It's important not to concede the premise that First Amendment protections are subordinate to the public interest at all. Carr argues in his statement, after all, that the FCC has to take action because the public is losing faith and confidence in the media altogether. But even if the FCC can produce a detailed, convincing explanation of how American democracy will suffer if they're not allowed to block certain viewpoints from the airwaves, they still can't do it.
