions in the forms E, I, and O, which have the same subject and predicate, and are related to it respectively as its contrary, contradictory, and subaltern, in the way shown in the Logical Square (p. 31) and explained In many Manuals. These opposite propositions may be thus stated

Subaltern

(I)-Some metals are conductors. Contradictory (0)-Some metals are not conductors. Contrary (E)-No metals are conductors.

The first of these (I) may be inferred from the original; the other two (O and E), so far from being inferrible, are inconsistent with its truth.

4. Given that a particular negative proposition is true, is the following chain of inferences correct? O is true, A is false, I is false, and therefore E is true. If so, the truth of O involves the truth of E.

There is a false step in this argument; for the falsity of A does not involve the falsity of I. It may be (and is

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materially false) that 'all men are dishonest;' but it nevertheless may remain true that some men are dishonest.' Observe, then, that the falsity of A does not involve the truth of I, nor does the truth of I involve the truth or falsity of A. But the truth of A necessitates that of I. As stated in the Elementary Lessons (p. 78), "Of subalterns, the particular is true if the universal be true: but the universal may or may not be true when the particular is true."

5. How do you convert universal affirmative propositions?

They must be converted by limitation or per accidens, as it is called, that is to say, while preserving the affirmative quality, the quantity of the proposition must be limited from universal to particular. Thus A is converted into I, as in the following more or less troublesome instances, the Convertend standing first and the Converse second in each pair of propositions :

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All organic substances contain carbon.

Some substances containing carbon are organic.
Time for no man bides.

Something biding for no man is time.

The poor have few friends.

Some who have few friends are poor.

A wise man maketh more opportunities than he finds. Some who make more opportunities than they find are wise men.

They are ill discoverers who think there is no land, when they can see nothing but water.

Some ill discoverers think there is no land, &c.

Great is Diana of the Ephesians.

Some great being is Diana of the Ephesians.

Warm-blooded animals are without exception air-breathers. Air-breathers are (with or without exception) warmblooded animals.

6. How would you convert 'Brutus killed Cæsar?'

The strictly logical converse is 'Some one who killed Cæsar was Brutus.' For, though a man can only be killed once, and Brutus is distinctly said to be the killer, yet in formal logic we know nothing of the matter, and Cæsar might have been killed on other occasions by other persons. An absurd illustration is purposely chosen in the hope that it may assist to fix in the memory the all-important truth that in logic we deal not with the matter.

7. How do you convert particular affirmative propositions?

To this kind of proposition simple conversion can be applied; that is to say, the converse will preserve both the quantity and the quality of the convertend. In other words, I when converted gives another proposition in I; thus either of the following pairs is the simple converse of the other :

Some dogs are ferocious animals.

Some ferocious animals are dogs.

Some men have not courage to appear as good as they are. Some, who have not courage to appear as good as they are, are men,

Some animals are amphibious.

Some amphibious beings are animals.