Contraposition, Partial and Full

In one line

The partial contrapositive is obtained by obverting a proposition and then converting the result; the full contrapositive is obtained by obverting once more.

In the wording a student can write in an examination: contraposition is the immediate inference in which the contradictory of the original predicate becomes the subject. In the partial contrapositive the predicate is the original subject; in the full contrapositive it is the contradictory of the original subject.

The shape

Partial contrapositive: non-P as subject, S as predicate.

Full contrapositive: non-P as subject, non-S as predicate.

That is what "partial" and "full" mean, exactly as sequence 440 said: partial negates one term, full negates both.

Deriving them for A

Original: All S is P.

Step one, obvert. Change quality, negate the predicate. No S is non-P. An E proposition.

Step two, convert. E converts simply. No non-P is S. An E proposition, and this is the partial contrapositive of A.

Step three, obvert again. Change quality, negate the predicate. All non-P is non-S. An A proposition, and this is the full contrapositive of A.

In words. "All contracts are agreements" gives, partially, "no non-agreement is a contract", and fully, "every non-agreement is a non-contract". Both say: anything that is not an agreement is not a contract.

Notice that the full contrapositive of A is an A, which means contraposition of A loses nothing at all. This is the only eduction that takes a universal to a universal while replacing both terms, and it is why it is worth so much.

Deriving them for E

Original: No S is P.

Step one, obvert. All S is non-P. An A proposition.

Step two, convert. A converts only by limitation. Some non-P is S. An I proposition, and this is the partial contrapositive of E.

Step three, obvert. Some non-P is not non-S. An O proposition, and this is the full contrapositive of E.

In words. "No minor's agreement is enforceable" gives, partially, "some non-enforceable things are minors' agreements".

Notice the loss. A universal has become a particular, because the chain passed through a conversion by limitation. Contraposition of E is therefore possible only in a weakened form, and only on the traditional reading.

Why I has no contrapositive

Original: Some S is P.

Step one, obvert. Some S is not non-P. An O proposition.

Step two, convert. O has no converse, as sequence 450 proved. The chain stops.

So I has no contrapositive at all, and the reason is the same proof that stopped the obverted converse of O.

Deriving them for O

Original: Some S is not P.

Step one, obvert. Some S is non-P. An I proposition.

Step two, convert. I converts simply. Some non-P is S. An I proposition, and this is the partial contrapositive of O.