I'm trying to learn Aristotle's Organon but am finding Chapter 7 (volume I) of Prior Analytics difficult to comprehend, i.e., providing validity indirect (reductio ad impossibile) reduction.
Indirect Reduction
The process seems to be:
- Find contradiction of a syllogism's conclusion
- Use contradiction of conclusion as minor premise with original syllogism's major premise
- Check to see if result contradicts original syllogism's minor premise (contradiction: Valid, no contradiction: invalid)
For example, with a valid (AOO-2) Syllogism:
| AOO-2 (Valid) | Major Premise and Contradictory | Result |
|---|---|---|
| All N are M | All N are M | Some O is not M |
| Some O are not M | All O are N | Contradicted by |
| Therefore, Some O are not N | Therefore, All O are M | All O are M |
There's a clear contradiction in this case (and a perfect AAA-1 syllogism), So AOO-2 seems to be proven valid. However, I'm not sure about contrary results (although both can't be true, so should still count?), e.g., with another valid (EAO-3) syllogism:
| EAO-3 (Valid) | Major Premise and Contradictory | Result |
|---|---|---|
| No S are P | No S are P | No R are S / No S are R |
| All S are R | All R are P | Contrary to |
| Therefore, Some R are not P | Therefore, No R are S | All S are R |
Is this contrary still a proof of validity?
Finally, it seems invalid syllogisms tested this way results in invalid reductions with logically undetermined conclusions, e.g. AEE-1:
| AEE-1 (Invalid) | Major Premise and Contradictory | Result |
|---|---|---|
| All B are A | All B are A | No C are B |
| No C are B | Some C are A | Not Contradicted by |
| Therefore, No C are A | Therefore, Some C are B (invalid) | Invalid conclusion |
So, although the conclusion of the Major Premise and Contradictory seems to contradict the minor premise of the original syllogism, as the conclusion is invalid, there is no contradiction, thus providing AEE-1 to be invalid.
Is this how reductio ad impossibile works? Much thanks!
