This is the translation to SQL for your formula that starts with
{S|∃ D ∈ Doctor (∃ C ∈ Duty (D.Doc_id = C.Doc_id ^ (etc..) ) ) }
SELECT * FROM S
WHERE EXISTS (SELECT * FROM Doctor D
WHERE EXISTS (SELECT * FROM Duty C
WHERE (D.Doc_id = C.Doc_id)
AND (etc...)));
So I guess the answer is no, this is not a left outer join.