To make dirty or soil; to stain or smudge something with dirt or filth.
Old formation combining 'be-' (to make) with 'soil' (earth, dirt, or to make dirty). 'Soil' comes from Old French 'soille' and Latin 'solum' (ground), and the be- prefix productive in creating new verbs in Middle English.
The dual meaning of 'soil' as both 'earth' and 'to dirty' reveals something deep about human language—we metaphorically associate contact with earth itself as making things unclean, a pattern found in many cultures' moral vocabulary.
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