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For example, "word" is a word, "adjectival" is adjectival, etc. Is there a term for this? Something like a set that includes itself, but not quite. I want to call it "autoinclusive"; would that work linguistically?

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Perhaps the word you're looking for is autological (or homological).

An autological word (also called homological word) is a word expressing a property which it also possesses itself. Source

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    From which we have a variant of the Grelling–Nelson paradox; "Is autological, autological?". If it is, then we can prove that it is. But if it isn't, then we can prove that it isn't. (The original Grelling-Nelson paradox applies to the antonym, heterological, asking "Is heterological, heterological?"; if it is then it isn't, and if it isn't then it is).
    – Jon Hanna
    Commented Jan 6, 2015 at 14:56

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