The way "subset of properties" is being used here is loose and that's where the confusion is arising from. If we want to be a bit more rigorous:
* Define O as a set of possible (finite) objects
* Define properties as functions P: O -> {0, 1}
* A given property then is a function of the form p(o) = 1 for all o in O that satisfy the property p.
You can use property functions to generate subsets of O ({ o for o in O where p(o) = 1 }). The intersection of these generated subsets of multiple property functions will be strict subsets (strictly "smaller" as O is finite) of O as long as at least one property function is not of the form P(o) = 1 for * in O. Generally assuming that each property function generates strict subsets, applying more properties derives a smaller intersection set. That's all.
Thanks for that, it makes sense using your definition. I'm curious as to whether you agree with the parent comment that the subset could contain 'more' properties than the superset.
My understanding from your stricter definition is that (by definition) all potential properties (o for o in O where p(o) = 1) must be contained within O.
If the answer is that the superset O has fewer properties where every p(o) = 1 then I agree, and I was overthinking it
* Define O as a set of possible (finite) objects
* Define properties as functions P: O -> {0, 1}
* A given property then is a function of the form p(o) = 1 for all o in O that satisfy the property p.
You can use property functions to generate subsets of O ({ o for o in O where p(o) = 1 }). The intersection of these generated subsets of multiple property functions will be strict subsets (strictly "smaller" as O is finite) of O as long as at least one property function is not of the form P(o) = 1 for * in O. Generally assuming that each property function generates strict subsets, applying more properties derives a smaller intersection set. That's all.