This is expressed as “For all X in S, X is true”. If S is empty, there are no X. However, the truth statement remains True, because for all X, X was true… there just aren’t any X!
Here is a explanation using logic.
Consider two sets A and B where A+B is the union of the two sets.
If any(A+B) = True -> any(A) or any(B) = True but we cannot assert
either any(A)=True or any(B)=True.If any(A+B) = False -> any(A) = False and any(B) = False.
If all(A+B) = True -> all(A)=True and all(B)=True
if all(A+B) = False -> all(A)=False or all(B)=False but we cannot assert either all(A)=False or all(B)=False.
Now instead of B, let’s add the empty set Ø to A. We want to come up
logic such that adding the empty set does not change the values of
all() or any(), since A+Ø=A.any(A+Ø) = any(A) or any(Ø)
any(Ø) must be False, so that if any(A) is True, any(A+Ø) is True, and
if any(A) is False, any(A+Ø) is False.all(A+Ø) = all(A) and all(Ø)
if all(A) is True, all(A+Ø) is True. Therefore, all(Ø) is True.