Etymology
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fog (n.1)
"thick, obscuring mist," 1540s, a back-formation from foggy (which appeared about the same time) or from a Scandinavian source akin to Danish fog "spray, shower, snowdrift," Old Norse fjuk "drifting snow storm." Compare also Old English fuht, Dutch vocht, German Feucht "damp, moist." Figurative phrase in a fog "at a loss what to do" first recorded c. 1600. Fog-lights is from 1962.
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fog (n.2)
"long grass, second growth of grass after mowing," late 14c., probably of Scandinavian origin; compare Norwegian fogg "long grass in a moist hollow," Icelandic fuki "rotten sea grass." A connection to fog (n.1) via a notion of long grass growing in moist dells of northern Europe is tempting but not proven. Watkins suggests derivation from PIE *pu- (2) "to rot, decay" (see foul (adj.)).
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fog (v.)
1590s (transitive), from fog (n.1). Intransitive use from 1849. Related: Fogged; fogging.
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befog (v.)
c. 1600, from be- + fog. Related: Befogged; befogging.
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defogger (n.)

"mechanism that clears condensed water vapor from the window of an automobile," by 1962, from agent noun from defog (v.) which is attested from 1945 (implied in defogging); see de- + fog.

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fogey (n.)
also fogy, "an old, dull fellow," 1780, Scottish foggie, originally "army pensioner or veteran," perhaps connected to fogram (1772) "old-fashioned," also "old-fashioned person;" or from fog (n.2) in an obsolete senses of "moss," or from foggy "bloated, fat" (1520s), which perhaps is an extended sense of fog (n.2). Related: Fogeydom; fogeyish; fogeyism.
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haze (n.)
"opaqueness of the atmosphere," 1706, probably a back-formation of hazy (q.v.). Sense of "confusion, vagueness" is 1797. The differentiation of haze, mist, fog (and other dialectal words) is unmatched in other tongues, where the same word generally covers all three and often "cloud" as well; this may be an effect of the English climate on the English language.
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foggy (adj.)

1540s, of the air, "full of thick mist," perhaps from a Scandinavian source, or formed from fog (n.1) + -y (2). Foggy Bottom "U.S. Department of State," is from the name of a marshy region of Washington, D.C., where many federal buildings are (also with a suggestion of political murkiness) popularized 1947 by James Reston in the New York Times, but he said it had been used earlier by Edward Folliard of The Washington Post.

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footle (v.)
"to trifle," 1892, from dialectal footer "to trifle," footy "mean, paltry" (1752), perhaps from French se foutre "to care nothing," from Old French futer "to copulate with," from Latin futuere "have sex with (a woman)," originally "to strike, thrust" (which is perhaps from PIE root *bhau- "to strike"). But OED derives the English dialect words from foughty (c. 1600), from Dutch vochtig or Danish fugtig "damp, musty;" related to fog (n.).
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