This is an interesting point. I haven't really thought about the connection between intuitionistic logic and three-valued logic before. They are different logi

This is an interesting point. I haven't really thought about the connection between intuitionistic logic and three-valued logic before. They are different logics in general though. 3-value logic is more like the A is True, False or None/Unknown. Another interpretation one can put on intuitionistic logic is game semantics. Here a statement is true if the verifier has a winning strategy to prove the proposition. It's false if the falsifier has a winning strategy. Then intuitionistic logic corresponds roughly to the idea that some games don't have a winning strategy for either player. So here (and in the proofs example I gave above) is that we get an idea of "neither true nor false" that is represented by the lack of an explicit rule for concluding truth or falsity in some cases. In contrast, 3-valued logic gets at the idea of "neither true or false" by introducing another symbol (e.g. "None" or "-1") together with new operators that operate on sentences with that symbol. So we get one from classical logic by deleting something (namely the law of the excluded middle) and the other by adding something.