AIT tried solving it? But AFAIK it's a lot of pretty results with not much real application.
A better approximation is something of a "shared model"; then you can actually state things like, the transfer of information sometimes is "trivial" because, well, it's right there in your compressor/decompressor.
An Outline of a Theory of Semantic Information by Carnap was the early attempt.
Fred Dretske wrote Knowledge and the Flow of Information in 1981.
Luciano Floridi has a few recent books.
I couldn't find much else. I don't think AIC really solves the problem of meaning either.
I think the Dretske book was the first time I really understood where Shannon was coming from but I gave up when it got to his actual semantic ideas.
I think I ran across a recent paper that motivated trying to back track what work had been done in this area but I don't recall the name of the paper.
I've have shelved all this for now as over my head.