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The conversion of a proposition is the changing the relative positions of the subject and predicate, so that the predicate may become the subject, and the subject the predicate of the new proposition: and propositions thus related, namely, having the subject of one the predicate of the other, and vice versa, are called converse propositions, thus, "Every equiangular triangle is equilateral," and "Every equilateral triangle is equiangular," are converse propositions. When the truth of the converse is implied by that of the original proposition, the conversion is said to be illative, but this is only the case under certain circumstances, namely, when the subject and predicate are precisely similar in quantity, or, in other words, are equally distributed, being either both universal or both particular. The following table shows the distribution of the subject and predicate in each kind of proposition:

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From this table it will be seen that in the universal negative proposition, and in the particular affirmative, both the subject and predicate are similar in quantity, in the former both being universal or distributed, and in the latter both particular or not distributed; both these forms of proposition may therefore be illatively converted, that is to say, if the truth of a universal negative or of a particular affirmative proposition be admitted, the truth of its converse is implied and cannot be denied. In regard, however, to the other two sorts of propositions, namely, the universal affirmative and the particular negative, since the subject and predicate are not equally distributed, they are not illatively convertible, that is to say, the truth of the converse is not implied by the truth of the original proposition, and therefore cannot be inferred from it, but must be the subject of separate proof. Thus, in the Elements, the truth of the converse of the universal affirmative proposition, "Every equilateral triangle is equiangular," cannot be inferred, but is made the subject of separate proof in the next proposition.

In the conversion of a universal negative or a particular affirmative proposition, both the quantity and quality of the proposition remain unaltered; thus, the converse of E gives E, and the converse of I gives I; and this kind of conversion is called simple

conversion. Although a universal affirmative proposition cannot be illatively converted by simple conversion, it may be by a process which is termed conversion per accidens, namely, by diminishing its quantity (its quality remaining unaltered) so that its converse may be only a particular affirmative proposition; for instance, the universal affirmative proposition, "Every X is 1," may be illatively converted, per accidens, into the particular affirmative, "Some YS ARE IS," the truth of this latter being positively implied by that of the former proposition.

Having thus briefly examined the nature of propositions, we must pass on to the second division of the subject, namely, the mode or form of reasoning to be adopted, in order that the truth of the conclusion may be as certain as that of the propositions employed.

Now there is but one regular form in which an argument can be logically stated, and that form is called a syllogism. A syllogism consists of three propositions so related that the truth of the last (termed the conclusion) is manifestly established, by admitting the truth of the two first (termed the premises).

In the proposition which is the conclusion of the syllogism, the predicate is called the major term and the subject the minor term, and these two are termed the extremes. In one of the other propositions or premises the major term is compared with some other term called the middle term, and this proposition is called the major premiss; in the other proposition the minor term is compared with the same middle term, and this is called the minor premiss. Now in both the premises the middle term may be either the subject or predicate, and hence arises what is termed the difference in the figure of the syllogism.

In the first figure the middle term is the subject of the major premiss and the predicate of the minor, for example* :

[Major Premiss] Every (plane figure) Is bounded by lines. Minor Premiss] Every triangle Is a (plane figure),

therefore

[Conclusion] Every triangle is bounded by lines.

In the second figure the middle term is the predicate of both the premises, as in the following example:

[Major Premiss] No circle Is (bounded by straight lines). Minor Premiss] Every square Is (bounded by straight lines),

therefore

[Conclusion] No square is a circle.

In the third figure the middle term is the subject in both premises, as in the syllogism

* In all these examples the middle term is enclosed in parentheses.

[Major Premiss] Every (square) is a parallelogram.
[Minor Premiss] Every (square) Is an equilateral figure,

therefore

[Conclusion] Some equilateral figures ARE parallelograms.

In the fourth figure the middle term is the predicate of the major premiss and the subject of the minor, as for example,

[Major Premiss] Every triangle is a (plane figure).

[Minor Premiss] Every (plane figure) is bounded by lines, therefore

[Conclusion] Some figures bounded by lines ARE triangles.

A syllogism consisting of three propositions, and there being four different kinds of propositions, it may be shown by the laws of combination that 64 different syllogisms (or Moods, as they are termed), may be formed with the same terms; thus, the major premiss may be either A, E, I, or O, and each of these may have any one again for its minor premiss, which gives 16 varieties, and each of these again may have any one of the four for its conclusion, giving in all 64 combinations.

There are, however, certain general rules or axioms relative to syllogisms, which exclude 53 of these combinations, and leave only 11 moods of the syllogism which can be legitimately employed in argument. And there are further special rules applying only to certain figures which exclude even some of these moods from being used in those figures, so that there are really only 19 legitimate modes of the syllogism. The axioms relating to syllogisms are as follows:

1. If two terms agree with one and the same third, they agree with each other.

2. If of two terms, one agrees and the other disagrees with one and the same third, those two terms disagree with each other.

3. If neither of two terms agree with the third, those two terms may either agree or disagree with each other.

From these axioms six general rules are deduced; we shall here only state the rules, since want of space will prevent our entering into their proof.

1. The middle term must not be taken twice particularly.

2. The extremes must not be taken more universally in the conclusion than in the premises.

3. From two negative premises no conclusion can be drawn.

4. A negative conclusion cannot follow from two affirmative pre

mises.

5. If one of the premises be negative, the conclusion must be negative; and if one of the premises be particular, the conclusion must be particular.

6. From two particular premises no conclusion can be drawn.

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