The new candidates bring the number of people hoping to compete in the June 3 election to 11, including incumbent President Bashar al-Assad.
Shortly before the new candidates were announced in a parliamentary session, state media said at least 14 people had been killed and dozens more injured when four mortar shells hit a religious school.
"Fourteen citizens were killed and 86 others wounded by terrorists who targeted the Shaghur neighbourhood in Damascus," the SANA news agency said.
The Syrian Observatory for Human Rights, a Britain-based NGO, put the toll in the attack at 17, adding that the number could rise because several of the injured were in critical condition.
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Observatory director Rami Abdel Rahman told AFP that the institute taught students as young as 14, but it was not immediately clear if there were children among the dead or wounded.
Rebel forces arrayed in positions on the outskirts of the capital regularly fire mortar shells and rockets into the heart of Damascus, often killing civilians.
Ali Wanous, Azza al-Hallaq, Talea Salah Nasser and Samih Mikhael Moussa are all relative unknowns, like most of the 11 candidates, except for Assad.
Syria's constitution requires that candidates for the presidency be Muslim, but a source in the constitutional court confirmed that Moussa is Christian.
"We receive all applications for presidential candidacy "In the five days after the candidacy period ends, on May 5, we will examine the candidates to see if they meet all requirements. On May 6, we will announce who has met the conditions," he added.