Traditional and Modern General Propositions Distinguished

What the disagreement costs each side

What the traditional scheme cannot do. It cannot make a true universal statement about an empty class, and there are many worth making. "All trespassers will be prosecuted" is not falsified by the absence of trespassers. "All persons convicted under the repealed section shall receive a refund" is not an assertion that anybody was convicted. A scheme that makes these assert existence has misdescribed them.

What the modern scheme loses. Three of the traditional square's relations. Subalternation disappears, since A no longer implies I. Contraries disappear, since A and E can now both be true when the subject class is empty. Subcontraries disappear, since I and O can now both be false for the same reason. Only the contradictories survive, A against O and E against I, and the square of opposition is reduced to its diagonals.

That is a real loss, and it is why the square is taught in the traditional form at sequence 410, with this qualification attached to it.

The empty-class test

The quickest way to see the difference, and a good way to answer an examination question, is to take an empty class and run the four propositions past it.

Let S be "unicorns in this courtroom". There are none.

A, "all unicorns in this courtroom are white." Traditional: false, because it asserts there are unicorns here. Modern: true, vacuously, because there is nothing to be a counterexample.

E, "no unicorn in this courtroom is white." Traditional: false, same reason. Modern: true, vacuously.

Both A and E true at once, on the modern reading, which is exactly why contraries fail: two propositions that traditional logic says can never both be true are both true here.

I, "some unicorn in this courtroom is white." False on both readings.

O, "some unicorn in this courtroom is not white." False on both readings.

Both I and O false at once, which is why subcontraries fail.

A worked example

A section provides: "Every person who was in occupation of the premises on 1 January 1990 shall be deemed to be a tenant."

Read it traditionally. The section asserts that there were persons in occupation on that date. If it turned out that the premises were vacant, the section would be false, which is an odd thing to say about a piece of legislation: a statute is not false, it simply does not apply.

Read it modernly. For anything whatever, if it is a person who was in occupation on 1 January 1990, then it shall be deemed a tenant. If nobody was in occupation, the provision is true and applies to nobody. That is the correct account of what a legislature does, and it matches how courts treat such provisions.

Now the practical question. A party asserts rights under the section. What must he prove?

On either reading, he must prove that he was in occupation on the date, which is a particular proposition and carries existential import on both schemes. The section itself never proved existence for him. The distinction therefore has a practical edge: a party cannot rely on the enacting words to establish that anybody was in occupation, because the section, properly read, asserts nothing of the kind.

And the drafting lesson. A legislature that wishes to assert existence must do it separately, which is why Acts contain recitals and declarations. The operative words do not carry it.

Distinctions that carry marks

What this does not mean

The modern scheme does not say the subject class is empty. It says the proposition does not assert that it has members. The question is left open, exactly as it should be.

The dispute does not affect particular propositions. Both schemes agree that I and O assert existence.

Denying existential import is not a verbal trick. It falls out of reading a universal as a conditional, which is itself forced, as sequence 360 showed.

Quick revision

The one difference: traditional universals assert that their subject class has members; modern universals do not.

Traditional reason: subalternation, the inference from A to I, requires it.

Modern reason: ∀x(Sx ⊃ Px) is true when nothing is S, because a conditional with a false antecedent is true.

Cost to the traditional scheme: it cannot state a true universal about an empty class, and legal provisions are full of them.

Cost to the modern scheme: subalternation, contraries and subcontraries all fail, and only the contradictories of the square survive.

Empty-class test: with no S, both A and E come out true and both I and O come out false on the modern reading.

No dispute about I and O: both schemes agree that particular propositions assert existence.

Test yourself

1. State the distinction between traditional and modern general propositions.

On the traditional view a universal proposition asserts that its subject class has at least one member, so "all S is P" implies "some S is P". On the modern view a universal proposition is a conditional about anything whatever, ∀x(Sx ⊃ Px), which is true when nothing is S and therefore asserts nothing about existence. The two schemes agree entirely about particular propositions, which assert existence on both.

2. Why does the traditional scheme need existential import?

Because it accepts subalternation, the inference from A to I. If "all S is P" did not assert that some S exists, that inference would carry one from a true universal to a false particular whenever the subject class was empty, which no valid inference may do. Existential import is therefore not an add-on but a consequence of an inference the traditional square relies on.

3. Why does the modern scheme deny existential import?

Because it symbolises "all S is P" as ∀x(Sx ⊃ Px), and a conditional with a false antecedent is true. If nothing is S, the conditional holds of everything, so the universal is true. The symbolisation is itself forced, since writing the universal as a conjunction would assert that everything in the universe belongs to the subject class, so the denial of existential import follows rather than being chosen.

4. What does the modern scheme lose from the square of opposition?

Three of its four relations. Subalternation fails, since A no longer implies I. The contrary relation fails, since A and E can both be true when the subject class is empty. The subcontrary relation fails, since I and O can both be false for the same reason. Only the contradictory relations survive, A against O and E against I, so the square is reduced to its diagonals.

5. Apply the empty-class test to the four forms.

Take a subject class with no members, such as unicorns in this courtroom. On the modern reading "all unicorns here are white" and "no unicorn here is white" are both true, since there is nothing to falsify either, which shows the contrary relation failing. "Some unicorn here is white" and "some unicorn here is not white" are both false, since neither can find an instance, which shows the subcontrary relation failing. On the traditional reading the two universals are both false, since each asserts that unicorns are present.

6. What practical difference does the distinction make to reading a statute?

A provision such as "every person in occupation on a stated date shall be deemed a tenant" does not, properly read, assert that anybody was in occupation. It states what shall follow if anyone was. A party claiming under it must therefore prove his own occupation as a particular proposition; he cannot rely on the enacting words to establish that occupation existed. A legislature that wishes to assert existence has to do so separately, which is one reason Acts carry recitals and declarations.