The Fourfold Classification: A, E, I and O

Singular propositions count as universal. "Ravi is a minor" speaks of a single individual, and the whole of the subject class, which has one member, is covered. Traditional logic therefore treats singular propositions as A, and singular negatives as E. This matters at sequence 430, where MU sets the opposition of singular propositions as a topic of its own.

The four forms are about categoricals only. A conditional proposition is not an A, E, I or O proposition at all, and forcing it into one is an error.

Where the four forms appear in statutes

Reading the four forms off legislative language is the practical skill, and it is worth doing once here.

A form. "Every person is competent to contract who is of the age of majority." All S is P.

E form. "No suit shall lie against the Government except after notice." No S is P.

I form. "Some of the provisions of this Act shall come into force at once." Some S is P.

O form. The rarest in drafting, because a legislature that means "some S is not P" usually says it as an exception to a universal. "Nothing in this section shall apply to a transfer by operation of law" creates an O proposition indirectly, by excepting part of the class from a rule stated universally.

That last observation is the useful one. Legislative drafting overwhelmingly uses A and E forms with provisos and exceptions, rather than I and O forms directly. A proviso carves a part out of a universal, and the result is that the universal now holds of the remainder. Recognising this shape saves a great deal of confusion when reading a section whose main limb and proviso appear to contradict each other.

A worked example

Classify each and give its letter.

"Every agreement made without consideration is void." Quantity universal, quality affirmative. A.

"No agreement made by a minor is enforceable." Quantity universal, quality negative. E.

"Some agreements without consideration are valid." Quantity particular, quality affirmative. I. This is the proposition section 25 of the Indian Contract Act 1872 makes true, since it saves natural-love-and-affection promises in writing and registered, promises to compensate for past voluntary services, and promises to pay a time-barred debt.

"Some registered documents are not compulsorily registrable." Quantity particular, quality negative. O.

Now notice a relation. The A proposition and the I proposition just given appear to conflict, and they do: if every agreement without consideration is void, no such agreement can be valid. What resolves it is that the statute states the universal in section 25 and then carves out the exceptions in the same section, so the true A proposition is about agreements without consideration other than the three excepted classes. The apparent contradiction was produced by taking the main limb without the exception, which is exactly the drafting shape described above.

The logical lesson. An A proposition and an I proposition with the same subject and predicate can both be true, and an A proposition and an O proposition cannot. Working out which pairs can hold together is the square of opposition, at sequence 410, and this example is a preview of it.

Distinctions that carry marks

What this does not mean

"Some" does not mean "not all". It means at least one, and it leaves open whether all are.

"No S is P" is not a particular proposition. It is universal in quantity and negative in quality.

A singular proposition is not a particular one. "Ravi is a minor" is treated as universal, because it covers the whole of a one-membered subject class. Students confuse "singular" with "particular" every year, and the words are not synonyms in this subject.

The classification is not about sentences. "Not a single agreement made by a minor is enforceable" is an E proposition however it is worded.

Quick revision

Quantity: universal or particular, settled by the quantifier. Quality: affirmative or negative, settled by the copula.

Four forms: A, all S is P; E, no S is P; I, some S is P; O, some S is not P.

Letters from AffIrmo and nEgO, first vowel universal, second particular.

"Some" means at least one, and does not exclude all.

Singular propositions are treated as universal, A if affirmative and E if negative.

In statutes: A and E forms with provisos and exceptions are the normal drafting shape; I and O forms appear indirectly, by exception.

Test yourself

1. On what two grounds are categorical propositions classified, and what are the four resulting forms?

On quantity, which is settled by the quantifier and is either universal or particular, and on quality, which is settled by the copula and is either affirmative or negative. Crossing the two gives four forms: the universal affirmative or A, "all S is P"; the universal negative or E, "no S is P"; the particular affirmative or I, "some S is P"; and the particular negative or O, "some S is not P".

2. Where do the letters A, E, I and O come from?

From two Latin words. AffIrmo, meaning "I affirm", gives A and I to the two affirmative forms, the first vowel to the universal and the second to the particular. NEgO, meaning "I deny", gives E and O to the two negative forms in the same order. The mnemonic is traditional and is worth stating in an answer, since it also fixes which letter goes with which quantity.

3. What does "some" mean in logic, and how does that differ from ordinary speech?

It means "at least one", and it leaves entirely open whether all are. In ordinary speech saying "some" carries a suggestion that not all are, so that "some agreements are void" would be heard as denying that all are. In logic that suggestion forms no part of what is asserted, so an I proposition is true even where the corresponding A proposition is also true.

4. Is a singular proposition particular?

No. A singular proposition such as "Ravi is a minor" speaks of a subject class with one member and speaks about the whole of it, so traditional logic treats it as universal: an A proposition if affirmative and an E proposition if negative. The word "singular" describes the subject, while "particular" describes the quantity, and confusing the two produces errors in the square of opposition.

5. Classify "No suit shall lie against the Government except after notice" and explain the exception.

The main limb is an E proposition, universal in quantity and negative in quality: no S is P. The exception carves a part out of that universal, so the proposition actually enacted is that no suit for which notice has not been given shall lie. This is the standard drafting shape: a universal stated in the main limb with a proviso or exception restricting the subject class, and the universal then holds of the remainder.

6. Can an A proposition and an I proposition with the same subject and predicate both be true?

Yes. "All contracts are agreements" and "some contracts are agreements" are both true, since "some" means at least one and does not exclude all. What cannot both be true is an A proposition and the corresponding O proposition, since one says the predicate belongs to every member of the subject class and the other says it fails of at least one. The full set of such relations is the square of opposition.