In its other ruling last Thursday, United States v. Alvarez, the Court held, by a 6-3 vote, that the Stolen Valor Act (which makes it a crime to falsely claim to have received military decorations or medals) violates the First Amendment. There was no majority opinion. Rather, a four-justice plurality (opinion by Kennedy, joined by the Chief, Ginsburg, and Sotomayor) combined with two justices concurring in the judgment (opinion by Breyer, joined by Kagan). Alito wrote the dissent, joined by Scalia and Thomas.
In brief:
Kennedy would treat bars on false statements as content-based speech restrictions subject to strict scrutiny, and he concludes that the Stolen Valor Act cannot survive such scrutiny.
Breyer offers some standard of intermediate scrutiny.
Alito would hold that false statements merit no First Amendment protection “in their own right” but instead receive such protection only if their prohibition “would chill other expression that falls within the Amendment’s scope.”
For more, I encourage you to read Eugene Volokh’s post, where he explains more fully that “lurking behind [the 6-3 holding] was a more complicated 4-2-3 split that was in some ways a 5-4 split in favor of treating lies as generally less constitutionally protected.”