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For example, if the Republican party wanted to hold a new primary in 2020 and pick someone other than Trump, is there any specific reason they could or could not do so?

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Absolutely, it happens all the time. An incumbent losing to a challenger in their own party's primary election is sometimes called "getting primaried".

Search for "getting primaried" or "primaried out", you can find lots of articles and books:

It has happened to several presidential incumbents as well, according to NPR's When Has A President Been Denied His Party's Nomination?

  • John Tyler, Whig, 1844
  • Millard Fillmore, Whig, 1852
  • Franklin Pierce, Democrat, 1856
  • Andrew Johnson, Democrat, 1868
  • Chester Arthur, Republican, 1884

Of these 5, only Pierce was elected prior to his primary loss. The other 4 took their office in other circumstances (death of the prior president).

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    Also worth noting that the parties can technically just change their rules and run anyone they want. Would obviously cause a big scandal and very severe repercussions from voters, but technically possible. – David says Reinstate Monica Mar 7 '17 at 17:49
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    Right. The "traditional" method, of course, is for another candidate to get more votes within whatever primary method has been already established. But primaries/caucuses are pretty much up to the state party to begin with, so yes, they could just change the rules. – BradC Mar 7 '17 at 17:51
  • Teddy Roosevelt had a somewhat weird situation as well. The Republicans probably should have primaried Taft in favor of him, but they didn't and it (may have) cost them the WH. – Kevin Mar 7 '17 at 22:24
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    Just having someone run against the incumbent usually is a sign that there are issues with the president, in general popularity. Being primaried, whether successful or not, is usually close to a guarantee of change of party in the White House, I believe. – PoloHoleSet Mar 7 '17 at 22:25

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