I doubt they will exchange the 20 inch monitor.
I doubt they would exchange the 20 inch monitor.
Which is correct, and why?
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Both are acceptable, but the usage is slightly different. The following extensions to the sentence illustrate the most common usage:
I doubt they will exchange the 20 inch monitor when I take it back.
I doubt they will exchange the 20 inch monitor if I take it back (but I may try to).
I doubt they would exchange the 20 inch monitor if I took it back (but I'm not going to).
In general, would is used in situations hypothetical, or contingent on some unfulfilled condition; will is used in situations that are more certain. The first sentence takes will because the speaker definitely intends to take the monitor back. The second uses will because, although there is a condition, there is a definite possibility of it being fulfilled. The third sentence takes would because the speaker has no intention of taking it back—and therefore the condition is entirely hypothetical.
Well, it is difficult to distinguish as the context is missing, I mean, what does a monitor has to do with anything mentioned in the previous sentences?
Furthermore, the last sentence is broken, or at least an article is missing there. Either they "will exchange a 20-inch monitor", or "they will exchange the 20-inch monitor".
Grammatically, both sentences are correct, but depending on the context, which we don't know here, one of them would be more suitable.