What references should I follow if I want to learn more about semantic entailment in multi-valued logics?

For fuzzy logic (where a truth value is a real number in the continuum interval $[0,1]$) I suggest the following book: Metcalfe, Gabbay, Olivetti: Proof Theory for Fuzzy Logics (2009). It has a proof-theoretical approach but the first chapter is about semantics; the definition of semantic entailment is on page 33.

For a quick but accurate introduction to many-valued logics (note that there are many many-valued logics) and their semantics, I suggest this article (2015) on Stanford Encyclopedia of Philosophy, and this paper by Gottwald (2005).

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Fuzzy Logic

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Nonclassical Logic

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