John Stuart Mill made a famous argument for free speech in On Liberty. Famous, but not very good.
The gist of Mills argument is as follows (and of course I simplify):
1) We are not infallible in our judgments, therefore the ideas whose expression we would ban might just be true; and
2) Even if, per impossibile, we are certain that some ideas are false, banning their expression is still bad, because true ideas are only really known when they have to be regularly defended against falsehood.
There are several problems here:
1) What Mill is doing here is of course expressing an idea. And per his own doctrine, we can't really be sure that the idea is true. So the notion of banning the expression of certain ideas always has to be kept on the table as a live option. But Mill clearly wants to rule censorship completely out of bounds, something that per his doctrine can't be done with any idea!
2) Mill's arguments seem to extend seamlessly to the world of actions. After all, often the very point of expressing some idea is to prompt some action. Mill is obviously not in favor of, say, making genocide legal. But it is not at all clear to me why, given the case Mill is making for free speech, it is okay to ban genocide but absolutely wrong to ban the advocacy of genocide. Surely the advocacy of genocide is usually the first step in any actual genocide. If we have to allow the advocacy because, after all, can we really be certain that the idea is wrong, then by similar reasoning shouldn't we allow the advocates to try it out on occasion? Do we really know if it's a bad idea until we test it out in practice? Of course I am not in favor of genocide, but then I also don't think it's a bad idea for Germany to ban the expression of Nazi ideas, given certain events that occurred there last century.
3) The last problem I wish to note is practical rather than theoretical, but nonetheless rather pressing at the moment. Per Mill, we should never ban the expression of political ideas by groups like the Muslim Brotherhood. But expressing those ideas is a step on the way to getting them enacted, and when they are enacted, books like Mill's On Liberty will be banned. In order to be good sports, do liberals have to allow this to happen?