"An X of an Y" or "The X of an Y"?

Important: All of the examples below are assumed to be used when we say something like "a cat" and not "the cat". That is, for example, when we are introducing something for the first time.

The 1st part of the question: "x of y" where x is unique:

• a name of a file
• the name of a file

Which version is correct, and why?

• Logically, this should be "the name of file". This is because each file can only have one name.
• But then, logically, this means that we should never say "a file name", and we should always say "the file name" instead. But nobody follows this logic: people say "the file name" only in those cases where they would say "the cat" instead of "a cat".

The 2nd part of the question: "x of y" where x is not unique:

• a fragment of a document
• the fragment of a document

Which version is correct, and why?

• 'A name of a file' would be at best unusual. 'A filename' would be far more likely for an unspecified filename. "We need to come up with a filename." 'The filename' is reserved for a specified filename.

• 'The name of a file' is idiomatic; the definite article is licensed/required because of post-specification (it's the name of a file). "Isn't it annoying when you can't remember the name of a/some file!"

• 'A fragment of a document' shows that both the fragment and the document are (as yet) unspecified.

• 'A fragment of the document' shows that the document in question is known, while the fragment is hitherto unspecified.

• 'The fragment of a document' is again unusual. If the fragment has been specified but the document is unidentified (it has of course been specified to a degree if the fragment has) 'the document fragment' or 'the fragment' would be used.