Contraposition combines the swap of conversion with the complements of obversion.
Key idea
To contrapose: swap the subject and predicate and replace both with their complements. "All S are P" becomes "All non-P are non-S". Contraposition is valid for A and O, not for E or I.
| Original | Contrapositive | Valid? |
|---|---|---|
| All S are P | All non-P are non-S | yes |
| Some S are not P | Some non-P are not non-S | yes |
| No S are P | No non-P are non-S | no |
| Some S are P | Some non-P are non-S | no |
Why it works
Contraposition is three steps in one: obvert, convert, obvert. Start with "All S are P". Obvert: "No S are non-P". Convert (E converts validly): "No non-P are S". Obvert again: "All non-P are non-S". Every step kept the meaning. Try the same with "No S are P": obverting gives an A, and A does not convert, so the chain breaks. That is why A and O contrapose, and E and I do not.
Worked examples
Worked example: Contraposing an A
Contrapose "All coders are skaters".
- Swap: skaters, coders.
- Complement both: non-skaters, non-coders.
- A contraposes validly.
Answer: All non-skaters are non-coders (equivalent)
Worked example: An invalid contraposition
Does "No non-jugglers are non-knitters" follow from "No knitters are jugglers"?
- It is the contrapositive of an E statement.
- E does not contrapose: the original shades the knitter–juggler overlap, while the contrapositive shades the region outside both circles.
Answer: no
On the traditional reading, an E statement can be contraposed by limitation: from "No S are P" to "Some non-P are not non-S", just as an A statement converts by limitation. Like every limitation, it needs the assumption that the classes involved have members.
Common mistake
Using contraposition on E or I, or swapping the terms without complementing them (which is only conversion). Check: both terms changed places, both gained or lost "non-", and the form is A or O.