| All M is P, | → | No M is not-P, |
| All S is M, | → | All S is M, |
| ↓ | ||
| All S is P. | ← | No S is not-P. |
Conversely, Celarent is reducible to Barbara ; and in a similar manner, by obversion of major premiss and conclusion, Darii and Ferio are reducible to one another.
It will now suffice if we can shew that Barbara and Darii are mutually reducible to one another. Clearly the only method possible here is the indirect method.
Take Barbara,
| MaP, | |
| SaM, | |
| ⎯⎯ | |
| ∴ | SaP ; |
325 for, if not, then we have SoP ; and MaP, SaM, SoP must be true together. From SoP by first obverting and then converting (and denoting not-P by Pʹ) we get PʹiS, and combining this with SaM we have the following syllogism in Darii,—
| SaM, | |
| PʹiS, | |
| ⎯⎯ | |
| ∴ | PʹiM. |
PʹiM by conversion and obversion becomes MoP ; and therefore MaP and MoP are true together; but this is impossible, since they are contradictories. Therefore, SoP cannot be true, i.e., the truth of SaP is established.
Similarly, Darii may be indirectly reduced to Barbara.[349]
| MaP, | (i) | |
| SiM, | (ii) | |
| ⎯⎯ | ||
| ∴ | SiP. | (iii) |