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In Esperanto, why does ĉiu mean "every" when ĉi serves elsewhere as a proximal prefix (e.g. ĉi-flanke) and -u is a suffix meaning "one"? Why does this not mean "this one" just as ti+ -u means "that one"?

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No, for the correlatives holds: ĉi-, ki-, ti-, neni-, i- and -u, -am, -el and such are not independent parts, roots, and have nothing to do with roots ĉi or imperative verb suffix -u.

The definition of -u is adjective selection of one of some, with persons used as noun.

The definition of ĉi- is all/every.

The correlatives "happen" to form a table with similarities. In fact this holds for natural languages too, to some degree.

For ki- the indoeuropean Q: q (Italian) / k (Russian) / w (German). So what, who, which, where. The same for ti-: that, there, this.

Mind that this table of 5 x 9 is more elaborate than most languages, and not every entry has the same frequency, even if -u and ki- serve different cases -u for persons too; ki- for both questions and subsentences. Linguists sometimes find this not a positive feature of Esperanto, though IMHO natural languages provide the foundation. Ido removed the table.

And of course on may say

ĉiuj ĉi, ĉi ĉiuj = all these
ĉial ĉi = for all these reasons
tiu ĉi = this

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