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.

This comes from the mathematical logic.

"everything is true of the elements of the empty set" (http://en.wikipedia.org/wiki/Empty_set)

See also http://en.wikipedia.org/wiki/Vacuous_truth

Because all elements are True. When there are no elements, you can say that 'all elements are ... anything'