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Divide into possible cases and prove each separately.
If A → C and B → C, then (A ∨ B) → CEither he lied, was deluded, or was truthful; only truthful fits.
(Liar ∨ Deluded ∨ Truthful) ∧ ¬Liar ∧ ¬Deluded → TruthfulThe Prophet ﷺ must be one of three: Case 1 (Liar): Liars seek gain, but he rejected wealth, power, and comfort; he was persecuted and could have stopped at any time. Case 2 (Deluded): The deluded lack wisdom, but he established laws, led armies, judged disputes, and displayed supreme rationality. Case 3 (Truthful): His character, consistency, and fulfilled prophecies fit only a true prophet. Only truthfulness explains the evidence.
This is proof by cases (Lord, Liar, or Lunatic trilemma applied to prophethood): we enumerate all possibilities, eliminate two through evidence, and conclude the remaining option must be true.
This proof demonstrates the logical foundations of Islamic beliefs through rigorous philosophical methodology.