How should we interpret:
$\forall k\in \emptyset$ proposition on k,
and
$\exists k \in \emptyset$ proposition on k.
I would say the first is always true, for all k in empty set, I tend to read it as If there's a k in empty set, then something. The second I would read as there exists a k in empty set in which we have the proposition as true.
This is just my intuition. I'm looking for a formal reasoning (not too much though) of why this is so, or not...
Any help would be appreciated.
Your intuition is correct. Let $\varphi$ denote a property and $X$ be a set. Then, formally, the statement $\forall x\in X (\varphi(x))$ is a shortcut for $\forall x(x\in X \to \varphi(x))$, which reads as 'every element of $X$ has the property $\varphi$'. Similarly for the existential quantifier: $\exists x\in X (\varphi(x))$ is a shortcut for $\exists x(x\in X$ and $\varphi(x))$. This is usually called a bounded quantification.
Now it is easy to determine the truth of these statements assuming $X=\emptyset$ simply applying knowledge from (classical) propositional logic. E.g. the first formula is always true because the premise of the implication is always false (one of the paradoxes of material implication).