Why 'a friend of mine' is not 'my friend's friend'?

Solution 1:

This is perhaps best explained by providing the relevant extract from the ‘Longman Student Grammar of Spoken and Written English’:

The double genitive is a special construction in which either the independent genitive or a possessive pronoun occurs in an of-phrase:

This was a good idea of Johnny’s.

There’s a talk by this lady from Boulder who’s a student of Sandy’s.

The woman who owns Harte’s is a friend of ours.

As these examples show, the main noun phrase typically begins with the indefinite article. In fact, the definite article does not normally combine with the double genitive: *the good idea of Johnny’s is unlikely to occur.

The meaning of the double genitive can sometimes be alternatively expressed by other constructions. Thus, a friend of ours could alternatively be expressed as one of our friends.

Here is a further explanation from ‘The Cambridge Guide to English Usage’:

The double genitive seems to serve two purposes:

emphasis. This is the effect of paraphrasing “not Jo’s fault" as no fault of Jo’s, or turning “our friend” into a friend of ours. The double genitive unpacks the phrase and foregrounds the noun rather than the person. In conversational examples such as That book of Bill Bryson’s is his best yet, the construction helps to adjust the topical focus.

clarification. Clearly a painting of Lady Rich’s and a painting of Lady Rich mean different things. The first (a possessive) makes the painting part of Lady Rich’s collection, while the second (technically an objective genitive) says that it is a portrait of the Lady herself. The duplication of the genitive is thus not redundant but clarifies the fact that the first construction is a possessive genitive.

Solution 2:

Barrie England's answer is useful, but doesn't address

(a) the reason that the double genitive is used

(b) the sum total of the restrictions on its use

I can't begin to answer the first of these points, but have some additional remarks to make about the second:

(1) The Longman Student Grammar of Spoken and Written English says correctly:

As these examples show, the main noun phrase typically begins with the indefinite article. In fact, the definite article does not normally combine with the double genitive: *the good idea of Johnny’s is unlikely to occur.

However, it misses the fact that a specific determiner (of the demonstrative subclass) may be freely used: I think we really should consider that good idea of Johnny's.

(2) Usage is idiosyncratic. We say a friend of mine but an enemy of Britain

The double genitive seems to be used more for more intimate / personal relationships than for more distant ones:

A friend / book of mine but Friends of the Earth

A friend of the president's far less common than A friend of the president*

NOT An enemy of France's but An enemy of France.

There seems to be a grey area:

NOT A friend of the family's but A friend of the family, and

A friend of the Smith's far less common than A friend of the Smiths.

(Non-possessive usage, but for close relationships.)