The difference between Maxim and Saying
When used as nouns, maxim means a self-evident axiom or premise, whereas saying means a proverb or maxim.
check bellow for the other definitions of Maxim and Saying
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Maxim as a noun (now, _, rare):
A self-evident axiom or premise; a pithy expression of a general principle or rule.
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Maxim as a noun:
A precept; a succinct statement or observation of a rule of conduct or moral teaching.
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Saying as a verb:
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Saying as a noun:
A proverb or maxim.
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Saying as a noun (obsolete):
That which is said; a statement.