Inversion, Partial and Full

Why I and O cannot be inverted

I: Some S is P. Obverting gives an O, which cannot be converted. Converting gives an I whose obverse is an O, which again cannot be converted. Every route runs into an O and stops, and no route ever puts non-S in the subject place.

O: Some S is not P. Obverting gives an I, converting gives an I, obverting gives an O, and the chain stops. Again no route reaches non-S as subject.

So only A and E can be inverted, and the reason in both cases is that the particular propositions cannot supply the universal step that the chain needs.

The table

The dispute about inversion

This is the part most books leave out, and a full answer needs it.

Look at what the inverse asserts. From "all contracts are agreements" it concludes that some non-contracts are not agreements. But the original proposition says nothing whatever about non-contracts. It confines itself to contracts and places them inside the agreements. How can a conclusion about a class the premise never mentioned follow from it?

The traditional answer is existential import, twice over. The chain passes through a conversion by limitation, which assumes the subject class has members; and the conclusion assumes that the class of non-S has members too. Grant both assumptions and the inference goes through. Deny either and it fails.

The modern position is that inversion is invalid. Since universals carry no existential import, step four of the A chain and step three of the E chain both fail, and there is no inverse at all.

What to write. State the chains, give both inverses, and add that inversion depends on the assumption that both the subject class and its contradictory have members, so that it is valid on the traditional reading and not on the modern one. That answer is complete, and it is the honest state of the topic.

A concrete illustration of the difficulty. Take "all trespassers will be prosecuted" from sequence 380. Its full inverse would be "some non-trespassers will not be prosecuted". If there are no trespassers and nobody is prosecuted at all, the original is true, on the modern reading, and the inverse asserts the existence of unprosecuted non-trespassers, which may be entirely correct in fact but does not follow from a notice on a gate.

A worked example

A rule provides: "All applications supported by an affidavit shall be admitted."

Reduce. A proposition. S is "applications supported by an affidavit", P is "applications admitted".

Invert, by the A chain.

Obvert: no application supported by an affidavit is a non-admitted application. E.

Convert: no non-admitted application is an application supported by an affidavit. E. This is the partial contrapositive, from sequence 480, and it is a genuinely useful proposition.

Obvert: every non-admitted application is an application not supported by an affidavit. A. The full contrapositive.

Convert by limitation: some applications not supported by an affidavit are applications not admitted. I, and this is the full inverse.

Obvert: some applications not supported by an affidavit are not admitted. O, the partial inverse.

Assess what has been obtained. The contrapositives, at steps two and three, are strong and useful: every application not admitted was unsupported by an affidavit. The inverses, at steps four and five, say only that some unsupported applications are not admitted, which is much weaker and, on inspection, is not something the rule ever said. The rule tells us what happens to supported applications and is silent about unsupported ones, which may be admitted on some other ground entirely.

That comparison is the lesson. Contraposition preserves the strength of an A proposition; inversion does not, and it reaches a class the original never mentioned. It is worth performing to answer an examination question and worth treating with suspicion in an argument.

Distinctions that carry marks

What this does not mean

Inversion is not contraposition with the terms the other way round. The chains are different lengths and start with different operations.

"Partial" does not mean it comes first. In the A chain the full inverse is reached before the partial one, which is the reverse of contraposition.

The dispute does not make inversion wrong for this paper. MU sets it, and the traditional derivation is what is examined. The qualification belongs in the answer, not instead of it.

Quick revision

Inverse: the contradictory of the original subject becomes the subject. Partial keeps P as predicate; full uses non-P.

Only A and E can be inverted. I and O run into an O proposition on every route and stop.

A chain: obvert, convert, obvert, convert by limitation, obvert. Full inverse at step four, partial at step five.

E chain: convert, obvert, convert by limitation, obvert. Partial inverse at step three, full at step four.

First step matters: obvert first for A, convert first for E. The other order dead-ends.

The dispute: the conclusion is about a class the premise never mentions, and the inference needs existential import for both S and non-S. Valid traditionally, invalid on the modern reading.

Test yourself

1. Define the partial and the full inverse.

The inverse of a proposition has the contradictory of the original subject as its subject. In the partial inverse the predicate is the original predicate, so its form is non-S and P. In the full inverse the predicate is also replaced by its contradictory, so its form is non-S and non-P. Only A and E propositions can be inverted.

2. Derive both inverses of "All S is P".

Obvert to get "no S is non-P", an E. Convert simply to get "no non-P is S", an E. Obvert to get "all non-P is non-S", an A. Convert by limitation to get "some non-S is non-P", an I, which is the full inverse. Obvert to get "some non-S is not P", an O, which is the partial inverse. The full inverse therefore arrives before the partial one.

3. Derive both inverses of "No S is P".

Convert simply to get "no P is S", an E. Obvert to get "all P is non-S", an A. Convert by limitation to get "some non-S is P", an I, which is the partial inverse. Obvert to get "some non-S is not non-P", an O, which is the full inverse. Note that the chain begins with a conversion, not an obversion as it does for A.

4. Why can I and O not be inverted?

Because every available route runs into an O proposition, which has no converse, and stops there. For I, obverting gives an O immediately; converting gives an I whose obverse is an O. For O, obverting gives an I, converting gives an I, and obverting again gives an O. In neither case does any route ever place the contradictory of the original subject in the subject position.

5. Why is inversion disputed?

Because the conclusion is about a class the premise never mentions. "All contracts are agreements" says nothing about non-contracts, yet its inverse asserts something about them. The traditional derivation goes through only because a conversion by limitation assumes that the subject class has members and the conclusion assumes that the contradictory class has members too. On the modern reading, where universals carry no existential import, both assumptions fail and inversion is invalid.

6. Compare what contraposition and inversion yield from an A proposition.

Contraposition yields a universal full contrapositive, "all non-P is non-S", which is equivalent to the original and is the form in which a necessary condition is actually applied. Inversion yields only particulars, "some non-S is non-P" and "some non-S is not P", which are much weaker and concern a class the original said nothing about. Contraposition preserves strength; inversion loses it and reaches beyond the premise.