The Square of Opposition

"All contracts are agreements" and "some contracts are not agreements" cannot both hold, and one of them must. This is the strongest of the four relations and it is the only one the modern reading preserves.

Contraries: A and E. Both universal, differing in quality.

Rule: they cannot both be true, but they may both be false.

"All the defendants were served" and "no defendant was served" cannot both hold. Both are false if two of three were served. This is precisely the relation between contrary terms at sequence 200, and the middle ground is what makes it possible for both to fail.

Subcontraries: I and O. Both particular, differing in quality.

Rule: they cannot both be false, but they may both be true.

"Some defendants were served" and "some defendants were not served" are both true where two of three were served. They cannot both be false, because if no defendant was served the second is true, and if all were served the first is true.

Subalterns: A and I; E and O. Same quality, differing in quantity. The universal is called the superaltern and the particular the subaltern.

Rule, and it runs in one direction only: truth descends and falsity ascends.

If A is true, I is true: if all contracts are agreements, some are. If I is false, A is false. But if A is false, nothing follows about I, and if I is true, nothing follows about A.

Remembering the rules

Four relations and four rules is more than most students hold under pressure. Two lines carry all of it.

Contradictories: exactly one true. Contraries: not both true. Subcontraries: not both false. Subalterns: truth down, falsity up.

And a check that catches errors: contraries are at the top, where universals are strong, so they conflict; subcontraries are at the bottom, where particulars are weak, so they can coexist. A student who can reconstruct that much can rebuild the square from the diagram.

Which of the four really deserve the name

Added after the past-paper check. MU sets this as a question in terms, and it is a good one, because the four relations are called opposition and only some of them are.

Opposition, strictly, means incompatibility: two propositions are opposed when they cannot both hold. Test the four against that.

Contradictories are opposition in the fullest sense. They cannot both be true and cannot both be false, so exactly one holds and the pair is completely incompatible. If any relation deserves the name, this one does.

Contraries are opposition in a real but weaker sense. They cannot both be true, so there is genuine incompatibility, but they can both be false, so the incompatibility is partial.

Subcontraries are barely opposition at all. They can both be true, and usually are: some contracts are written and some contracts are not written hold together every day. All that is excluded is their both being false. Many writers say the relation is misnamed for this reason.

Subalternation is not opposition on any view. The universal and its particular are not incompatible in any respect: when A is true I is true as well. It is a relation of inclusion, not of conflict, and it is grouped with the others only because it joins two of the four corners.

The answer to write. Contradiction fully deserves the name; contrariety deserves it in a qualified sense; subcontrariety barely does; and subalternation does not deserve it at all, being a relation of implication rather than of incompatibility. And on the modern reading only contradiction survives, which makes the point twice over.

The modern qualification

The subaltern, contrary and subcontrary relations all depend on universals asserting the existence of their subject, and the modern reading denies that they do. Sequence 380 works this out in full and the conclusion is repeated here because a complete answer needs it.

Take an empty subject class. On the modern reading A and E are both true, so contraries fail. I and O are both false, so subcontraries fail. A is true and I is false, so subalternation fails.

Only the contradictories survive, which is why the modern square is only its diagonals.

What to write in an examination. Set out the traditional square in full, since that is what MU's syllabus teaches and what the questions are set on, and add that on the modern reading only the contradictory relations hold, with one sentence of reason. That answer is complete and honest, and leaving the qualification out is the commonest way to lose a mark on this topic.

A worked example

A partner in a firm asserts: "All the partners signed the guarantee." That is an A proposition, with subject "the partners" and predicate "persons who signed the guarantee".

What may be said against it?

Its contradictory, O: some partner did not sign. This is the cheapest and strongest denial, because it needs only one instance and, if made out, it establishes that the assertion is false.

Its contrary, E: no partner signed. This is a stronger claim and a worse tactical choice: it asserts far more, it requires proof about every partner, and if it fails the original assertion is not thereby established, because contraries may both be false.

Its subaltern, I: some partner signed. This does not contradict the assertion at all. It follows from it.

The lesson for a pleader. To deny a universal, plead its contradictory, not its contrary. The instinct is the other way: a defendant who wants to deny that all the partners signed feels the pull of saying that none did. That is a heavier burden for no additional benefit, and it hands the other side a case to attack. This is a genuine and common drafting error, and the square explains it exactly.

Distinctions that carry marks

What this does not mean

Opposition requires identical terms. Two propositions with different subjects or different predicates are not opposed, whatever they say.

Contrary is not the same as contradictory. The words are close and the rules are different, and this is the same distinction as at sequence 200, now applied to propositions rather than terms.

Subalternation is not a two-way street. Truth descends and falsity ascends, and neither reverse holds.

Quick revision

Opposition: same subject, same predicate, differing in quantity, quality or both.

Square: A top left, E top right, I bottom left, O bottom right.

Contradictories are the diagonals, A with O and E with I: exactly one true.

Contraries are the top edge, A with E: not both true, may both be false.

Subcontraries are the bottom edge, I with O: not both false, may both be true.

Subalterns are the sides, A with I and E with O: truth descends, falsity ascends.

Modern qualification: only the contradictories survive, because the other three depend on existential import.

Which deserve the name: contradiction fully; contrariety in a qualified sense; subcontrariety barely, since the pair usually both hold; subalternation not at all, being inclusion rather than incompatibility.

Pleading rule: to deny a universal, plead its contradictory, never its contrary.

Test yourself

1. What is opposition, and what does it require?

Opposition is the relation between two propositions having the same subject term and the same predicate term but differing in quantity, in quality, or in both. Both conditions on the terms are necessary: propositions that do not share a subject and a predicate are not opposed at all, and nothing follows from one about the other however similar they may look.

2. Name the four kinds of opposition and state the rule for each.

Contradictory, between A and O and between E and I, where exactly one of the pair is true. Contrary, between A and E, where both cannot be true though both may be false. Subcontrary, between I and O, where both cannot be false though both may be true. Subaltern, between A and I and between E and O, where the truth of the universal carries down to the particular and the falsity of the particular carries up to the universal.

3. Distinguish contraries from subcontraries.

Contraries are the two universals and cannot both be true, since one asserts the predicate of the whole class and the other denies it of the whole class; they may both be false where the predicate holds of some members only. Subcontraries are the two particulars and cannot both be false, since if neither some are P nor some are not P, the class would have to be both wholly P and wholly not P; they may both be true, and usually are.

4. State the rule of subalternation and its two limits.

Truth descends and falsity ascends: if the universal is true the particular is true, and if the particular is false the universal is false. The two limits are the reverses. From the falsity of the universal nothing follows about the particular, since some members may still have the predicate; and from the truth of the particular nothing follows about the universal, since the rest of the class is untouched.

5. Which relations survive on the modern reading, and why?

Only the contradictories. The other three depend on a universal proposition asserting that its subject class has members, and the modern reading denies that it does. With an empty subject class both universals come out true, so contraries fail; both particulars come out false, so subcontraries fail; and the universal is true while its subaltern is false, so subalternation fails.

7. Which of the four forms of opposition really deserve the name, and why?

Opposition strictly means incompatibility. Contradictories cannot both be true and cannot both be false, so they are fully opposed and the name is fully deserved. Contraries cannot both be true but can both be false, so the incompatibility is real but partial. Subcontraries can both be true and usually are, so only their both being false is excluded and the name is barely deserved. Subalternation involves no incompatibility at all, since the truth of the universal carries the particular with it: it is a relation of inclusion and does not deserve the name. On the modern reading only contradiction survives in any case.

6. Why should a pleader deny a universal by its contradictory rather than by its contrary?

Because the contradictory is both weaker to prove and stronger in effect. To defeat "all the partners signed" it is enough to show that one did not, which is the contradictory; asserting that none signed is the contrary, requires proof about every partner, and if it fails leaves the original assertion unestablished, since contraries may both be false. The instinct to deny sweepingly is a common drafting error and the square explains exactly what it costs.