I am curious to hear people's thoughts on the question in the title. There has been a lot of philosophical work done on the idea that a sentence can be strictly speaking false, yet nonetheless approximately true (or 'truthlike' or 'verisimilar'). For example: I am 5'11", but if someone said 'Greg is 6 feet tall,' we want to say that that claim is approximately true or something like that. But what if the claim was (strictly speaking) neither true nor false? (Readers may insert their own favorite truth-valueless sentence here.)
I ask because, as I mentioned in an earlier post, I'm toying with the idea that the Pessimistic Induction over the history of science plus something like Kuhnian incommensurability (esp. untranslatability) will lead us not to the conclusion that current science is likely to be false, but rather is likely to lack a truth-value. For if we cannot translate the claims of a pre-revolutionary language into the post-revolutionary one, then the pre-revolutionary language (from our current point of view) is truth-valueless, not false.
I ask the question in the title because one common realist response to the Pessimistic induction is: "Well, yes, our current scientific theories are probably not exactly true, but they are approximately true." If truth-valueless sentences cannot be approximately true, then this response is not available to the realist.