With a general picture in mind, and judging partially from an historical perspective, we can identify two main families of non-classical logics: Modal Logics, and many-valued logics (which can be understood in a loose way as logics complete with respect to classes of algebras different from the Boolean ones). Modal logics, and a large class of MV Logics are still decidable, while offering a rich expressive level.
From the 60’s, modal expansions of Intuitionistic Logic, as well as modal versions of other finite many-valued logics were explored. While the approach to Modal IL has been tightly related to its relational semantics, for other many-valued logics (whose relational semantics has not an intuitive meaning) different definitions have been considered. One of the most studied approaches was that of defining the logics from valued Kripke models, that take a classical frame and enrich it with an evaluation that assigns truth-values different from {, } to the variables (and possibly, further valuing the accessibility relation of the frame too).
In this tutorial we will go through the systematic study developed in the latter years for the previous family of logics, focusing in the modal logics defined over residuated lattices and paying special attention to the three widely-studied cases of the infinite Gödel, Łukasiewicz and Product logics. We will show some interesting results (and open questions) in the area, ranging from basic properties of these logics, axiomatizations (and so, the new kind of canonical models we need to use), decidability (these logics do not necessarily enjoy the FMP with respect to the above semantics) and enumerability.