A demonstrable judgment is self-evident, as in mathematical axioms, e. g.

If equals be added to equals, the wholes are equal.

Things which are equal to the same thing are equal to one another.

Or it may be a deduction from self-evident principles; as,
The circumference of a circle is thrice its diameter.
The angles at the base of an isosceles triangle are equal.

SECTION III.

THE OPPOSITION OF PROPOSITIONS.

Propositions have hitherto been treated in their absolute character, we now proceed to consider them in their relative character or affections. The relative affections of propositions are of three kinds, viz., subalternation, conversion, and opposition. A knowledge of these is subservient to the doctrine of syllogisms; for by subalternation we infer several conclusions from the same premisses; by opposition we reduce imperfect moods to perfect, while conversion aids in both.

1. Subalternation.

Subalternation is the deduction of a particular or singular proposition from a universal,-both, namely, the premiss and deduction, having the same subject and the same predicate. Subalternation is either between A. and I. or E. and O.

The two propositions compared together are the one a universal proposition, called the subalternans; the other, deduced from the former, and called the subalterna,-a particular or singular proposition, with the same subject and the same predicate. In all cases the subalternans and subalterna must agree in quality, e. g.

A. All diseases are contagious.
I. Some diseases are contagious.

The principle on which subalternation proceeds is, that universals include particulars, but that particulars do not include universals.

From any universal proposition we may infer a particular subalterna; but before we can infer a singular subalterna, the subalternans must be metaphysically universal. The following is an example of a proposition metaphysically universal; viz.,

Every effect has a cause;

and as it is metaphysically universal (that is, admitting of no exceptions), we can infer from it not only the particular proposition,

Some effects have causes,

but also the singular proposition—

Gravitation has a cause.

The relation between a subalternans and a subalterna can scarcely be called opposition, in the common acceptation of the term, though it comes within the range of the definition of that term as here used.

In pronouncing on the truth or falsehood of propositions thus related, we have to consider how far we may infer the truth or falsehood of one of them from the truth or falsehood of the other; and in reference to subalternation, we have the four following axioms :—

1. The truth of the universal infers the truth of the particular, for if it be true that

All islands are surrounded by water,

it must be true that

Some islands are surrounded by water;

and it is also true that

Sicily is an island surrounded by water.

In this example, however, we infer a singular subalterna, because the subalternans is metaphysically universal.

A sign of universality prefixed to the subject of a proposition does not, let it be observed, necessarily render that proposition metaphysically true, e. g., the following proposition:All children reverence their parents,

is only morally universal, and therefore admits of some exceptions; for all children do not reverence their parents; and hence, although from such a proposition we can infer with truth that

Some children reverence their parents,

we cannot infer this with regard to any particular individual contained in the subject; for it may be among the exceptions.

The truth of this axiom is equally valid in negative as in affirmative propositions; for if in affirmative propositions the predicate contains the whole extension of the subject, it contains a part; and if in negative propositions it excludes the whole, it excludes a part.

It will be seen from the foregoing, that the subalternans and the subalterna, deduced from it, differ in quantity but agree in quality.

2. The truth of the particular does not infer the truth of the universal.

The predicate of a proposition may agree with a part of the extension of the subject, yet it may not agree with the whole extension of the subject; for although it may be true that Some wars are just,

it is not true that

All wars are just;

and although it may be true that

Some men have not an affectionate disposition,

it is not true that

No men have an affectionate disposition.

In reference to the terms of a proposition, it is to be laid down that in any inference no term can be changed from a particular to an universal; in other words, that an argument a particulari ad universale is invalid; for if any term is particular, a part of it only is said to agree or disagree with the other term. About the remaining part of it, therefore, nothing can be inferred.

3. The falsehood of the particular infers the falsehood of the universal.

In illustration of this axiom, it may be stated that if it be false that the predicate contains or agrees with any part of the extension of the subject, it must also be false that it contains the whole; for instance, if it is false that

Some men are patriotic,

it must be false that

All men are patriotic.

4. The falsehood of the universal does not infer the falsehood of the particular.

It may be false, for instance, that the predicate contains or excludes the whole extension of the subject, yet it may contain or exclude a part. Thus it may be false that

All men are logicians,

or that

No island is fertile ;

yet it may be true that

Some men are logicians,

or that

Some islands are not fertile.

It must be observed, in relation to the four axioms, that we

are to understand by the universal and particular propositions mentioned, a universal and a particular deduced from it by subalternation.

On examining the four foregoing axioms, it will appear that they may be reduced to two, for the third is contained in the first, and the fourth in the second.

The third and fourth axioms differ from the first and second, in being their converses by contraposition.

It may be mentioned that Aristotle takes no notice of subaltern propositions. Its laws were first laid down by Apuleius, and the name given by Boëthius.

2. Contrary Opposition.

Contrary Opposition is between two universal propositions differing in quality only, viz., between A. and E.; as,

All trees possess vegetable life.

No trees possess vegetable life.

With regard to the truth of contrary propositions, both of them may be false, but they cannot both be true, or the one may be false and the other true.

When the opposed propositions are in necessary matter, the affirmative will be true and the negative false; e. g.—

A. All human institutions are imperfect.
E. No human institutions are imperfect.
A. All bad habits should be avoided.
E. No bad habits should be avoided.

When the opposed propositions are in impossible matter, the negative will be true and the affirmative false; e. g.—

E. None of the planets are stationary.
A. All the planets are stationary.

E. No island is under water.

A. All islands are under water.