The difference between As a rule and Chiefly
When used as adverbs, as a rule means in general, whereas chiefly means especially or primarily.
Chiefly is also adjective with the meaning: of, or relating to a chief.
check bellow for the other definitions of As a rule and Chiefly
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As a rule as an adverb (idiomatic):
In general; most often.
Examples:
"We go fishing every Sunday as a rule, but today the lake is frozen."
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Chiefly as an adverb (focus):
Especially or primarily; above all.
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Chiefly as an adverb (focus):
Mainly or principally; almost entirely.
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Chiefly as an adjective:
of, or relating to a chief