What standard of law must be applied if the judge chooses to rule between non-Muslim parties?

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If the judge chooses to rule between them, he must rule only according to the rulings of Islam. This is mandated by the divine command: "And judge, [O Muhammad], between them by what Allah has revealed and do not follow their desires" (Quran 5:49) and "And judge between them by what Allah has sent down" (Quran 5:42).