Is there a sign for "according to?"

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There are signs in mathematics for "therefore" and "because," which are $\therefore$ and $\because$ respectively.

However, another frequently-used expression is "according to." Is there any symbol for that?

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No, there isn't, and it isn't good practice to use mathematical/formalistic symbols in the middle of a sentence.

For instance:

"Now, Eq. (1) $\rightarrow$ that Eq. (2) is $\sim$ true,"

instead of

"Now, Eq. (1) implies that Eq. (2) is false,"

only makes the sentence more obscure. You could write out every sentence as a logical statement, but ask yourself, "what text would I rather read?" Your audience probably would choose the same.

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There is a Latin abbreviation sec. from secundum, meaning "following" or "in accordance with", sometimes used as "sec. Smith", which means "according to Smith".