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The linked question explains why writing isn't language, but the embolded sentence is alleging something different, that writing isn't even "part of language". Why?

Linguistics usually divides language into various levels. In Figure A.1, meaning, sound, and writing are considered part of the outside world, not part of language. Language is the cognitive structure linking meaning and sound. We can see from this figure that semantics is the part of language most closely related to meaning, and that phonetics is the part of language most closely related to sound. Writing has a more complicated relationship to language in that units of writing are commonly related to both morphology and phonology, but not generally to semantics, syntax, or phonetics. To illustrate the different levels, we will analyse the sentence Mary purchased a new bookcase; this analysis will be followed by other examples for each level.

Henry Rogers, Writing Systems (2004), p 281.

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It is somewhat puzzling, since this reflects the viewpoint of generative grammar that "language" is a specific innate cognitive faculty (made up of phonetics, phonology, morphology, syntax and semantics). I never met the gentleman and I have no idea what he thought of the theory: perhaps someone who knew him can address the question of his beliefs as to what "language" is.

More specifically, the term "language" is used to refer to two different kinds of things. One is to refer to a component of an individual mind, the (mental) grammar of their language – nowadays referred to as "I-language". The other is "the external manifestations of a collection of I-languages", that is, E-language, for example "English" or "Korean". E-language includes the actual utterances performed by one or more speakers, and they may be performed in writing or in speaking.

He does clearly endorse the position that "Language is a complex system residing in our brain which allows us to produce and interpret utterances", which is the generative theory of I-language. He also says (emphasis added) "Although writing is not language, writing does represent language, and in our definition, only language". As he points out, people often say that Hebrew has no vowels, which is false applied to the Hebrew language but close enough to true as a statement about the system for writing the Hebrew language.

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    Not generative grammar as such, but Chomskyan MITish generative theories. Other generative theories never accepted the mysticism of an entirely mental discipline. We generative semanticists, for instance, were always happy to distinguish between what people actually said (in detail, with measurements), and what they thought they said.
    – jlawler
    May 30 at 15:49
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    We've had this confusion here for some years, where the popular view of "generative" is "whatever Chomsky says" but the actual definitions by Chomsky in Aspects are more convoluted, and it actually hinges on his peculiar definition of "grammar". This is one of those lose-lose propositions, where there is no safe answer that both informs the popular audience and satisfies the reader who knows the non-popular details. However, your comment intrigues me because it suggests something about Gen. Sem. that I had not assumed, and motivates a question.
    – user6726
    May 30 at 15:56
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    I'm not sure that failing to inform the popular audience should be viewed as a loss. It's more of a general limitation on abstractions; the popular audience doesn't usually see the point, and there isn't much one can do about that. Mysticism, on the other hand, if delivered with sufficient panache and rigamarole, often works better.
    – jlawler
    May 30 at 16:04

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