The Paradox of Tolerance is an idea created by the philosopher Karl Popper that argues, basically, that tolerating the expression of speech that is intolerant of other ideas will lead to the destruction of the tolerant environment that allowed it to form, as intolerance gains prominence. I've noticed that it's commonly appealed to by progressives online as an explanation for why silencing nazis, bigots, and the like is acceptable despite the importance of free speech.
The central issue with the Paradox is that "intolerance" in speech just can't be objectively determined. Very few people that actively support a belief will believe that it is "intolerant" - it will always be their belief that is tolerant, and the opposing belief the intolerant one.
An example here that's particularly relevant for progressives is anti-zionism/anti-semitism. Over the last couple years especially, we've seen firsthand how accusations of intolerance, in this case antisemitism, have been weaponized in order to attempt to silence legitimate anti-zionist speech. Numerous countries and institutions have adopted the IHRA's ridiculous [antisemitism definition](https://holocaustremembrance.com/resources/working-definition-antisemitism) which includes "claiming that the existence of a State of Israel is a racist endeavor" and "applying double standards by requiring of Israel a behavior not expected of any other democratic nation" as examples of supposedly antisemitic speech. A society that accepted that intolerant, bigoted speech ought to be silenced would be deeply vulnerable to this sort of twisting of definitions being used to censor genuine speech that is controversial.
We can see some of this twisting already happening in the proscription of the anti-zionist group Palestine Action in the UK. Palestine Action is a protest group that engages in direct action through trespassing, vandalism, and property destruction of companies and organizations that support Israel's military actions. Last year, they trespassed onto a military base and sprayed paint into the engines of planes, causing significant damages, and pissing off the government enough that they decided to declare them a proscribed terrorist organization. Nevermind that any reasonable person wouldn't consider what the group did to rise to the level of "terrorism" a la flying planes into buildings or mass killing civilians - the decision was up to the government. In fact, courts have now determined that the proscription was inappropriate, but it is still in effect for now.
And, because the UK lacks broad speech protections and has laws against speaking in favor of proscribed groups, simply holding up a cardboard sign with the words "I support Palestine Action" has become enough to get you arrested - and it has gotten many people arrested, in huge numbers, tons of peaceful protestors and aging grandmas hauled away into the paddy wagon for supporting "terrorism" in scenes that couldn't make it more obvious that the UK doesn't care one bit about allowing its citizens to express their beliefs. "Surely there's no problem with banning people from advocating for intolerant, dangerous terrorists" seems like a sensible enough statement, but somebody has to decide who's a terrorist, and who's intolerant, and that somebody in this case was a government that had already taken a side on the issue and sneers at anyone who opposes its view.
Apart from the fact that intolerance is subjective, the Paradox gives unearned power to intolerant speech through it's assumption that intolerant beliefs will, if allowed to fester, always spread and prevail over other beliefs. It argues that any opposing ideology to bigotry or fascism is simply too unconvincing to win out in any environment where both are present - that the forbidden fruit of hateful speech is just so overwhelmingly alluring that it has to be sealed away in order to protect our society. It's a argument that's both cowardly and condescending - it's a refusal to combat hate speech directly on an even playing field, and an assumption that an idiotic populace will always be drawn in by bigotry if it is allowed to be present.
The utility for progressives of appealing to the Paradox is in giving a way to justify silencing reprehensible beliefs while also pretending to still adhere to ideals of free expression. You cannot have it both ways. Censoring intolerant speech or hate speech *is* a condemnation of the principle of free speech, and it's one that actively works against any activist who supports controversial causes like anti-zionism. It is not worth it to compromise on free expression just to get rid of bigots or fascists that you'd rather not see espousing their repulsive ideas, and the Paradox of Tolerance does not give you an excuse to do so. CMV.