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rationally has no recorded synonym. 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. Its one recorded opposite is irrationally.
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.
in a rational manner
“we must act rationally”
Synonyms, antonyms and the broader and narrower relations for rationally come from Princeton University’s WordNet. Full sources and licences.