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Check all possible cases exhaustively.
P(x) is true for every possible xEvery alternative explanation for his mission has been refuted.
∀M ∈ {worldly motives}: ¬Applies(M, Muhammad) → Divine_commissionWe exhaustively examine all worldly motives: (1) Political power—he rejected kingship when Quraysh offered it. (2) Economic gain—he died with his armor pawned, living simply when he could have had wealth. (3) Psychological disorder—he was universally acknowledged as sound-minded, called "Al-Amin" (The Trustworthy). (4) Social reform—he challenged Arab customs rather than appeasing them. All worldly explanations fail; divine mission alone remains.
This is proof by exhaustion: we systematically examine every proposed worldly motivation, demonstrate each one fails to explain the evidence, and conclude divine commission is the only remaining explanation.
This proof demonstrates the logical foundations of Islamic beliefs through rigorous philosophical methodology.