# Existential import of particular proposition

I saw that in categorical logic the proposition, some $$A$$ is $$B$$ is considered to have existential import meaning we assume there exists a particular in class $$A$$ but the proposition, "All $$A$$ is $$B$$ mean that if $$A$$ is empty" the proposition is true, why is that saying "some unicorns are pink" is a false statement while saying "All unicorns are pink" is a true statement?

## 1 Answer

The issue of so-called existential import of categorical proposition has a long history.

In modern logic "Some A is B" is symbolized with $$\exists x (Ax \land Bx)$$ and thus it is true only when there are $$A$$s and $$B$$s.

The universal, instead, is symbolized with $$\forall x (Ax \to Bx)$$ which is true also when there are no $$A$$s.

[Why] saying "All unicorns are pink" is a true statement?

Because the conditional $$Ax \to Bx$$ is true when the antecedent $$Ax$$ is False.

According to Arsitotle's Logic and the so-called Square of Opposition, we have that the universal imples the corresponding particular, but in modern logic the imference:

$$\forall x (Ax \to Bx)$$, therefore $$\exists x (Ax \land Bx)$$

is not valid, exactly due to the counter-examples like yours.

For some references, see: