The Figures of the Syllogism

As described by Petrus Hispanius.

I Barbara all M is P; all S is M: all S is P
I Celarent no M is P; all S is M: no S is P
I Darii all M is P; some S is M: some S is P
I Ferio no M is P; some S is M: some S is not P
   
II Cesare no P is M; all S is M: no S is P
II Camestres all P is m; no S is M: no S is P
II Festino no P is M; some S is M: some S is not P
II Baroko
Fakofo
all P is M; some s is not M: some S is not P
   
III Darapti all M is P; all M is S: some S is P
III Disamis some M is P; all M is S: some S is P
III Datisi all M is P; some M is S: some S is P
III Felapton no M is P; all M is S: some S is not P
III Bocardo
Dokamok
some M is not P; all M is S: some S is not P
III Ferison no M is P: some M is S: some S is not P
   
IV Bramantip all P is M; all M is S: some S is P
IV Camenes all P is M; no M is S: no S is P
IV Dimaris some P is M; all M is S: some S is P
IV Fesapo no P is M; all M is S: some S is not P
IV Fresison no P is M; some M is S: some S is not P

The vowels indicate the type of statements:
A - Universal affirmative
E - Universal negative
I - Particular affirmative
O - Particular negative

Conversions of II, III, IV to corresponding I:
S - simple
P - per accidens
M - transpose premises
N - reductio ad absurdum

Daniel Seely Gregory's "Practical logic: or, The art of thinking" (1881) says: "The initial consonant, B, C, D or F, in the last three Figures indicates the mood in the first Figure to which the syllogism reduces. Thus, a syllogism in the mood Cesare, reduces to Celarent. The inserted consonants, s, p, k, f, m, indicate the various processes in reduction. S indicates that the proposition symbolized by the vowel preceding it is to be converted simply; p, by limitation or per aociden; k, by contraposition; f, by infinitation or obversion. The letter m (mutari) indicates that the premises of the preceding judgment are to be transposed. The p in Bramantip shows that, after converting simply, the premises warrant a universal conclusion. The other consonants, b, d, l, n, r, t, are not significant, but are inserted for the sake of euphony, or of the metre in the mnemonic hexameters invented, to keep the moods and figures in mind, by Petrus Hispanus, who died in 1277 as Pope John XXII."

For more information, consult the lecture by R. J. Kilcullen of Macquarie University on the  Abbreviatio Montana.

Another useful classification.