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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.

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  • $\begingroup$ I'm not sure what are called "multi-valued logics" are the "other" logics you should be most interested in. You can have semantics for non-classical logics that nevertheless only have two values in the semantic domain. Also, things like constructive logic aren't really about having multiple truth values. Similarly, there are plenty of semantics for classical logic with more than two truth values. Any Boolean algebra will do. $\endgroup$ Aug 10, 2019 at 1:37

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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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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).

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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.

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