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For my linguistics course I was asked to explain the difference in meaning of the following:

1. I would like to have lunch with you on Friday
2. I would like to have lunch with you on a Friday

Also, a similar example:

3. We will have a test the next week
4. We will have a test next week

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What did you say the differences were? – Andrew Leach Jan 29 '13 at 11:17
How are the examples similar? They are not similar at all. Anyway, related: “The end of next week” or “the end of the next week”, “In the next two weeks” vs. “next two weeks” – RegDwigнt Jan 29 '13 at 11:22
Agh, you are right! Moreover, I've solved the second set: "the next week" (which by the way is grammatical) means seven days from now. "Next week" means next week starting on Monday. – Paulina Mazur Jan 29 '13 at 11:26
I disagree on "the next week". Also, whether a week starts on Sunday, Monday, sunset of Friday evening, sunset of Saturday evening, or any other possibility depends on cultural differences and convention in different contexts, not the English language. – Jon Hanna Jan 29 '13 at 11:33
See Fillmore's Deixis Lectures on Time. – John Lawler Jan 29 '13 at 18:03

"On Friday". Normally next Friday. It could be also the Friday which occurs within a given context ("I'll visit for Christmas day and then see you on Friday" would mean the Friday after Christmas).

"On a Friday". On some day which is a Friday, but not stating which.

"We will have a test the next week". This only makes sense if there is already a context. Hence:

"We will be studying calculus until the 8th of February. We will have a test the next week". means there will be a test some time in the period 9th-15th of February.

With a stated context, the the anchors the timeframe to that context. Otherwise it could be unclear whether "next week" mean "next after what we just discussed" or "next after now".

"We will have a test next week". Means the week after now.

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With the first example, it's saying you would like to have lunch on that coming Friday. The next one is saying lunch could be on any Friday.

The second example would be better worded as 'we will have a test IN the next week, but (b) is also correct.

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Actually...my friend's notes say that the situation is quite the opposite with reference to the first example. (which puzzled me too) Also, it's not a matter of correctness but of being able to point out the difference. – Paulina Mazur Jan 29 '13 at 11:20

In the first pair, on Friday means next Friday, and on a Friday means any Friday.

In the second pair, (b) is grammatical and, in the context, (a) is not.

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  1. I would like to have lunch with you on Friday means that , that persons wants to have lunch with you on the upcoming Friday.

  2. I would like to have lunch with you on a Friday that the person might be available on Fridays , and says to you that he can have lunch with you on any Friday in which you are available.

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