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Since this seems to be a pressing issue--especially at my university where politically conservative or religiously sympathetic events are under constant threat of being cancelled by angry student protesters--I would like to figure out a proper Thomist understanding of free speech if there is one.
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I can't speak for Aquinas specifically, but in general the Ancient and Medieval view concerning free-speech is that speech serves a purpose, namely expressing the truth. If someone says something false, and offers no argumention for their statements, I can't imagine someone like Aquinas or Aristotle defending one's right to do so. This seems to based on a fundamentally non-liberal conception of the state, which is so far removed from our contemporary idea of the state, that Aquinas and others probably wouldn't recognize free-speech as an issue.
An issue that does concern Ancient and Medical philosophers (and which is immensely overlooked in discussions of protests and political speech) is that of rational discussion vs. rhetoric. Often times when people claim, "I have a right to free speech" what they mean is "I want to express an irrational opinion in a highly emotional manner and not be questioned in any way." This is more properly a question of rhetoric and sophistry the it is of free speech.
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It's correct that it is not a topic which Aristotle and Aquinas wrote about.
It's in some ways a major theme of Alasdair MacIntyre's work. MacIntyre has theorized generally about conflicts of tradition. His inquiries led him to become a Thomist, but conflict is also part of his own metaphilosophy (in some ways Aristotelian and Thomistic, in other ways inspired by Karl Popper and C.S. Peirce). He thinks people have to work from within traditions, but they have to be working to respond to the best objections from without the tradition as well. He thinks that was what Aristotle was doing and what Aquinas was doing, and that is what he thinks the Thomist has to do today (albeit he will have to respond to different critics).
So despite Aristotle and Aquinas's relative illiberalism, and MacIntyre's illiberalism for that matter, one does find him arguing that what traditions need is to engage each other, which means that people need to be free to engage each other without being deterred from conversation by motives such as fear. So he will think that some form of toleration is necessary although he rejects Millian arguments for freedom of thought.
Two essays on these themes come to mind, both included in the second volume of his collected essays, Ethics and Politics: "Aquinas and the extent of moral disagreement" and "Toleration and the goods of conflict".
On the other hand, I know Thomists who basically accept Mill's views in On Liberty. One doesn't have to be a utilitarian to do so; similar arguments might be acceptable to anyone whose ethics gives some role to consequences, even if not an exclusive role.
I tend to think that Mill's empirical claims are somewhat post hoc. He wants to defend freedom of thought and speech, so he claims that there's always a good reason to doubt the truth of one's own view, and that even if there were not, the truth is always made more vivid by conflict with error. There are clearly cases where this is just not true (phrenology, say). But there's something to his view; I don't think it's unfair to say that generally arguments in philosophy are made stronger in response to opposing arguments, even if those are unsound. And disciplines in which there is not enough disagreement do (I think) produce weaker arguments. Those tendencies seem to be sufficient to justify procedural freedom of speech, if not in societies at large then certainly in universities.
I'm also inclined to think that love for our neighbors can support a defeasible obligation to meet them where they are, intellectually. It might be that I have the power to prevent someone from saying something I think is wrong. And that will prevent others from hearing him, perhaps. But there is still the person who is wrong, and one should try to talk to him for his own sake, even if one thinks one really has nothing to learn from him.
What I like about this sort of consideration is that it does not underwrite mere procedural requirements on avoiding suppression, but seems to call for an "ethics of discourse". If, with every person, we have at least an interest in sharing the truth with him, then any engagement with him should be aimed at persuasion. That doesn't mean that we should trick him--since what we want for him is knowledge, and not mere true belief. But it does mean that we should be courteous, sensitive to his concerns, honest about the shortcomings of our own arguments, etc. People are probably more inclined to recognize that you have the truth, when you have it, if you also show by your actions that you care for the truth as such.