To damage the reputation or honor of someone or something; to soil or stain, especially metaphorically.
From be- plus smirch (origin of smirch unclear, possibly from Middle English smorche or related to smudge). Unlike purely literal 'besmear,' besmirch evolved early to mean damaging reputation through association with dirt or disgrace.
Besmirch is unique because it skipped staying concrete—while 'besmear' still means literally covering with grease, 'besmirch' leaped almost immediately to metaphorical use. In Renaissance texts, it meant damaging honor through scandal, making it one of English's most psychologically sophisticated words.
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