Strictly speaking, there are only three words in the English Language that are called articles Webster, 1994. These are 'a' (or sometimes 'an', although these two are simply the same word inflected differently), 'the' and sometimes 'no'. Articles indicate the way in which a noun is being referenced, so that we know whether the noun in question is indicating a singular specific thing or a genus of things. If the reference is being made to a specific individual article, we say that the noun is being referenced definitely. For example:
I just kicked the cat.
The phrase above is talking about a specific, 'definite' cat. We therefore use the definite article 'the'. Whereas:
I just kicked a cat.
Although the statement above is talking about a singular cat, it is not specific - or in other words, we do not know the identity of the cat. Since the specific cat cannot be readily inferred, we say that the reference to the noun 'cat' is indefinite. We thus use the indefinite article 'a', or in some cases 'an'. On the other hand:
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No cat has been kicked.
The above statement uses the word 'No' as a negative article. That is, it is used to indicate that the noun 'cat' references the lack of a cat.
There is a fourth type of article, although it does not use a word. Rather, it uses the lack of an article to suggest a definite article as in, 'like a duck to water'. When the zero article is used in English, it is a substitute for the definite article, and so the phrase 'like a duck to the water' would mean the same thing. Generally, the zero article may be used interchangeably with the definite article in cases where the noun refers to an organization or mass noun.
The decision of whether to use 'a' or 'an' as the indefinite article, uniquely among grammatical protocols, is based on the way in which the following word is pronounced (Ibid.). If the following word starts with a vowel sound, or an unstressed H, the article must be 'an'. If the word starts with a stressed H or a consonant sound, the article must then be 'a'.
In the above letter, the articles are used thusly:
I went to the(1) library yesterday in order to obtain the(2) book you recommended in your lecture yesterday
The first two articles are examples of the definite article 'the' being used to show that the following noun (in this case 'library' and 'book') is a subject that the reader will be acquainted with. That is to say that the author assumes that the recipient will know which library and book the letter refers to. Since a definite book and library can be identified, the article used is the definite.
but I was told that I had to fill in a(3) reservation form. That is why I am writing to you now.
The author then uses the indefinite article 'a' to refer to a single, nonspecific example of a group of things called reservation forms. A good test of whether to use the indefinite article is whether the noun phrase 'a reservation form' can be replaced in the statement by the phrase 'a reservation form (any reservation form)' - which of course it can.
My name is Camara Kuffour and I am a Senegalese visiting scholar, studying at Tokyo University. I have a(4) library card from (zero article)(5) University of Dakar in Senegal.
The next article is once again indefinite and can pass the 'a library card (any library card)' test. After that, there is an instance of the zero article, which as we have said above functions as a definite article. It could instead be "the University of Dakar".
The(6) enclosed document shows my home university registration number. I understand that the(7) library card my home university provided cannot be used here, so I would be grateful if you could provide me with a(8) letter to obtain a university of Tokyo card. You may contact Dr Joseph Ndour who will be happy to provide the(9) reference he usually writes for me. I look forward to hearing from you.
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The remaining articles follow the rules we have outlined above to indicate when the noun phrase is or is not referring to a subject that is generally well-known. There is conspicuously no use of the word 'an', and correctly so.
There is a further article in the letter, used to reference 'a university of Tokyo card'. Although the article is not highlighted in the text, it is worth noting that even if a noun begins with a vowel its appropriate indefinite article may be 'a'. 'University' is a good example of this, and the reason that it is accompanied by 'a' is that it starts with a consonant sound rather than a vowel sound (phonetically, 'yoo-nee-vers-it-ee'.)