英语中为什么加冠词a？

请问这三句话语法上对嘛？可是查了很多资料也不好理解为什么加冠词？什么时候不加冠词…

Jack was president of the school.

John was a student of the school.

John was a better president than Jack.

Let me quickly go over the general rules of when to use
(a) “a”,
(b) “the”, and
(c) neither “a” or “the”.

You use “a” for any one in a group of 2 or more. In Chinese, it is “某” (没有指定的)一个。(Note: “a” can be used with a singular noun only).

You use “the” for a particular one (or more) you have in mind. In Chinese, it is “那” (指定的)。(Note: “the” can be used with singular or plural noun).

You use neither “a” or “the” when you discuss one or more items in general. In Chinese, it is “普遍来说”。(Note: not using “a” or “the” can be applied to singular or plural noun”.

Note that often you can choose to use “a”, or “no-a-or-the”, at your option.

These are all correct:

1.      A home is a man’s castle until the queen shows up. (任何一个家是任何一个人的城堡直至那皇后出现).

2.      Home is man’s castle until the queen shows up. (家【普遍来说】是人【普遍来说】的城堡直至那皇后出现).

3.      A home is man’s castle until the queen shows up. (任何一个家是人【普遍来说】的城堡直至那皇后出现).

Notice we use “the” for the “queen” because she is designated to be the “queen” of the particular home we just talked about.

Let’s apply what you have just learned to your examples:

1.      Jack was president of the school. (There is no “a” or “the” in front of “president” because “president” is a title. It would be easier to understand if you say: “Jack was President (or ‘president’) of the school. (Note 1)

2.      John was a student of the school. (Jack is one (没有指定的任何一个) of the students of the school)。

3.      John was a better president than Jack. (In the history of the school, there were many school presidents. Some were better than Jack, but some were worse than Jack. John was “a” (one of the many) better president than Jack.

Let me give you a couple of more examples.

You have two churches on your street. You went to one of them yesterday. So you say “I went to a church on my street yesterday”. If there is only one church on your street and you went there yesterday, you say “I went to the church on my street yesterday”.

Similarly if your brother has two sons and one of them came to visit you yesterday, you say “A son of my brother came to visit me yesterday”. If your brother only has one son and he came to visit you yesterday, you say “The son of my brother came to visit me yesterday”.

I hope it helps.  

Note 1: Example 1 (Jack was president of the school)

If there is only 1 president in that school, “Jack was the president of the school” is correct.

If there are more than 1 president in that school (e.g., President of Geography Club, President of Arts Club, etc.), then “Jack was a president of the school” is correct.

In either of those examples, “Jack was president of the school” (i.e., without “a” or “the” and not intended to be a title) is not that correct.