Then its two apagogical modifications are the second and third figures.
Fig. 2
Fig. 3
It is customary to enumerate six moods of the third figure instead of four, and the moods Darapti and Felapton appear to be omitted. But a particular proposition is asserted (actually and not merely virtually) by the universal proposition which does not otherwise differ from it; and therefore Darapti is included both under Disamis and Datisi, and Felapton both under Bocardo and Ferison. (De Morgan.)
The second figure, from the assertion of the rule and the denial of the result, infers the denial of the case; the third figure, from the denial of the result and assertion of the case, infers the denial of the rule. Hence we write the moods as follows, by allowing inferences only on the straight lines:—
Fig. 1
Fig. 2
Fig. 3
The symmetry of the system of moods of the three figures is also exhibited in the following table.
Enter at the top the proposition asserting or denying the rule; enter at the side the proposition asserting or denying the case; find in the body of the table the proposition asserting or denying the result. In the body of the table, propositions indicated by italics belong to the first figure, those by black-letter to the second figure, and those by script to the third figure.
If, as the denial of the result in the second and third figures, we put the form “Any N is N,” we have—
Fig. 2
Fig. 3
These are the formulæ of the two simple conversions. Neither can be expressed syllogistically except in the figures in which they are here put (or in what is called the fourth figure, which we shall consider hereafter). If, for the denial of the result in the second figure, we put “No not-N is N” (where “not-N” has not as yet been defined4) we obtain
In the same way, if we put “Some N is some-N” (where some-N has not been defined5) for the denial of the result in the third figure, we have
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
are reduced to
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
are reduced to the following:—
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:—
Baroco and its long reduction are
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 S is 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