Etymology
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Words related to fog

foul (adj.)
Old English ful "rotten, unclean, vile, corrupt, offensive to the senses," from Proto-Germanic *fulaz (source also of Old Saxon and Old Frisian ful, Middle Dutch voul, Dutch vuil, Old High German fül, German faul, Gothic füls), from PIE *pu- (2) "to rot, decay," perhaps from the sound made in reaction to smelling something bad (see pus).

Old English ful occasionally meant "ugly" (as contrasted with fæger (adj.), modern fair (adj.)), and this sense became frequent in Middle English. The cognate in Swedish is the usual word for "ugly." Of weather from mid-14c. In the sporting sense of "irregular, unfair, contrary to established rule or practice" it is first attested 1797, though foul play is recorded from mid-15c. Baseball sense of "out of play" attested by 1860.
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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.

aftermath (n.)

1520s, originally a second crop of grass grown on the same land after the first had been harvested, from after + -math, from Old English mæð "a mowing, cutting of grass," from PIE root *me- (4) "to cut down grass or grain."

Also known as aftercrop (1560s), aftergrass (1680s), lattermath, fog (n.2). Figurative sense is by 1650s. Compare French regain "aftermath," from re- + Old French gain, gaain "grass which grows in mown meadows," from Frankish or some other Germanic source similar to Old High German weida "grass, pasture."

When the summer fields are mown,
When the birds are fledged and flown,
      And the dry leaves strew the path;
With the falling of the snow,
With the cawing of the crow,
Once again the fields we mow
      And gather in the aftermath.  

[Longfellow, from "Aftermath"]
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.
befog (v.)
c. 1600, from be- + fog. Related: Befogged; befogging.
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.

fog-horn (n.)
1844, from fog (n.1) + horn (n.).
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.).
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.