Cursus Philosophicus Thomisticus, Prima Pars Artis Logicae

The twofold power of syllogisms is commonly assigned, although Aristotle in Prior Analytics 2¹ assigned six; for the present, however, it suffices for us to explain the two principal ones. The first is the power of concluding several things. The second is the power of reducing imperfect syllogisms to perfect ones.

The power of concluding several things is nothing other than this: by the very fact that some consequence or conclusion is concluded, one can also conclude what follows from that consequence; for example, if a universal affirmative or negative is concluded, one can also conclude the particular contained under it. Similarly, one can conclude the converse of one’s conclusion and its equivalent, and universally whatever follows from such a conclusion. All this is presupposed for ostensive reduction, as we shall say at once.

The power of reducing imperfect syllogisms to perfect ones consists in this: those syllogisms in which “being said of all” and “being said of none”—the regulative principles of the syllogism, as we shall say below in ch. 10—do not appear so evidently and perfectly are reduced to, and proved through, those in which these principles are preserved perfectly and evidently. The most perfect modes of concluding are the first four of the first figure, namely Barbara, Celarent, Darii, and Ferio. For all conclusions that can be inferred are either universal affirmatives, universal negatives, particular affirmatives, or particular negatives. The universal affirmative is inferred most perfectly in Barbara, the universal negative in Celarent, the particular affirmative in Darii, and the particular negative in Ferio, as we shall show below in ch. 10.

Therefore all the other modes, whether concluding indirectly or directly, can be reduced to these four and proved through them. Reduction is twofold: one kind is ostensive, the other is by impossibility. OSTENSIVE reduction uses two principles, namely the conversion of a proposition and transposition, that is, changing the major into the minor or conversely. For this reduction shows only that the premises can be arranged in a clearer way, so that the regulation “being said of all” and “being said of none” becomes more evident there. The whole procedure is grounded in the principle stated above: “Whatever follows from the consequent follows from its antecedent.”

REDUCTION BY IMPOSSIBILITY proves its case by deriving two contradictories as simultaneously true, which is impossible. For this purpose it assumes the contradictory of the proposition denied and, together with one proposition that has been granted, infers the contradictory of the other proposition granted, and consequently infers two contradictories as simultaneously true. It is grounded in this principle: “In a good consequence, from the contradictory of the consequent follows the contradictory of the antecedent.”

By this reduction by impossibility all the imperfect modes can be reduced to the perfect ones, because in every useful mode in which the consequence is denied, one must grant the contradictory of what is denied; from that granted contradictory and the other granted premise follows the contradictory of the other. Nevertheless, two modes in particular are said to be reduced by impossibility, namely Baroco and Bocardo, because they cannot be reduced in another way.

We shall give examples of both kinds of reduction. For example, I wish to reduce to the perfect mode Ferio the imperfect mode Fapesmo of the same first figure. I form a syllogism in Fapesmo as follows: “Every body is quantified; no soul is a body; therefore some quantified thing is not a soul.” I reduce it to Ferio by transposing the premises—that is, putting the major in place of the minor and the minor in place of the major—and by converting the affirmative proposition by accident and the negative simply. I thus form the syllogism: “No body is a soul; some quantified thing is a body; therefore some quantified thing is not a soul.” I wish to reduce Darapti, the first mode of the third figure, to Darii. I form the syllogism in Darapti thus: “Every animal is a substance; every animal is sentient; therefore some sentient thing is a substance.” I reduce it to Darii by converting the minor by accident, and say: “Every animal is a substance; some sentient thing is an animal; therefore some sentient thing is a substance.”

Likewise, for reduction by impossibility, I give an example in Baroco, the fourth mode of the second figure, which I reduce by impossibility to Barbara. I form a syllogism in Baroco thus: “Every animal is sentient; some stone is not sentient; therefore some stone is not an animal.” If you deny the consequence, I assume its contradictory and say: therefore the contradictory is true, namely, “Every stone is an animal.” Once this is granted, I take the major of the first syllogism, which had been granted, and this contradictory as the minor, and infer the contradictory of the granted minor. Thus I compel you to grant two contradictories in Barbara as follows: “Every animal is sentient; every stone is an animal; therefore every stone is sentient.” But you had granted that some stone is not sentient; therefore you grant two contradictories.

Likewise, I wish to reduce Bocardo, the fifth mode of the third figure, to Barbara by impossibility. I form a syllogism in Bocardo: “Some virtue is not natural; every virtue is a habit; therefore some habit is not natural.” If, after granting the major and minor, you deny the consequence, I take the contradictory of what was denied and say: therefore the contradictory is true, namely, “Every habit is natural.” I take this as the major while retaining the granted minor, and form a syllogism in Barbara that infers the contradictory of the major thus: “Every habit is natural; every virtue is a habit; therefore every virtue is natural”—which is the contradictory of the major, “Some virtue is not natural.”

To help you remember easily which modes are reduced to which others, and how, observe the device by which the verses given above—Barbara, Celarent, and so on—were formed. The first four begin with the initial consonants B, C, D, F; and all the other modes, whether in the first, second, or third figure, begin with one of these letters to signify that each mode is reduced ostensively to the mode beginning with the same letter. Thus Baralipton is reduced ostensively to Barbara. But Bocardo and Baroco are not reduced ostensively to Barbara, but by impossibility, because one must assume the contradictory of the conclusion, which is a universal affirmative, in order to put it into Barbara. This is reduction by impossibility, namely, by assuming the contradictory of the conclusion. Celantes, Cesare, and Camestres are reduced ostensively to Celarent. Dabitis, Darapti, Disamis, and Datisi are reduced ostensively to Darii. Fapesmo, Frisesomorum, Festino, Felapton, and Ferison are reduced to Ferio. Thus, from the letter with which each expression begins, one must reduce ostensively to the corresponding expression beginning with the same letter in one of the first four modes.

You will also find other consonants in these expressions, deliberately placed because they indicate either a conversion of a proposition to be made or a transposition of the premises. These are the three letters P, S, and M. Wherever you find the letter S, it signifies that the proposition is to be converted simply; wherever you find P, that it is to be converted by accident; wherever you find M, that the premises are to be transposed. For example, in Fapesmo you will find the letter P after the first A: it signifies that the universal affirmative is to be converted by accident. After the E you will find the letter S, signifying that the universal negative is to be converted simply. The letter M signifies that the premises are to be transposed, namely, the major into the minor. And so it is reduced ostensively to Ferio. You can readily observe the same in the other cases. When you find the letter C in the middle of an expression, it signifies that the mode cannot be reduced ostensively, but only by impossibility; this occurs only in Baroco and Bocardo.

In the ostensive conversion of Baralipton in particular, note that it yields an indirect particular affirmative, and that it has the letter P not beside either premise but beside the conclusion. This indicates that it is reduced to Barbara by retaining the very same premises of Baralipton and immediately and directly inferring the conclusion of Barbara, which is a universal affirmative. Since this universal affirmative is converted by accident through a good consequence, whoever immediately infers it in Barbara can also infer, by adding conversion by accident, the indirect particular affirmative that was the conclusion in Baralipton. Thus Baralipton is not reduced by converting or transposing it into Barbara, but rather Baralipton is derived from Barbara by conversion by accident of its conclusion, which is a universal affirmative. For example, this syllogism is in Baralipton: “Every animal is a substance; every human being is an animal; therefore some substance is a human being.” I reduce it to Barbara by proving that such a conclusion is inferred mediately from Barbara, as follows: “Every animal is a substance; every human being is an animal; therefore every human being is a substance.” And by conversion by accident: “Therefore some substance is a human being,” which was the conclusion of Baralipton.

As for the reduction of syllogisms by impossibility, it is certain that all the imperfect modes, whether of the first, second, or third figure, can be reduced by impossibility to one of the first four. But they are not all reduced in the same way as in ostensive reduction, where each expression is reduced to a mode among the first four that begins with the same letter—except Baroco and Bocardo, which are reduced to Barbara by taking the contradictory of the inferred conclusion, which is a universal affirmative. To reduce all the other modes by impossibility, always attend to the inferred conclusion, take its contradictory, put it with the other premise in place of one premise, and infer the contradictory or contrary of the other granted premise. In this way you derive an impossibility, namely, the granting of two contradictories or contraries.

To do this quickly and readily, note that in the five imperfect modes of the first figure—Baralipton, Celantes, Dabitis, Fapesmo, and Frisesomorum—the contradictory of the conclusion must be taken and put in place of the major, the granted major put in place of the minor, and the contrary of the granted minor inferred. But make an exception for Celantes: because the contradictory of its conclusion is particular, it must be put in place of the minor, and the minor of Celantes put in place of the major; thus in Darii one infers the contradictory of the granted major. In the modes of the second figure, if one forms a syllogism and wishes to reduce it by impossibility, take the contradictory of the conclusion, put it in place of the minor, and with the major of the syllogism infer the contrary or contradictory of the granted minor. In the modes of the third figure, take the contradictory of the conclusion, put it in place of the major, and with the same minor infer the contradictory or contrary of the granted major.

Although these rules alone, in their actual application, lead to the reduction by impossibility of each syllogism to its corresponding mode among the four perfect ones, a verse has nonetheless been devised to correspond to all the imperfect modes and to contain the four vowels A, E, I, O. A signifies reduction by impossibility to Barbara, E to Celarent, I to Darii, and O to Ferio. The verse is this:

Febifecaxis obit terras spherámque quotannis.

Others set down these expressions in this order: Nesciebatis, Odiebam, Laetare, Romanis; they amount to the same thing.

Febifecaxis has five vowels, corresponding to the five imperfect modes of the first figure. The first vowel, E, signifies that the first imperfect mode, namely Baralipton, is reduced to Celarent. The second vowel, I, signifies that the second mode, Celantes, is reduced to Darii. The third vowel, E, that the third mode, Dabitis, is reduced to Celarent. The fourth vowel, A, that the fourth mode, Fapesmo, is reduced to Barbara, and so on. Obit terras corresponds to the four modes of the second figure, and spherámque quotannis to the six modes of the third figure. The first vowel, O, signifies that the first mode of the second figure is reduced to Ferio. The second, I, that the second is reduced to Darii, and so on.