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rigidity has 2 distinct synonyms spread across 2 of its 2 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
the physical property of being stiff and resisting bending
the quality of being rigid and rigorously severe
Synonyms, antonyms and the broader and narrower relations for rigidity come from Princeton University’s WordNet. Full sources and licences.