How can you explain this articles rule?

Solution 1:

Yes, you've mentioned the book before.

What you haven't mentioned is that it belongs in the category "interesting book". That category happens to use the same noun, but it's a different book altogether. The one book and the other book are two different books.

To better see that, replace the second book with a different noun.

• This book is a disgrace.
• This book is a delight.
• This book is a revelation.

And so, by the same token,

• This book is a book about science.
• This book is a book by Joe Smith.
• This book is an interesting book.

If instead you straight away say "This book is the interesting book", that means that it's already been established not what book you're talking about, but what interesting book you're talking about.

Solution 2:

In your example, the sentences about the book might just be separate sentences that serve as separate examples. However, if we read them as forming a single paragraph, they can describe two books: “that” book and “this” book. The interesting book refers to “this” book, which happens to be the first reference to it. By this measure, using an indefinite article satisfies your heuristics.

However, another use of the indefinite article is to signify “one element of a class”. (Singleton classes are an interesting case that still work with “a/an” in that construction, but that’s a sidetrack.) Your computer example works well here: “This is a computer .... It is a good [one].” Here, “a” functions functions to highlight the adjective good. You can think of this mechanically as “this is one of those in the class of good things”. Saying it is the good one goes a step further to apply good exclusively to that instance.

This is consistent with your intuition. In your question, you use “a” and “an” this way (I’ve changed the formatting to show this more clearly here):

• That book isn't thin. - "That" is a demonstrative pronoun.

• This is an interesting thick book. - Why is there an indefinite article?

You’ve already referenced the demonstrative and the article within the immediate context, but still use “a” and “an”. That’s because you’re not using the indefinite article(s) to introduce new nouns, but to qualify or to highlight/describe them. That’s what’s happening with the computer and book examples you quoted.