One might (why? I don't know why, but I did think this at some point) think that a distinction between thought and language, together with the idea that the only meaning language has is speaker meaning expressive of a thought, would help with the liar paradox. After all, it seems that the liar sentence is a merely linguistic curiosity, and does not express a thought.
But a variant of the liar escapes this. I am going to describe this variant through a story that seems to make it very plausible that there is a thought being expressed. Frank has two diaries. A blue one for insightful ideas and a gray one for humdrum observations. He feels that his insights haven't been very good lately. He thus thinks to himself, on good empirical evidence, that today is a day on which nothing true gets written down in the blue diary. He then reaches for his gray diary, and writes down this pessimistic thought: "Today is a day on which nothing true gets written down in the blue diary." However, although he had reached for the gray diary, he ended up writing down his thought in the blue one, because he was rather absentminded. So now his blue diary contains the entry: "Today is a day on which nothing true gets written down in the blue diary." Moreover, let us suppose that he writes nothing else in the blue diary today.
What I like about this story is it makes the paradoxical sentence have a genuine use in our language, rather than just being an excrescence in the way the standard liar sentence is.
It seems that what is written in the blue diary expresses a genuine, and even epistemically justified, thought that Frank had. But what is written in the blue diary is true if not true and false if true, according to his intentions. So the move to speaker meaning doesn't help.
If we add the further assumption that what is in the blue book successfully expresses Frank's thought, then Frank's thought is true if not true and false if true. So unless we abandon classical logic, we need to say that Frank's thought was not expressed by what he wrote. But why not? Had he written it in the gray book, it would have expressed the thought. And he wrote the same thing in the blue book. So there doesn't seem to be a mismatch between word and thought—the words seem just the right ones to do justice to the thought. Maybe, though, some variety of future-based externalism is true: maybe what, if any, thought has been thought can depend on what will transpire (e.g., on whether Frank writes something or not). That's a weird idea, but even apart from the Hegelianism that was rampant where I did my PhD, there is independent reason to accept that. E.g., I might dub Kenya's next child "Patrick." And then I might think to myself what seems to be the thought that Patrick will be a girl. But if Kenya doesn't have another child, then I haven't thought anything. But this line of thought leads to the very disquieting conclusion that introspection is not a perfect guide to whether I am having a thought—and that undercuts the cogito.
It seems that in these cases we must go for the least paradoxical claim. Denying classical logic is most paradoxical. So we either need to say that the meaningfulness of the sentence depends on where I happened to write it down, or I have to say that the meaningfulness of the thought depends on future events. Oddly, my present intuitions pull me towards the second, though the first seems less weird. Epistemic akrasia?