What references should I follow if I want to learn more about semantic entailment in multi-valued logics? I am simply curious about this subject, and would like to learn more about it.
I am an undergraduate student and in our studies we've always tackled classical logic and simply mentioned that other "logics" exist.
As such I would like to be pointed towards an undergraduate friendly reference on this topic.
 A: 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).
A: There are a lot of different versions of "multivalued" logic.  One that has been studied quite a lot recently is known as "continuous first-order logic".  This logic takes truth values in [0, 1] instead of {T, F}, and has structures that are based on (complete) metric spaces rather than just sets.  The interesting thing about this logic is that a lot of the theorems from classical model theory carry over to the continuous setting (though often with trickier proofs).
The standard reference for this logic is http://math.univ-lyon1.fr/~begnac/articles/mtfms.pdf.  It's not really aimed at undergraduate readers, but if you have some background in first-order model theory (and a bit of real analysis) you will probably be able to get a reasonable idea of what is going on.  The approach in that paper is entirely semantic (there is a proof theory for this logic, but the model theory was developed first).
A: The essential concept of semantic entailment is easily lost in verbiage and complex symbolism. The two-valued material conditional has the properties of a mathematical ordering relation on the truth values of propositions, so that P -> Q signifies that Q is at least as true as (or is not less true than) P.
This idea resolves most of the difficulties in understanding the two-valued conditional, but does not seem to be widely employed in either explanations of it or in multivalued logic.
