In a logagraphical script, from what I’ve understood (mainly from YouTube), every character represents a word or concept, with little to no attachment to the pronunciation of the idea itself. Theoretically, could one learn one of these scripts (for example Chinese) without learning the phonological aspects of the associated language(s) and communicate with pencil and paper?

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Probably, although there’s no reason that can’t be true for any language with a writing system. Even if you ignore phonology altogether you still have to learn that language’s rules for grammar and syntax. All that’s to say you’re basically just interacting with language the same way a deaf person would, and most deaf people know how to read.