The whole notice part is suppossed to be as a benefit to the employer not the employee. If the person agreed to quit that day there shouldn't be any need to pay them. Was there anything in the contract saying that the employee would be given notice before being let go. Did you give the employee the empression they would get payed those four days? When they quit did they ask to continue working the four days or were they just trying to give notice. If nothing else just say you fired them that day, rather than accept their four days notice.