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Possibly Deborah was daughter b. 1681 of Joseph and Bethiah (Macy) Gardner of Nantucket. See the Thomas Gardner genealogy in the references.
Edmund Janes Cleveland is noncommittal regarding Deborah's surname, "Deborah ___ whose surname is supposed to be Gardner, she d. bef. 1751".
'PALMER CLEVELAND, ... m. 1st, N. K., bef. 1717, Deborah -----, whose surname is supposed to be Gardner, she d. bef. 1751.'