These are the two ways of contraposing the Universal Affirmative. There are two ostensive reductions of each mood of the second and third figures. I shall distinguish them as the short reduction and the long reduction. The short reduction is effected by converting or contraposing that premise which is not the denial of the result. The long reduction is effected by transposing the premises, contraposing or converting the denial of the result, and contraposing or converting the conclusion. The alteration thus produced in the order of the terms is shown in the following figure:

The names bestowed by Shyreswood, or Petrus Hispanus, upon the moods indicate the possibility of the short reduction in the case of Cesare and Festino of the second figure, and of Datisi and Ferison of the third figure; also the possibility of the long reduction of Camestres of the second figure and of Disamis of the third.

The short reduction of Camestres and Baroco is effected by introducing the term not-P, and defining it as that which S is when it is not P. Hence for the second premise (Any or some S is not P) we substitute Any or some S is not-P"; and as the first premise, Any M is P, gives by contraposition Any not-P is not M, the

moods

The short reduction of Disamis and Bocardo is effected by introducing the term some-S, defining it as that part of S which is or is not P when some S is or is not P. We can therefore substitute for the first premise, Some S is or is not P, All some-S is or is not P; while, for the second premise, All S is M, can be contraposed into "Some M is some-S": and thus the forms

To reduce Cesare, Festino, and Baroco in the long way, it is necessary to introduce the terms not-P and some-S. Not-P is defined as that class to which any M belongs which is not P. Hence for the first premise of Cesare and Festino we can substitute "Any M is not-P." Some-S is defined as that class of S which is (or is not) P, when some S is (or is not) P. Hence for the second premises of Festino and Baroco we can first substitute "Any some-S is (or is not) P"; and then, by contraposition or conversion, we obtain “Any P (or not-P) is not some-S." Then, by the transposition of the premises, we obtain from Cesare, which is

And from the conclusion of this reduced form we obtain the conclusion of Cesare by simple conversion. So Festino and its long reduction are

and the conclusion of Festino is obtained from that of the reduced form by a substitution which may be made syllogistically thus:

and the conclusion of Baroco is obtained from the conclusion of the reduction in the same way as that of Festino.

In order to reduce Datisi, Bocardo, and Ferison in the long way, we must define Some-S as that S which is M when some S is M, and Not-P as that which some (or any) S is when it is not P. Hence for "Some Sis M" we can substitute "Any some-S is M"; and for "Some (or any) S is not P," "Some (or any) S is not-P." "Some S is not-P" may be converted simply; and "Any S is not-P" may be contraposed so as to become "Some not-P is some-S." Then Datisi and its long reduction are

And from the conclusion of the reduction, the conclusion of Datisi is

obtained by simple conversion. Ferison and its long reduction are

And from the conclusion of the reduction, the conclusion of Ferison may be obtained by a substitution whose possibility is expressed syllogistically thus:

And the conclusion of Bocardo is obtained from that of its reduction in the same way as the conclusion of Ferison.

The ostensive reduction of the indirect or apagogical figures may be considered as the exhibition of them under the general form of syllogism,

S is M; M is P:

.. S is P.

But, in this sense, it is not truly a reduction if the substitutions made in the process are inferences. But although the possibility of the conversions and contrapositions can be expressed syllogistically, yet this can be done only by taking as one of the premises,

Now, these are properly not premises, for they express no facts; they are merely forms of words without meaning. Hence, as no complete argument has less than two premises, the conversions and contrapositions are not inferences. The only other substitutions which have been made have been of not-P and some-S for their definitions. These also can be put into syllogistic form; but a mere modification of language is not an inference. Hence no inferences have been employed in reducing the arguments of the second and third figures to such forms that they are readily perceived to come under the general form of syllogism.

There is, however, an intention in which these substitutions are inferential. For, although the passage from holding for true a fact expressed in the form "No A is B," to holding its converse, is not an inference, because, these facts being identical, the relation between them is not a fact; yet the passage from one of these forms taken merely as having some meaning, but not this or that meaning, to another, since these forms are not identical and their logical relation is a fact, is an inference. This distinction may be expressed by saying that they are not inferences, but substitutions having the form of inferences.

Thus the reduction of the second and third figures, considered as mere forms, is inferential; but when we consider only what is meant by any particular argument in an indirect figure, the reduction is a mere change of wording.

The substitutions made use of in the ostensive reductions are shown in the following table. Where

e, denotes simple conversion of E;

i, denotes simple conversion of I;

a2, contraposition of A into E;

a, contraposition A into I;

02, the substitution of "Some S is not M" for " Any Mis not some-S"; 03, the substitution of "Some Mis not P" for "Some not-P is M";

e", introduction of not-P by definition;

i", introduction of some-S by definition.