The Obverted Converse

In one line

The obverted converse of a proposition is what you get by converting it and then obverting the result.

In the wording a student can write in an examination: the obverted converse is the obverse of the converse. The given proposition is first converted, so that its predicate becomes the subject, and the result is then obverted, so that its quality is changed and its predicate replaced by the contradictory. The result has the original predicate as its subject and the contradictory of the original subject as its predicate.

The shape of the result

Reading the definition into a shape saves a great deal of confusion in a chain.

Start with S and P. Convert, and the subject becomes P. Obvert, and the predicate becomes non-S.

So the obverted converse always has P as its subject and non-S as its predicate.

Compare the partial contrapositive of the next chapter, which always has non-P as its subject and S as its predicate. The two are mirror images, and they are what the two orders of the same pair of operations produce. Anybody who can hold those two shapes will never confuse the two eductions again.

The four propositions

A: All S is P.

Convert, by limitation, since A converts only to a particular: some P is S, an I proposition.

Obvert the I: some P is not non-S, an O proposition.

So the obverted converse of A is "Some P is not non-S". Working example: "all contracts are agreements" gives "some agreements are not non-contracts".

E: No S is P.

Convert simply: no P is S, an E proposition.

Obvert the E: all P is non-S, an A proposition.

So the obverted converse of E is "All P is non-S". Working example: "no minor's agreement is enforceable" gives "every enforceable agreement is a non-minor's-agreement", that is, is not an agreement made by a minor.

I: Some S is P.

Convert simply: some P is S, an I.

Obvert: some P is not non-S, an O.

So the obverted converse of I is "Some P is not non-S", the same form as for A, which is unsurprising since both converted to an I.

O: Some S is not P.

O has no converse, as sequence 450 proved. So O has no obverted converse, and the reason is not a further rule but the same proof.

The table