Logic of proof: vacuously true

Question

In a proof, I defined a term, say T, if a subobject of class of objects however the subobject may not exist in general. Do objects , which do not have the subject, still classify as T? I think it should still as it follows the logic of the empty set being a subset of all sets. Eg. Consider the scenario : Define the property S on people by , a person is S if they have a child , then the child is a boy. Question: is a person with no children S? I would say it is vacuously true .

Answer 1

Yes, your argument is correct.

The implication $$P\implies Q$$ is false only if $P$ is true and $Q$ is false.

Thus in case of $P$ being false, the implication is true regardless of the truth value of $Q$