Etymology
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batty (adj.)

1580s, "pertaining to or resembling a bat or bats," from bat (n.2) + -y (2). Slang sense "nuts, crazy" is attested from 1903, from the colloquial expression (to have) bats in (one's) belfry "not be right in the head" (1899).

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Definitions of batty

batty (adj.)
informal or slang terms for mentally irregular;
Synonyms: balmy / barmy / bats / bonkers / buggy / cracked / crackers / daft / dotty / fruity / haywire / kooky / kookie / loco / loony / loopy / nuts / nutty / round the bend / around the bend / wacky / whacky
From wordnet.princeton.edu