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When does one correctly use the colon symbol $:$, to be read as "such that"? Since I would not exprect there to be one correct answer, what are the conventions used?

To give a concrete example, where would you insert a colon symbol here?

$$\forall a ~ \exists b ~ (\forall c, ~ \forall d ~ (c \in a \wedge d \in c ) \rightarrow d \in b))$$

Thank you in advance!

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There are many conventions in use for quantifiers, but if you would like to use a $:$, it should probably be after the $\exists b$.

Keep in mind that no notation is universal: Some use commas for all quantifiers, some use commas for universal quantifiers and $:$ or something like $\ni$ for existential ones. Some avoid any of those and recommend nothing (see this answer), or parentheses after the quantifier.

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I would avoid to insert any colon symbol as it is unnecessary.

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I think it is readable as is. Every set of parenthesis represents a "such that" phrase (so I would say, use one $:$ right after the $\exists b$ and one right after the $\forall c, \forall d$). It is also a good one, because one realizes where the conditions imposed upon those $b,c,d$ start, and where they end.

To convery that (``such that'') message, I would rather abbreviate it as $\text{s.t.}$ or write explicitly $\text{such that}$ rather than use colon. Colons are used alot, when defining functions or defining sets, so they get pretty confusing pretty fast. Also, one sees vertical bars quite alot (when defining sets, e.g., $\mathbb{N} = \{n\in \mathbb{Z} \mid n\geq 0 \})$.

A slightly more general perspective: We use symbols and words (BTW, what is the difference between them?) to convey messages and ideas. They are measured by their readability, by the rate of successful interpretations of them. Sometimes it's hard to say how to use a symbol or a word correctly.

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