Key notes on direct and indirect reduction

Aristotle’s dictum applies directly only to the first figure. So the first figure is called the perfect figure where as second, third and fourth figures are called imperfect figures. Consequently, the valid moods of first figure (BARBARA, CELARENT, DARII and FERIO) are known as perfect moods and the valid moods of other figures are called imperfect moods.

However, it is possible to transform any imperfect mood into perfect mood by a process called reduction. We thus define reduction as a process of transforming an imperfect mood into perfect mood. In reduction we prove the validity of an imperfect mood by using the validity of perfect moods. Reduction is of two types, namely, direct and indirect.

Direct reduction is a process of transforming an imperfect mood into perfect mood by the application of either conversion, observing or interchanging the place of major and minor premises.

On the other hand, indirect reduction consists of proving the validity of imperfect mood by using “the principle of reduction ad absurdum” together with the validity of perfect Moods; we discuss and explain below each of them in detail.

Note that Aristotle’s dictum is an intuitively valid principle. Since it applies directly to the moods namely ‘AAA’, EAE, All and EIO (i.e. BARBARA, CELARENT, DARII and FERIO) of first figure such moods are valid moods. Then Aristotle proved the validity of moods belonging to the imperfect figures by using the valid moods of first figure.

The method of reduction was used by Aristotle to prove the imperfect syllogistic moods. Earlier we tested the validity of syllogistic moods either by using “General syllogistic rules” or by “Special syllogistic rules”. All these ways of proving syllogistic moods are nothing but alternative ways of proving the same thing. Moreover, since imperfect moods are reduced to perfect moods, we may say that reduction as a method of proof reveals the unity of all ‘forms of syllogistic inference.

For easy remembering of valid syllogistic moods, the mnemonic verses have been composed. These are composed of fictitious words indicating valid syllogistic moods and at the same time they carry instructions with respect to reduction of syllogistic moods of second, third and fourth figure to those of the first figure. As already discussed, the first figure and the second figure have each four valid moods. The third figure contains six valid moods and the fourth figure has five valid moods. For convenience we state below all the valid syllogistic moods in a mnemonic verse.

First Figure- BARBARA, CELARENT, DARII, FERIO

Second Figure- CEMESTRES, CESARE, BAROCO, FESTINO

Third Figure- DARAPTI, DATISI, DIS AMIS, FELAPTON, FERISON, BOCARDO

Fourth Figure- BRAMANTIP, CAMENES, DIMARIS, FESAPO, FRESISON

Here each word in our mnemonic verse represents a valid syllogistic mood in the figure written by its side. For example, the word ‘BARBARA’ is the name of a valid syllogistic mood in the first figure.

Similarly ‘DARAPTI’ is the name of a valid mood in third figure and so on. In each of these words we find three vowels. The first and second vowel respectively denotes the major and minor premise. The last vowel represents the conclusion.

For example the mood CELARENT in first figure contains three vowels namely “E A E”. The first vowel “E” indicates that the major premise in the above mood is an E-proposition; the second vowel ‘A’ indicates that the minor premise is an A-proposition, and the third vowel “E” indicates that the conclusion is an E-proposition. Analogously we understand the presence of three vowels in other syllogistic moods.

The initial letters of the moods of the first figure are the first four consonants ‘B’, ‘C’, ‘D’ and ‘F’. The initial consonant of the moods of the imperfect figures (except B AROCO and BOCARDO) indicates that the mood is to be reduced to the perfect mood having the same initial consonant. For example, the first initial consonant “D” of the mood DARAPTI in third figure indicates that the mood in question is to be reduced to the mood of the first figure having the same initial consonant.

In other words, the mood DARAPTI in third figure is to be reduced to DARII in first figure. Similarly, since the initial consonant of CESARE in second figure is same as that of the initial consonant of CELARENT in first figure. So CESARE is to be reduced to CELARENT; in first figure and so on.

We interpreted other consonants such as ‘S ‘P’, ‘M’, ‘K’ and ‘C’ occurring in the syllogistic moods as given below. But the letters such as ‘R’, ‘T’, ‘L’, ‘B’, ‘D’ and ‘N’ have no role in reduction. They have been used only to help pronunciation.

‘S’ denotes simple conversion of the proposition indicated by the preceding vowel.

‘P’ denotes conversion per limitation of the proposition indicated by the preceding vowel.

We note that if ‘S’ or ‘P’ occurs after the third vowel then the conclusion of the new syllogism admits either simple conversion or conversion per limitation according to situation.

‘M’ indicates the interchange of place of major and minor premises i.e. the major premise of the given syllogism becomes the minor premise of our new syllogism in first figure and the minor premise of the given syllogism becomes the major premise of the new syllogism in first figure.

‘K’ denotes obversion applied to the preceding proposition.

Now we may understand ‘KS’ as first apply obversion to the preceding proposition then apply simple conversion. Similarly, ‘SK’ indicates those first apply simple conversion then apply obversion to the result. Note that although ‘KS’ and ‘SK’ do not occur in the mnemonic verse, yet they will occur in the renaming of BAROCO and BOCARDO as FAKSOKO and DOKSAMOSK. (See page 164 below.)

Further if the consonant C is used not as an initial letter then the letter ‘C’ indicates that the mood in question is to be reduced indirectly. This situation holds only with respect to BAROCO in second figure and BOCARDO in third figure. After explaining technical role of the occurrence vowels and consonants in the names of the moods in our mnemonic verses, let us illustrate direct reduction of imperfect moods.

(a) Direct reduction of second figure valid moods to first figure.

CASARE:

Note that the mood CESARE is an imperfect mood belonging to the second figure. Since the initial letter of this mood is ‘C’, it is to be reduced to the mood ‘CELARENT’ in first figure. Moreover, as the letter ‘C’ does not occur in the mood except as the initial letter, it is to be reduced directly to ‘CELARENT’. The direct reduction of CESARE is shown as given below.

CESARE (Second figure)		CELARENT (First figure)
E No P is M	——————–> by S	E No M is P
A All S is M	——————–> taken as it is	A All S is M
E No S is P		E No S is P

Note that the major premise of CESARE is an E-proposition of the form is “No P is M”, which is followed by the letter ‘S’. So we apply simple conversion to it yielding an E-proposition “No M is P”. This is indicated by the arrow starting from the major premise of CESARE and ending with the major premise of ‘CELARENT’ as shown above.

To construct the mood in first figure, there is no need to apply anything to the minor premise of CESARE i.e. the minor premise of CESARE is taken as it is to form the minor premise in first figure. As a result the new mood “E A E” (i.e. CELARENT in first figure) is formed as shown above. Therefore, we notice that the imperfect mood CES ARE is directly reduced to “CELARENT” in first figure.

Analogous explanation of direct reduction of an imperfect mood to a perfect mood can easily be given. In what follows we carry out direct reduction without giving explanation. The explanatory part is left for students to practice.