I In the case of the first sentence, this is due to the mere usage that has been made of the noun "knowledge". It happens that for a given meaning this word belongs to two categories, that of the uncoutable nouns and that of the singular nouns (cannot be used in the plural).
(OALD) [uncountable, singular] the information, understanding and skills that you gain through education or experience
♦ scientific/technical knowledge and skills
♦ intimate/first-hand/basic knowledge
♦ a thirst for knowledge
♦ It will be an opportunity to gain knowledge and experience.
♦ knowledge of something
♦ She has acquired a detailed knowledge of the subject.
♦ They enjoy sharing their knowledge of the business.
♦ He had no prior knowledge of the language before visiting the country.
♦ He has a wide knowledge of painting and music.
It follows that you cannot say "She has acquired several/many/… detailed knowledge of various subjects.", but in the singular there is no problem.
However, you have a choice, and you can suppress the article; you are then considering "knowledge" as uncountable.
- The position requires a knowledge of German.
- The position requires knowledge of German.
requires knowledge of, page of examples
II In the case of the second sentence, the problem has nothing to do with the countability of nouns; it is a matter of determination.
CoGEL § 5.58
In English, noncount abstract nouns usually have no article when used
generically:
♦ My favourite subject is history/French/mathematics/music …
♦ Happiness is often the product of honesty and hard work.
♦ Theory must go hand in hand with practice.
Normally the zero article also occurs when the noncount abstract noun is
premodified:
[…] when the same noun is postmodified, especially by an of-phrase, the
definite article normally precedes it:
♦ She's studying *history of Europe.
♦ She's studying the history of Europe.
We thus find typical contrasts of the following kind:
human evolution - the evolution of man
medieval art - the art of the Middle Ages
Oriental philosophy - the philosophy of the Orient
18th century morality - the morality of the 18th century
III A chapter of a book. / The chapter of a book.
For the simple reason that there are usually several chapters in a book, if you can't be specific about any or if you don't want to be specific you use "a".
- A chapter of the book I studied last was difficult.
You can be more specific, but as long as there are other chapters of the sort you describe you use "a".
- I read an easy chapter and skip two. (There are probably other easy chapters.)
If you want to mention precisely a given chapter that is unique, then "the" is necessary.
- I read the first chapter, but it'll be a long time before I get to the last one.
By itself, "The chapter of a book" is meaningless; there must be some postmodification that will say what is unique about it.
- The chapter of the book that I have been talking about is the sixth.
- The chapter of the book you need to know well is that on transformations.
Addition from 11/04/2022 concerning a comment by user Edwin Ashworth (Your first two sentences are unqualified ('These are the facts ...'). Is the OALD classification 'singular nouns' (1) accepted by CGEL, (2) accepted generally by grammarians (note that the term 'noncount'/'uncountable' etc is accepted by OALD, CGEL, and as far as I know most grammarians [for usages of nouns]) and (3) helpful when 'The position requires a knowledge of German' is idiomatic but 'The position requires one knowledge of German' isn't? I believe you're quoting two contradictory sources and claiming them both as fact.)
As an explicit categorization of nouns the OALD classification "singular noun" is found uniquely in the OALD. According to this dictionary, it is a classification meant to comprise all nouns that are not used in the plural (from the point of view of a certain definition of what is the concept "plural"), and thus it incompasses a rather large variety of nouns. In the words of the dictionary, many nouns like this can be used only in a limited number of ways, for example some singular nouns must be or are often used with a particular determiner in front of them or with a particular prepostion in front of them. It might be useful to remark—a remark of mine which I shall hazard—that those words are singular by virtue either of the fact that they denote a unique entity ("sun") or of the fact that it wouldn't come easily to someone's mind to use them in the plural ("knowledge"). A short list of these nouns follows; some of the particularities are given.
[compulsory "the"] the Sun, the matter (only in "what's the matter?"), the unthinkable, the impossible, the moment (exact point in time, "at the moment"), the gun (signal by firing pistol), the background [C, usually singular, U], the chair
[compulsory "the", proper nouns] the Arts and Crafts Movement, the Ivy League, the British Council,
[no article, proper noun] Britain,
[compulsory "a"] time ("a long time", "for a time", exceptions: "at one time", "in my time" )
trunk (main part of the human body) [usually singular]
[no article, "the"] sunrise, (usually singular), market (ex.: "their share of the market", "market values"),
[no article, "the", "a"]
knowledge (of/about sth) [U, sing.] (definition: the information, understanding and skills that you gain through education or experience)
["a", "the"] A to Z, busman's holiday, wonderland [usually singular],
Because of the straightforwardness of its nature, it is clear that the category "singular" as used in the OALD is justified; however it fits into a scheme of classification based upon a given set of criteria that is different from that in the CoGEL; that does not mean that contradictions result but instead that different points of view form the basis of the choice of criteria; thus, whereas in the CoGEL uncount nouns are also considered singular nouns because of the simple fact that [the idea of plurality does not exist for that category / no plural form exist], in the OALD it is not so, and uncount nouns cannot be considered as singular because this concept is reckoned with as presupposing countability, the possible extent of a counting restricted to just 1 being no hindrance; in other words, on the whole, the singular classification is a refinement of the countable in which are set apart those nouns for which counting starts and stops at 1. Whereas according to CoGEL practices uncountable nouns (gold), abstract adjective heads (the impossible, the unreal) and most proper nouns (Henry) are SINGULAR INVARIABLE NOUNS, following the OALD, only the abstract adjective heads are SINGULAR (INVARIABLE) NOUNS. It could be said that if the notion of plurality in the grammar is morphological, in the dictionary it is more fundamental, and semantic.
This should answer the question about the acceptation, there is no common ground in virtue of which to accept or reject anything. Thus, I believed there is no ground for talking about contradiction. As to the second part of the question, there is not much I can say because my knowledge of the literature on the subject is rather scanty; I can say, however, that in all the explanations I have had an occasion to read I never found a discussion about the approach adopted in the OALD.
The third part is more difficult to answer; it is helpful to the extent that the inquirer is aware that the singular noun is a very special sort of noun as far as its determination is concerned; this is a warning to the enquirer that, relatively to the possibilities of determination, idiomaticity plays a much greater part than in the case of regular countable nouns; otherwise, it is not helpful and merely helps to seize the nature of the word, which in itself is very important to who wants to gain a deeper understanding of the nature of nouns. I believe nevertheless that it has to be mentioned because it eliminates from the scope of all possible determination all plural determiners.
