Why Python built in "all" function returns True for empty iterables?

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.

all() (documented to "Return True if all elements of the iterable are true (or if the iterable is empty).") is equivalent to the following:

def all(iterable):
    for element in iterable:
        if not element:
            return False
    return True

Since there are no elements, it will skip the loop and return True.