Question

Allow that ur-elements can count as purely set-theoretic, depending on which ones are introduced (we might say: an ur-element is pure if the only extralogical facts about it feature just the elementhood relation and no other relations). Now, if it is contingent whether some pure ur-element is an element of some set, does that set have to count as impure?

Answer 1

A pure ur-element's contingent existence in a set does not make the set impure.

Step-by-step explanation:

According to the question, if it is contingent whether a pure ur-element is an element of some set, the question is asking if that set has to count as impure. A pure ur-element is defined as an element that only has extralogical facts about its elementhood relation and no other relations.

If it is contingent whether a pure ur-element is an element of some set, it means that the set can exist with or without the ur-element as an element. In this case, the set does not have to count as impure because the ur-element's elementhood relation is the only extralogical fact about it.

So, a set can still be considered pure even if it is contingent whether a pure ur-element is an element of that set.