David Marjanović mentioned the archaic verb mote ‘may/might,’ (obsolete) ‘must,’ and when I went to that Wiktionary page I saw “Related to empty,” which surprised me. The etymology at that last link read:
From Middle English emty, amty, from Old English ǣmtiġ, ǣmettiġ (“vacant, empty, free, idle, unmarried”, literally “without must or obligation, leisurely”), from Proto-Germanic *uz- (“out”) + Proto-Germanic *mōtijô, *mōtô (“must, obligation, need”), *mōtiþô (“ability, accommodation”), from Proto-Indo-European *med- (“measure; to acquire, possess, be in command”). Related to Old English ġeǣmtigian (“to empty”), ǣmetta (“leisure”), mōtan (“can, to be allowed”). More at mote, meet.
The odd thing is that that Proto-Indo-European link does not, so far as I can see, provide any way to get to empty. AHD says “Middle English, from Old English ǣmtig, vacant, unoccupied, from ǣmetta, leisure; see med- in the Appendix of Indo-European roots”; the appendix entry, under “7. Possibly lengthened o-grade form *mōd‑,” has:
2. empty, from Old English ǣmetta, rest, leisure, from Germanic compound *ē-mōt-ja‑ (prefix *ē‑, meaning uncertain, from Indo-European *ē, *ō, to). Both a and b from Germanic *mōt‑, ability, leisure.
The OED (entry revised 2014) has:
< Old English ǣmetta (also ǣmta) leisure, freedom (to do something), opportunity (< the Germanic base of e- prefix¹ + the Germanic base of mote v.¹ + a Germanic (dental) suffix causing i-mutation (compare -th suffix¹)) + -y suffix¹.
It all sounds a little handwavy to me, and I wonder how firmly established the etymology is.
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