Existential import of particular proposition

Question

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?

Answer 1

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:

• Joseph Wu, The problem of existential import (1969), and:

• John Corcoran & Hassan Masoud, Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions (2015).