The fifteen valid forms assume nothing about existence. On the traditional reading, where classes are assumed to have members, nine more forms become valid. They are called conditionally valid.
| Form | Name | Class that must have members |
|---|---|---|
| AAI-1 | Barbari | S |
| EAO-1 | Celaront | S |
| AEO-2 | Camestros | S |
| EAO-2 | Cesaro | S |
| AAI-3 | Darapti | M |
| EAO-3 | Felapton | M |
| AEO-4 | Camenos | S |
| EAO-4 | Fesapo | M |
| AAI-4 | Bramantip | P |
Key idea
A conditionally valid form commits the existential fallacy on the modern reading, but it is valid if one particular class has at least one member. The Venn test shows which class: the one whose circle has its only open part inside the region the conclusion needs.
Why it works
Each of these forms has two universal premises and a particular conclusion. The premises shade regions but draw no X, and the conclusion needs an X. Now assume a class has members. If the shading leaves that circle open only inside the conclusion's region, the member must be there, which puts the X right where the conclusion needs it. Assuming a different class may not help, so the choice of class matters.
Worked examples
Worked example: Barbari
"All mammals are animals. All whales are mammals. So some whales are animals." Valid on which reading?
- AAI-1: two universals, a particular conclusion. On the modern reading: existential fallacy.
- Assume there are whales (S). The shading leaves only the whale–mammal–animal region open in the whale circle, so a whale must be there.
- So it is valid on the traditional reading, if there are whales.
Answer: conditionally valid: S must have members
Worked example: Darapti
For AAI-3, "All M are P. All M are S. So some S are P", which class must have members?
- Assuming S helps? The S circle still has open parts outside P (region 1), so no.
- Assume M has members: both premises shade every part of M except the region inside S and P (region 5). The X goes there.
Answer: M
Common mistake
Counting AAI-1 as valid on the modern reading. It needs the assumption that the minor term's class is not empty, or it commits the existential fallacy. Check: two universal premises with a particular conclusion always need an existence assumption.