The Obverted Converse

Which of them is worth anything

The obverted converse of E is the useful one, and it is worth learning as a form in its own right.

"No unregistered document is admissible" gives "Every admissible document is a non-unregistered document", which unwinds to: every document that is admissible is one that is registered.

That is a real and useful proposition and it is exactly the direction a court needs when the question is not "may this unregistered document be used" but "what must a document be if it is to be used". The original states a bar; the obverted converse states the requirement.

The other two are chiefly steps. "Some agreements are not non-contracts" is valid, correct and unusable, and it exists so that a longer chain can pass through it.

A worked example

A section provides: "No appeal shall lie from an order passed with the consent of parties."

Reduce and label. E proposition. S is "orders passed with the consent of parties", P is "orders from which an appeal lies".

Convert simply, since E converts simply. "No order from which an appeal lies is an order passed with the consent of parties." Still an E, and this is the statement of the bar seen from the other side.

Obvert. Change quality, negate the predicate. "Every order from which an appeal lies is an order not passed with the consent of parties." An A proposition, and this is the obverted converse.

What has been gained. The original tells you what you cannot appeal. The obverted converse tells you a property that every appealable order has, which is what a registry needs when scrutinising a memorandum of appeal. The two say the same thing and only one of them is a checklist.

And a warning about the order of operations. Had the two steps been taken the other way about, obverting first and then converting, the result would have been a proposition about orders not passed with consent, which is a different and much larger class, and it would have said something quite different. That is the partial contrapositive, and it is the subject of the next chapter.

Distinctions that carry marks

What this does not mean

The obverted converse is not the converse of the obverse. That is the partial contrapositive. The names are close and the order is opposite, and the two produce propositions about different classes.

The absence of an obverted converse for O is not a separate rule. It follows from O having no converse at all, which was proved from the distribution rule at sequence 450.

Conversion by limitation carries its usual qualification. The obverted converse of A depends on the traditional reading, since it goes through a conversion by limitation. On the modern reading it fails with it.

Quick revision

Obverted converse: convert first, then obvert.

Shape of the result: P as subject, non-S as predicate.

A gives "some P is not non-S", an O, going through conversion by limitation. E gives "all P is non-S", an A. I gives "some P is not non-S", an O. O gives nothing, because it has no converse.

The useful one is E's: a bar stated as a requirement, "every admissible document is registered".

Do not confuse it with the partial contrapositive, which is the same two operations in the other order and has non-P as its subject.

Test yourself

1. Define the obverted converse and state the shape of its result.

The obverted converse is the obverse of the converse: the given proposition is converted, so that its predicate becomes the subject, and the result is then obverted, so that the quality changes and the new predicate is replaced by its contradictory. The result therefore always has the original predicate P as its subject and the contradictory of the original subject, non-S, as its predicate.

2. Give the obverted converse of each of the four propositions.

For A, "all S is P", conversion by limitation gives "some P is S" and obversion gives "some P is not non-S", an O. For E, "no S is P", simple conversion gives "no P is S" and obversion gives "all P is non-S", an A. For I, "some S is P", the result is "some P is not non-S", an O. O has no converse and therefore no obverted converse.

3. Why does O have no obverted converse?

Because the first step cannot be taken. O has no converse at all: its subject is undistributed and its predicate distributed, and every candidate converse would distribute the original subject, which the distribution rule forbids. Since the obverted converse is defined as the obverse of the converse, the absence of a converse is the whole reason, and no further rule is needed.

4. Which obverted converse is practically useful, and why?

That of E. "No unregistered document is admissible" gives "every admissible document is a non-unregistered document", that is, every admissible document is registered. The original states a bar and the obverted converse states a requirement that every member of the permitted class satisfies, which is the form needed when the question is what a document must be rather than what it must not be.

5. Distinguish the obverted converse from the partial contrapositive.

They are the same two operations in opposite orders. The obverted converse converts first and then obverts, giving a proposition whose subject is P and whose predicate is non-S. The partial contrapositive obverts first and then converts, giving a proposition whose subject is non-P and whose predicate is S. The two therefore speak about different classes, and reversing the order of the steps is not a matter of convenience.

6. What qualification attaches to the obverted converse of A?

That it passes through a conversion by limitation, which depends on the universal asserting that its subject class has members. On the modern reading, where universals carry no existential import, that step fails, and with it the obverted converse of A. The obverted converses of E and I are unaffected, since both go through simple conversions.