What is the common underlying reason for performing the Hajj on behalf of the deceased and settling their debts, and is it to alleviate the burden on someone who is unable to fulfill an obligation, or the necessity of making up for an obligatory duty, or for another reason?

The four pillars of analogical deduction (Qiyas) are summarized as follows: The branch (paying Allah's debt on behalf of another), the origin (paying a human's debt), the effective cause ('Illah) (each being a right owed by another), and the ruling (the permissibility of paying Allah's debt by analogy with the permissibility of paying a human's debt). Ibn Daqeeq al-Eid adduced evidence for the effective cause by stating that the Prophet, peace and blessings be upon him, analogized the obligation of fulfilling Allah's right to the obligation of fulfilling the rights of people. Thus, the comprehensive effective cause is that each of them is a right owed by the deceased.