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Here's my situation: I'm currently proof-reading something a colleague of mine has written. The content is all okay, but his use (or non-use) of "the" is all over the place, because I assume his native language (Persian) doesn't have definite articles. (I'm not a native speaker either, but German has very similar grammar, so I guess it's easier for me.)

Anyway, I realized that I don't have a quick and easy rule for when "the" gets used and when it doesn't. Especially in the context of scientific writing. Some techniques or procedures get a "the" and some don't. For example you'd say "We apply Principal Component Analysis to our data" but you'd say "We then use the Fourier transform to...".

Is there a good source I could refer my colleague to so he can brush up a bit on his usage of "the"?

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    It's a reasonable question with an unpleasant answer. No, there is no quick and easy rule for articles. Instead there are many slow and difficult rules, almost all idiomatic, arbitrary, and limited to specific circumstances, like the use of the in I dialed the wrong number (*a wrong number is ungrammatical). These each have to be learned, like separate words do. – John Lawler Oct 31 '18 at 21:49
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    @JohnLawler What about “I dialed a wrong number yesterday”? – Dan Bron Nov 4 '18 at 2:38
  • That's OK. But the idiom is dialed the wrong number, and deviations from idiomatic language draw attention. – John Lawler Nov 4 '18 at 16:13
  • I'd love to know a good answer, but I suspect there is no such thing, at least "quick and easy"… The best textbook written in Japanese exclusively about the use of the articles in English is over 300 pages long (by Glenn Paquette), despite the fact it assumes the readers have advanced knowledge about English, and I am afraid the book is yet far from complete. So I suspect a "quick and easy" rule wouldn't cover anything more than superficial. – Masa Sakano Nov 5 '18 at 9:14
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The use of the definite article is not what you want. What you want is to learn about the quality of Definiteness, and this is not wrapped up in the use of the definite article but in the use of definite noun phrases. No one has yet come up with a single theory that accounts for all uses of definite noun phrases in English. Much of it has to do with what the speaker assumes about his interlocutor (hearer).

For a start, you could consider this answer on the main site; which is partially based upon Birner & Ward's Uniqueness, Familiarity, and the Definite Article in English (pdf); Abbott's Definiteness and indefiniteness (pdf); Epstein's The definite article, accessibility, and the construction of discourse referents (pdf); important also is Indefinite "this" and the givenness hierarchy (pdf); (while teaching helps such as Over Usage of the Definite Article Among Speakers of English as a Second Language in Sri Lanka: Reasons and Solutions and other resources here may also be of help); after that, you can do searches for other resources.

The plethora of resources and the linguistic and metaphysical theories that inform them is why this is not a topic that can be summarized in a pithy manner.

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