The standard line about private free speech is that it doesn't exist, that as soon as a private venue operator decides that someone has to shut up, they have to.
That's not entirely accurate.
Free speech principles extend beyond the legal protections against government censorship and speech control. It's not simply a matter of free speech ending where the law ends. Free speech is also a social value. If an organization dedicates itself to free speech, or has a forum with a particular topic, it is a violation of the free speech principles of that organization for it to deny the forum to someone who has a "wrong" opinion. This is even more true for organizations that have free speech and inquiry as a reason to exist.
Let me contrast two hypothetical examples:
(This is inspired by the article here, which covers a different, but related, subject.)
1) A florist shop that refuses to provide services to a gay wedding, on the grounds that such an event violates their beliefs on what marriage is.
2) A scientific forum with multiple speakers denying a speaker a slot expressly because they have a religious belief regarding the requirements of marriage.
The florist shop is expressing it's speech by refusing to endorse, by providing service, to a gay wedding. It's refusal is limited expressly to the topic with which it disagrees, and does not extend to any other aspect of the refused customer's life. It does not deny the refused customer the chance to perform their ceremony, but rather simply refuses to be part of it.
The forum, on the other hand, is taking a scientist who was invited to speak and rescinding the invitation because the forum does not agree with the unrelated views of the scientist on an unrelated topic. The forum claims it supports free inquiry, as that is the cornerstone of science, but denies a scientist a slot because he holds the "wrong" views. Here, the forum violates it's own free-speech principles.
Colleges do this with some frequency. Their promotional materials will boast about free inquiry being the element of university life, and how all opinions are welcome, and then ban all speech from nearly all of campus, and allow speech in one tiny area but only with administrator pre-approval. While public universities violate the First Amendment when they do this, private universities violate the principles that justify their existence when they do.
Reddit, also, was long a free speech-dependent and -centered venue, but recently they have begun enforcing, at least in some areas. Despite the allowance of some truly creepy subforums, they've begun looking the other way when high-level moderators begin co-ordinating censorship campaigns. They haven't violated the First Amendment, but they have violated their principles of free-speech.
A final note on the word "censorship": Censorship is not just government suppression of speech. When an organization suppresses information, either by redacting it as [ablist slur] or simply refusing to release the information because they don't want it out, that counts as censorship.
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