The Logic of the Qur’an: A Six-Valued Quasi-Matrix Deontic Logic of Religious and Legal Norms

Standard deontic logic has one notion of permission: the absence of a prohibition. Von Wright observed long ago that this conflates two normatively distinct situations — a deed the authority has expressly left free, and a deed it has simply never considered. The distinction matters: expressly permitted deeds compose safely, gap-permitted deeds do not. Yet no truth-value semantics has been able to keep the two apart — until now. We construct a six-valued quasi-matrix deontic logic, QK, in which the two kinds of permission are genuine truth values of deeds, not meta-level constructs. The system splits the classical "indifferent" value into iₘ (textually grounded permission, ibāḥa sharʿiyya) and iₙ (default permittedness — silence or legal gap, barāʾa aṣliyya). The split is vindicated by a sharp formal asymmetry: it is provable that the sequential composition of two textually permitted deeds can never turn out censured-yet-legal, while the analogous claim for gap-permitted deeds is refuted by an explicit countermodel. The proof is purely formal, derived from three abstract properties of any "closed" normative code; the Islamic jurisprudential terminology supplies the intended interpretation, not the argument. The result is a system that can locate normative risk precisely: wherever ¬S holds — wherever a status rests on default rather than on a ruling — compositional guarantees fail. The operator S ("status established by a source") is the formal address of that vulnerability.