The quality or state of being eligible; eligibility.
From eligible + -ness (a Germanic suffix meaning state or quality). This is a native English formation rather than a Latin one, showing how English speakers created synonyms for Latin-derived words.
English speakers had a choice: use the Latinate 'eligibility' or the Germanic 'eligibleness,' and they chose both! This shows how English is fundamentally bilingual, mixing Romance and Germanic word-building strategies.
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