PART III.

OF ILLATION.

WE restrict the term Illation to that kind of inference which is direct and immediate, and we confine the term Syllogism to that kind of inference which is indirect and mediate. When, from the consideration of one or more propositions, involving only two terms, we are enabled to arrive at a judgment not formally identical with the judgments expressed in such proposition or propositions, the process of arriving at the second judgment is called immediate inference or Illation.

Illation by Opposition.

The simplest instance of this, (excluding Conversion, which, when the predicate is quantified, is no inference at all), are the inferences depending on logical Opposition. Two propositions are said to be in opposition, when the terms of each are the same while the one proposition is affirmative and the other negative. Accordingly it follows that when any one of the following propositions, viz.:

1. All-X is All-Y

2. All-X is Some-Y

Affirmative,

3. Some-X is All-Y

4. Some-X is Some-Y

is asserted simultaneously with any one of the following, viz.:

5. Some-X is not Some-Y

6. Some-X is not Any-Y

7. Some-Y is not Any-X

Negative,

8. Any-X is not Any-Y

the two are in opposition.

The following are all the pairs that can be formed by taking one proposition from the affirmatives, and one from the negatives.

1+8. 1+7. 1+6. 1+5. 2+8. 2+7. 2+6. 2+5. 3+8. 3+7. 3+6. 3+5. 4+8. 4+7. 4+6. 4+5, sixteen in all.

Those pairs that exclusively contradict each other are (as we have already seen) called Contradictories, the one proposition denying exactly that which the other affirms. In Contradictories moreover, there are always two terms indefinite, and two definite. Those propositions in any pair which are inconsistent with each other, without necessarily excluding each other, are called Contraries. Those propositions in any pair which are consistent with each other, but whose Contradictories are Contraries, are called Subcontraries.

Those propositions in any pair, which may be simultaneously either inconsistent or consistent, are called Subalterns. The remaining combinations of propositions, two and two, viz. those in which there is no difference of quality, both propositions being affirmative or both negative, are not cases of opposition at all. They will be treated of under another head. (Illations by Fluxion.)

The cases of opposition above enumerated, viz. four Contradictories, five Contraries, five Subcontraries, and two Subalterns, (that is, sixteen in all,) appear in detail in the following tables. In all such pairs, Some of course means Some at least, perhaps All. However, in investigating the conditions of their solution, we replace every 'Some' by Certain.

I. Exclusives (or Contradictories).

(All-X is All-Y
Some-X is not Some-Y

(All-X is Some-Y
Some-X is not Any-Y
Some-X is All-Y
Some-Y is not Any-X

(Some-X is Some-Y
Any-X is not Any-Y

In each pair, the propositions cannot both be true at once, or both false at once. One of the two must be true, and the other false.

II. Inconsistents (or Contraries).

III. Consistents (or Subcontraries).

IV. Subalterns.

(All-X is Some-Y Some-Y is not Any-X

(Some-X is All-Y Some-X is not Any-Y

In each pair, both propositions may be true at once; or both be false at once.

may

These tables may be greatly simplified. It is observable that in the table of Contraries, the propositions of each pair are the Contradictories of those in a corresponding pair in the table of Subcontraries. Thus:

Similarly, the propositions in either of the two pairs of Subalterns, are the Contradictories of those,

each to each, in the other. Thus:

All-X is Some-Y.

Some-X is not Any-Y.

Some-Y is not Any-X.

Some-X is All-Y.