Define what is meant by "subaltern-superaltern pair of propositions" (only on the basis of their truthfulness properties)

Question

I am having a hard time understanding what is meant by a subaltern-superaltern pair of propositions.

My textbook describes such pair as

A categorical proposition is a subaltern of another iff it must be true if its superaltern is true, as well as the superaltern must be false if the subaltern is false.

The textbook also uses the Aristole's square of opposition to classify the pair of propositions.

For example:

i. ∀x (B(x) → W(x)) is A in Aristotle's square

ii. ∀x (B(x) → ¬W(x)) is E

iii. ∃x (B(x) ∧ W(x)) is I

iv. ∃x (B(x) ∧ ¬W(x)) is O

Based on the square of Aristotle I know that

• (I) is a subaltern of (A)
• (O) is a subaltern of (E)

But what does it mean to be a subaltern/superaltern of something ?

Answer 1

If you have the following two statements:

$\forall x \: P(x)$

and

$\exists x \: P(x)$

then the first is the superaltern of the second, and the second is the subaltern of the first.

So it is really a relationship between two statements where the one is an eistential, and the other a universal, but that are otherwise the same. Thus, for example, even though something like $P$ implies $P\lor Q$ we wouldn't call $P$ the superaltern of $P\lor Q$ .... Indeed, there would be no 'its (one) superaltern' in suc a case. So agai this is really about two quantificational statements with the only difference the type of quantifier.

In the square, the superaltern will be at the top, and the subaltern at the bottom, which is where I think the terminology comes from.