The steps of contraposition, obversion and conversion can be chained, and combined with the square of opposition. A chain lets you decide whether one statement follows from another even when the two look nothing alike.
Key idea
A chain is valid only if every step is valid for the form it is applied to: obversion for any form, conversion for E and I, contraposition for A and O (and, on the traditional reading only, subalternation and conversion by limitation).
| Step | Works on |
|---|---|
| Obversion | A, E, I, O |
| Conversion | E, I |
| Contraposition | A, O |
| Contradictory (square) | any form, any reading: it gives the opposite truth value |
| Subalternation, limitation | traditional reading only |
Why it works
Each valid step produces a statement with the same meaning (or, for the traditional steps, one that is guaranteed true when the original is). Linking statements that each mean the same as the one before gives a last statement that means the same as the first. One invalid step anywhere breaks the guarantee, even if the remaining steps are fine.
Worked examples
Worked example: A two-step chain
If "No poets are campers" is true, must "All campers are non-poets" be true?
- Obvert the original (any form): "All poets are non-campers".
- That is an A, so contraposition is allowed: "All non-non-campers are non-poets", that is, "All campers are non-poets".
Answer: yes, by obversion then contraposition
Worked example: A broken chain
Does "Some non-P are S" follow from "All S are P"?
- Obvert: "No S are non-P". Convert (E is fine): "No non-P are S".
- That says no non-P are S, the opposite of what was asked.
- No valid chain reaches "Some non-P are S"; in fact it is the contradictory of a statement that does follow.
Answer: no
Common mistake
Losing track of the form and slipping in an invalid step, such as converting an A halfway through a chain. Check: write the letter of each line before choosing the next step.