"Barbara Celarent" are the first two words of a chant, which was a mnemonic device invented in the middle ages to remember the valid syllogistic forms:
Barbara, Celarent, Darii, Ferio que prioris;
Cesare, Camestres, Festino, Baroko secundae;
Tertia, Darapti, Disamis, Datisi, Felapton,
Bokardo, Ferison, habet; Quarta in super addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison
On the face of it, this is simply a way to remember the valid forms in every figure. This is gleaned from the first three vowels of each name, ie. "Barabara" means that AAA is valid for the first figure (see categorical syllogism for the definition of these figures). Thus, for the second figure, EAE, AEE, EIO, and AOO are valid.
However, this chant also shows to which first figure form each of the valid sylllogistic forms can be reduced to (by applying immediate inference rules), from the first letter of each name. For example, Bokardo (OAO) in the thrid figure, can be reduced to Barbara (AAA) in the first figure.
But wait, there's more! This device also tells us *how* to reduce the syllogism to first figure forms. These are indicated by the consonants following the vowels
If a vowel is followed by 's' then the statement has to be converted (see immediate inference rules for how conversion happens). If an 'a' is followed by a 'p', the A is to be converted by limitation. If an m follows a vowel, it means that the major and the minor premises are to be exchanged. If an 'i' precedes a 'p' the conversion by limitation is to be done on the conclusion of the First figure form. A k indicates that a direct reduction is not possible, and an indirect reduction must be used.
The sheer amount of information that this verse conveys, makes it probably one of the best mnemonic device ever invented.
Harlan Miller, course notes Philo 5505 (virginia tech)