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Is there a word for when a word is used as both the superset and subset?

The example comes from the word atheist (putting aside none of this is universally agreed). In this world we refer to

  • Atheist, or weak atheist — does not believe in god
  • Strong atheist — believes there is no god

Atheist is both the superset encompassing weak and strong atheist, but it’s also a subset in that every atheist is at a minimal level a weak atheist (but possibly a strong atheist on top of that). Many atheists will refer to themselves as just atheists, and the subset only when needed to differentiate or stress their view.

Hope that makes sense and that there is an obscure word for this.

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  • But 'atheist' can also mean 'strong atheist', so I don't think yours is the best example :)
    – Joachim
    Commented Jun 12, 2022 at 10:02
  • A strong atheist is an atheist, yes. But they would usually differentiate themselves due to the more extreme version of their position. We could make up an example. Lets say Chairs in our new world are called fourleggies. But fourleggies is also a group including tables. Done, what's a word that describes this word "fourleggies" when it's used to name the subset and the superset.
    – Paul
    Commented Jun 12, 2022 at 10:30
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    If the same word is used with two different meanings, it is ambiguous. I cannot think of any word for this particular type of ambiguity.
    – Peter
    Commented Jun 12, 2022 at 13:04
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    'Animal' has senses meaning 'mammal' (check in dictionaries before you disagree) as well as 'member of the kingdom animalia', thus one can say, if one's feeling pernickety or perverse, that some animals aren't animals (fish, for instance). These conflicting usages of 'animal' (etc) are polysemes rather than homonyms, and the confusing occurrence is called hypernymy with polysemy.
    – Edwin Ashworth
    Commented Jun 12, 2022 at 15:21
  • Improper subset.
    – aparente001
    Commented Jun 13, 2022 at 4:27

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If you have a subset which is the same as a superset, it is not a subset.

[If you're unhappy with this analysis and can provide another real-world example without making up names, then I'll be happy to remove or adjust this answer.]

Let us define, from the Universal set ℰ, the set of atheists A, the set of weak atheists W and strong atheists S.

Your question states that S ⊂ A and that S ⊂ W; and contends that W ⊂ A and — because "every atheist is a weak atheist" — that W ∩ A = A.

In a Venn diagram, W and A would be concentric circles, W inside A, with nobody in the annular space outside W and inside A:
A − (W ∩ A) = ∅
But it is not just that there is noone in that region; there can be noone in that region. It is an impossibility.

You can even put the set A inside W with the same results.

This means that A and its putative subset W are identical; there is an identity between W and A.

OED defines identity as

1.a. The quality or condition of being the same in substance, composition, nature, properties, or in particular qualities under consideration; absolute or essential sameness; oneness.

I believe the word you are going to need in this example is the word identity.

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