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There is such a grammar

string_literal ::= " { graphic_character } "

The double-quotes that you see must appear in the user program and curly braces mean "any number of graphic characters in the string". The characters are defined further

graphic_character ::= basic_graphic_character | lower_case_letter | other_special_character
basic_graphic_character ::=
upper_case_letter | digit | special_character | space_character 

with special characters listed

" # & ' () * + , - . / : ; < = > ? @ [ ] _ `

I have encountered a situation where I need the double quote to be the part of the string. This is admitted indeed: double quote is a first special character and it enters the basic graphic characters and, thus, graphic characters. So, I can legally put it into my string. But, this means that terminating quotation mark will be considered a part of the string also and parsing will fail. The language specification says that we need to double the quotation marks to produce a single qmark in the string. I do not get why this requirement it is not a part of the grammar. How should proper grammar had to look in this case?

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Just replace " in the list of possible characters with "".

That might have consequences for other grammar productions which use the same non-terminals, in which case you might need to separate special_character into special_character_not_quote and ".

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