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It is not the case that Wittgenstein's family resemblance account shows that terms can legitimately cover heterogeneous cases without sharing a single essential property.
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Reasons For
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Reason for
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1.
Family resemblance requires some commonality to ground resemblance itself; unexplained likeness seems to invoke properties covertly.
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2.
The account struggles with clearly non-members: why 'rock collection' differs from 'game' without essential distinguishing features is unclear.
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3.
Logical and mathematical terms (prime number, triangle) seem genuinely essential; family resemblance cannot uniformly explain all legitimate categories.
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Reasons Against
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Reason against
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1.
Natural language categorization empirically works without shared essences: 'game' and 'sport' successfully organize heterogeneous activities.
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2.
Essentialist definitions often fail or over-constrain: any proposed necessary property excludes clear category members or includes non-members.
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3.
Overlapping similarity chains explain how disparate items cohere: chess and peek-a-boo count as games via successive resemblances, not common features.
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