To sink into or get stuck in a bog; to cause to become mired or trapped in swampy ground.
From the prefix 'em-' (meaning to put into a state) plus 'bog,' a wetland area. This follows the pattern of em- verbs that place something into a condition or location.
Medieval travelers feared being emboggled on journeys—getting stuck in bog meant you might sink deeper and deeper, which is probably why 'bog' became a metaphor for getting trapped in bad situations.
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