TWITTER HAS provoked outrage for its refusal to remove disgraceful tweets from President Trump that further a baseless conspiracy theory about the death of a former congressional staffer. Whether anger at the social media site is justified, the debate shouldn’t obscure the obvious: The bigger problem is the president.

Twitter provided only an apology this week to the husband of Lori Klausutis after he petitioned the company to take down the “vicious lies” Mr. Trump has spread about his wife, who passed away in 2001 when she fainted and hit her head in the office of then-Rep. Joe Scarborough (R-Fla.). Twitter did, however, take action against an entirely separate set of offensive tweets from the commander in chief: false claims that mail-in ballots are fraudulent, to which the platform applied a fact-checking label with a link to authoritative information.

The story is, admittedly, a little head-spinning. Twitter has rules for offensive posts; it exempts world leaders from most of them; but it claims that those exemptions aren’t relevant here, because the tweets related to Ms. Klausutis don’t violate any terms of service about harassment, as critics allege. Twitter is looking to its rules about misinformation instead. The platform rarely takes down misleading content, but it has started to attach fact-checking labels in areas where confusion can become dangerous. One is “civic integrity,” which is why the mail-in ballot tweets were addressed. Apparently, Twitter so far hasn’t developed labeling guidelines for false allegations of murder.

What ought to be more head-spinning than all of this, however, is that the man in the Oval Office spends his days slinging out lies so copious that Twitter has multiple options for policing in the first place. Equally appalling was Mr. Trump’s reaction to Twitter’s timid fact-check: He threatened to “close them down,” having cried earlier this month that the “Radical Left is in total command & control” of social media sites. Twitter’s policies are messy, but that’s in part because the company is scrambling to serve a country whose most powerful public messenger is also its least responsible — all while maintaining the dedication to free expression that makes it a space for everybody.

Believing in democracy means believing in the ability of the people to choose their leaders with full knowledge of who they are, and for America fully to know who Mr. Trump is, his most despicable broadcasts must remain within the citizenry’s reach. Yet his messages are not only instructive in their vileness: They also have potential to do great damage. Twitter should develop a more coherent framework to deal with speech by world leaders; with real-world harm, including harassment; and with misinformation — and to articulate how they knit together in this case and in others. But it’s no wonder that its policies are a muddle. This president is a world leader who harasses and who lies, and everyone else is stuck trying to live with it.

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