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Tuesday, Dorsey said neither Jones nor his InfoWars account will be removed.

"We didn't suspend Alex Jones or Infowars," Dorsey tweeted. "We know that's hard for many but the reason is simple: he hasn't violated our rules.

"We'll enforce if he does. And we'll continue to promote a healthy conversational environment by ensuring tweets aren't artificially amplified."

The social media chief acknowledged Jones' sensational claims but said they don't automatically cross the terms of Twitter's user policies.

"Accounts like Jones' can often sensationalize issues and spread unsubstantiated rumors, so it's critical journalists document, validate, and refute such information directly so people can form their own opinions," Dorsey said. "This is what serves the public conversation best."

Dorsey's tweet echoed an earlier statement from Twitter, which said it's "constantly moderating content on its platform and will take action if it needs to."

It also noted that some of the content banned from other platforms hasn't shown up on Twitter.

Jones, a self-professed conservative, recently threatened to "take down" special prosecutor Robert Mueller, the leader of the Justice Department's Russia investigation the radio host called a "demon."

Twitter's decision not to ban Jones follows efforts by the company to improve what it's called "conversation health" among users. Last month, it began removing millions of accounts deemed inactive or suspicious. The moves are partly motivated by scrutiny paid to Twitter and other social platforms over the spreading of phony news and politically-charged ad content aimed at influencing voter behavior in the United States.