And from the conclusion of the reduction, the conclusion of Ferison may be obtained by a substitution whose possibility is expressed syllogistically thus:—
Bocardo and its long reduction are
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,
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;
a3, contraposition of A into I;
o2, the substitution of “Some S is not M” for “Any M is not some-S”;
o3, the substitution of “Some M is not P” for “Some not-P is M”;
e”, introduction of not-P by definition;
i”, introduction of some-S by definition.
With the exception of the substitutions i” and e”, which will be considered hereafter, all those which are used in the reduction of the moods of either oblique figure have the form of inferences in the same figure.
The so-called reductio per impossibile is the repetition or inversion of that contraposition of propositions by which the indirect figures have been obtained. Now, contradiction arises from a difference both in quantity and quality; but it is to be observed that, in the contraposition which gives the second figure, a change of the quality alone, and in that which gives the third figure, a change of the quantity alone, of the contraposed propositions, is sufficient. This shows that the two contrapositions are of essentially different kinds, and that the reductions per impossibile of the second and third figures respectively involve the following formal inferences.6
FIGURE 2
The Result follows from the Case;
∴ The Negative of the Case follows from the Negative of the Result.
FIGURE 3
The Result follows from the Rule;
∴ The Rule changed in Quantity follows from the Result changed in Quantity.
But these inferences may also be expressed as follows:—
FIGURE 2
FIGURE 3
Now, the limitations in parentheses do not affect the essential nature of the inferences; and omitting them we have,
FIGURE 2
FIGURE 3
We have already seen that the former of these is of the form of the second figure, and the latter of the form of the third figure of syllogism.
Hence it appears that no syllogism of an indirect figure can be reduced to the first figure without a substitution which has the form of the very figure from which the syllogism is reduced. In other words, the indirect syllogisms are of an essentially different form from that of the first figure, although in a more general sense they come under that form.
§6. The Theophrastean Moods
It is now necessary to consider the five moods of Theophrastus, viz. Baralipton, Celantes, Dabitis, Fapesmo, Frisesomorum. Baralipton is included in Dabitis, and Fapesmo in Frisesomorum, in the same way in which Darapti is included in Disamis and Datisi, and Felapton in Bocardo and Ferison. The Theophrastean moods are thus reduced to three, viz.:—
Suppose we have, 1st, a Rule; 2d, a Case under that rule, which is itself a Rule; and, 3d, a Case under this second rule, which conflicts with the first rule. Then it would be easy to prove that these three propositions must be of the form,
These three propositions cannot all be true at once; if, then, any two are asserted, the third must be denied, which is what is done in the three Theophrastean moods.
These moods are resolved into one another by the contraposition of propositions, and therefore should be considered as belonging to different figures.
They can be ostensively reduced to the first Aristotelian figure in two ways; thus,
The verses of Shyreswood show how Celantes and Dabitis are to be reduced in the short way, and Frisesomorum in the