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7. Great is Diana of the Ephesians.

= Diana of the Ephesians is a great goddess.

5. Few men succeed in life.

= The number of men who succeed in life is-not a large

number.*

6. All my guesses but two were correct.

= The number of guesses which I correctly made is

the total number diminished by two.

8. An honest man's the noblest work of God.

Ε.

A.

A.

= Honest manhood is the noblest work of God.

A.

9. He envies others' wealth who has none himself.

= All non-wealthy persons are persons envious of

others' wealth.

A.

10. Only doctors understand this subject.

= All non-doctors are-not persons able to under

stand this subject.

11. The more, the merrier.

= All new-comers are mirth-increasers.

E.

A.

14. A little knowledge is a dangerous thing.

15. The Romans conquered the Carthaginians.

12. He has no home but Athens.

= All places outside of Athens are-not homes for him.

13. A few Greeks vanquished the vast army of Darius.

= A mere handful of Greeks is the force that van

quished the vast army of Darius.

= A smattering of knowledge is a dangerous thing.

= The Roman power is a power that conquered the

E.

A.

Carthaginians.

A.

16. The angles of a triangle are equivalent to two right angles.

= The sum of the angles of a triangle is a quantum

equal to two right angles.

A.

17. Two blacks won't make a white.

= All possible combinations of two blacks are-not com

binations producing whiteness.

E.

18. My friend plays golf.

= My friend is a golfer.

A.

19. Not all the gallant efforts of the veterans availed any

thing.

= The gallant fighting of the veterans is-not fighting

that availed anything.

E.

20. Few dogs are not fond of fetching sticks.

(Force Affirmative)

= All dogs not fond of fetching sticks are exceptional

dogs.

A.

* As the original proposition is negative in meaning, the logical form must be sympathetically negative.

21. A few dogs are fond of cats.

= Some dogs are fond of cats.

22. Only a few politicians are statesmen.

(Exponible)

= Some politicians are statesmen and some politicians
are-not statesmen.

23. Only ignorant persons hold such opinions.

Φ.

(Exclusive)

= All non-ignorant persons are-not persons holding

such opinions.

E.

Or

= All persons holding such opinions are ignorant
persons.

24. Some men are not incapable of telling falsehoods.

(Force Affirmative)

= Some persons are persons quite capable of telling
falsehoods.

25. Scotchmen are level-headed.

Is 'level-headed' part of the connotation of
'Scotchman'? Obviously not, unless by
'Scotchman' we mean 'typical Scotch-
man.' It appears to be a problematic pro-
perty. We therefore have as the strict
logical form:

= Some Scotchmen are level-headed persons.

26. To be or not to be, that is the question.

= Whether life is worth living or not is the question

I must answer.

27. Scarcely anyone got through.

= The number of persons who passed is a very small
number.

28. Men are not what they were.

= The manhood of to-day is-not manhood as it used

to be.

29. The side and diagonal of a square are incommensur-
able.

= The ratio of the side of a square to its diagonal is
an incommensurable ratio.

30. The only interested persons are candidates and ex-
aminers.

= All interested persons are either candidates or
examiners.

31. Am I my brother's keeper ?

(Rhetorical question implying negative state-
ment. The logical subject here is 'The
guardianship of my brother.')

= Looking after my brother is-not my business.

A.

I.

I.

A.

A.

E.

A.

A.

E.

of falling.

32. Fain would I climb, but that I fear to fall.

= My wish to climb is a wish checked only by my fear

(N.B.-' But' has the force of 'only.')

A.

Other alternative renderings are :

= All my impulses to climb are impulses inhibited
by the single fear of falling.

A.

Or

= My fear of falling is the only thing which prevents
me from climbing.

A.

33. All the travellers were not provided with pass

ports.

= Some travellers in the company specified are-not
travellers who were provided with passports.

0.

34. All but Noah and his family were drowned.

= All persons of Noah's day who were not of his house-
hold are persons who were drowned.

35. Afflictions are often salutary.

= Some visitations of affliction are salutary experi

ences.

I.

36. All these claims upon my time overpower me.

= This multitude of claims upon my time is a burden
that overwhelms me.

A

CHAPTER XIX.

V. (iv.) THE OPPOSITION OF PROPOSITIONS.

It is customary to say that two propositions are logically opposed when, having the same subject and predicate, they differ in quantity, in quality, or in both quantity and quality. From this point of view the relation between any two of the propositions A, E, I, and O is treated as an Opposition.

The definition, though convenient, is superficial and arbitrary, and it necessitates using the term 'Opposition' in an entirely technical sense, for the relation between 'All men are mortals' and 'Some men are mortals' is, according to the definition given above, an opposition.

A much sounder method is to guide ourselves by principle, and to hold that propositions are in opposition only when they violate the requirement of non-contradiction. Where this requirement is respected, as in the so-called 'subcontraries' and 'subalterns,' and the relation ceases to be one of opposition, we may suitably speak of Subcontrariety and Subalternation, but not of Subcontrary Opposition or of Subaltern Opposition. * The only two forms of genuine opposition between propositions are known as Contradictory Opposition and Contrary Opposition respectively. They are relations between genuine opposites, because in each case we have a pair of propositions juxtaposed which cannot logically be entertained together. If one of them is accepted, the other must be rejected.

We proceed now to the more minute consideration of the four propositions, A, E, I, O, in respect of those relations between them which are customarily known as Oppositions. The relations we have to deal with are the following:

(a) Contradictory Opposition :

A. All S's are P's. ) († O. Some S's are-not P's.
E. All S's are-not P's. )( I. Some S's are P's.

(b) Subcontrariety :

I. Some S's are P's. )( O. Some S's are-not P's. (c) Contrary Opposition :

A. All S's are P's. ) ( E. All S's are-not P's.

(d) Subalternation :

A. All S's are P's. )( I. Some S's are P's.

E. All S's are-not P's. )( O. Some S's are-not P's.

These last two pairs are known as Subalterns. A is usually called the Subalternans of I, and I the subalternate of A. Similarly, E is the subalternans of O, and O the subalternate of E.

The above-named relations may be diagrammatically represented in what is known as the Square of Opposition :

[blocks in formation]

* Mr. Joseph ('An Introduction to Logic,' chap. ix., p. 207, footnote) draws attention to Aristotle's own statement on this point: 'For some are is only verbally opposed to some are not' (Anal. Pri. B., xv. 636, 27). At the same time Mr. Joseph holds that if subcontraries are not opposed, they are anyhow contrasted, and that may justify their continued inclusion' among 'forms of opposition.'

† We adopt the grammatical sign )( as signifying any kind of so-called Opposition.

Contradictory Opposition.

The Rules or Laws of Contradictory Opposition are identical with the formulæ already given (p. 98) for the Principle of Non-Contradiction:

Rule 1. Contradictories cannot both be accepted.
Rule 2. Contradictories cannot both be rejected.

These rules cannot be proved by means of principles more ultimate than themselves. But they admit of the most cogent proof possible, in another sense of the term 'proof.' If the proof of a law lies in its indispensableness, then the Rules of Contradictory Opposition may be proved to the hilt. For if they do not hold good, there is no such thing as consistency, and it becomes unreasonable to think at all. Moreover, the very attempt to deny these laws refutes itself. The statement 'Contradictories can be accepted together' implies that the contradictory of this very statement may itself be accepted. Finally, as we have already seen, the Principle of Non-contradiction being a Law of Thought, to violate it is absolutely impossible. To have succeeded in doing so would be to have thought the unthinkable.

If a proposition is stated in the general form 'S is P,' its contradictory may be stated in the form 'Sis P' or 'Not (S is P).' Some care is needed in the interpretation of this contradiction-formula. It is misinterpreted, for instance, whenever the denial is directed upon some assumption which the proposition 'S is P'takes for granted, instead of being directed upon the proposition itself.

Thus, the denial of the statement that 'Japan's President is revered by his people' cannot be identified with the assertion that 'The ruler of Japan is not its president.' For this simply denies the implicit assumption that Japan is represented by a president; it does not deny the given proposition. So, again, 'The Mikado is the President of the Japanese Republic' is not expressed by asserting that the realm of Japan is not a presidency. This denies, not the proposition itself, but an assumption which the statement has taken for granted.

The denial of 'S is P' must then itself be a statement concerning the relation of S to P, and not the denial of some assumption which the assertion 'S is P' presupposes. But insistence on this should not be carried to the point of asserting that the denial of 'S is P'is simply 'S is-not P,' and that the clumsier form 'Not (S is P)' may therefore give way to the form 'S is-not P,' as the typical form of the denial of 'S is P.'

Those who support this equivalence appear to take the singular proposition as representative of all the rest. The statement that 'Socrates is-not wise' may be taken as the denial of 'Socrates is

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