A Brief History of Validity #2

The 19 Traditional Forms

In the first post in this series, we saw that Aristotle identified 16 valid forms of categorical syllogisms (though he formally acknowledged only the first three figures). Some thirteenth-century logicians such as William of Sherwood and Peter of Spain recognized nineteen valid forms, giving them Latin names as a mnemonic device for ease of memorizing:

Barbara, Celarent, Darii, Ferioque prioris.
Cesare, Camestres, Festino, Baroco secundae.
Tertia Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison habet.
Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.

The vowels in each name correspond with the mood, such that “Barbara” is AAA-1, “Cesare” is EAE-2, and so on. Thus the medievals recognized these valid forms:

Figure 1: AAA, EAE, AII, EIO
Figure 2: EAE, AEE, EIO, AOO
Figure 3: AAI, IAI, AII, EAO, OAO, EIO
Figure 4: AAI, AEE, IAI, EAO, EIO

The five forms not included in this list are AAI-1, EAO-1, EAO-2, AEO-2, and AEO-4. Why were these five not included? They are the forms in which the conclusion is the subimplication of moods with all universal statements, namely AAA-1, EAE-1, EAE-2, AEE-2,  and AEE-4. Thus they were seen as “weaker” forms of the syllogisms (why bother concluding the particular “Some S is not P”when you can conclude the universal “No S is P”?).

Defending the Missing Five

Interestingly, these five omitted forms can readily be shown to be equivalent to Bramantip (AAI-4) using immediate inferences, as follows:

AAI-4 (given)
All P is M
All M is S

∴ Some S is P

AAI-1 (taking the converse of the conclusion, correcting the premise order)
All M is S
All P is M

∴ Some P is S

EAO-1 (taking the obverse of the major premise and conclusion of the AAI-1)
No M is non-S
All P is M

∴ Some P is not non-S

EAO-2 (taking the converse of the major premise of the EAO-1)
No non-S is M
All P is M

∴ Some P is not non-S

AEO-2 (From the AAI-1, take the contrapositive of the major premise, obverse of the minor premise and conclusion)
All non-S is non-M
No P is non-M

∴ Some P is not non-S

AEO-4 (From the AEO-2, take the converse of the minor premise)
All non-S is non-M
No non-M is P

∴ Some P is not non-S.

This is one practical application of the immediate inferences learned in Lesson 27 of Introductory Logic.

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