Recently, though, a few legal scholars, too, have engaged in what others call First Amendment heresy. What does clear and present danger mean when terrorists are provoking violence over the Internet? Should not the government have a way, they ask, to block messages that facilitate terrorist acts?

The existing standard is often illustrated by the classic example of shouting “Fire!” in a crowded theater when there is no hazard. That is not protected speech because it could cause a deadly stampede. But an article praising the merits of causing stampedes, even offering phrases to shout, is not closely enough linked to an imminent, actual threat to be outlawed. In November, Cass R. Sunstein, a Harvard law professor and former Obama administration official, broached the subject in an article on Bloomberg View. He called the clear and present danger test “the greatest American contribution to the theory and practice of free speech.” In view of the Islamic State’s successful use of the Internet to nurture terrorists, he said, “it’s worth asking whether that test may be ripe for reconsideration.”

A more forceful case and a legislative proposal were put forth by Eric Posner, a professor of law at the University of Chicago, in an article for Slate. “Never before in our history have enemies outside the United States been able to propagate genuinely dangerous ideas on American territory in such an effective way,” Mr. Posner wrote. The Islamic State’s ability to spread “ideas that lead directly to terrorist attacks,” he said, “calls for new thinking about limits on freedom of speech.”

Mr. Posner supported urging companies like Facebook and YouTube to crack down on propaganda by the Islamic State, which is also known as ISIS or ISIL, but said that could never be fully effective. He proposed, in addition, passing a law to deter potential consumers from viewing dangerous sites. While the law would apply to all Internet users, his goal, admittedly limited, is to head off the radicalization of those he described as “naïve people” who research the Islamic State out of curiosity, “rather than sophisticated terrorists.”

His proposal would make it illegal to go onto websites that glorify the Islamic State or support its recruitment, or to distribute links to such sites. He would impose graduated penalties, starting with a warning letter, then fines or prison for repeat offenders, to convey that “looking at ISIS-related websites, like looking at websites that display child pornography, is strictly forbidden.”