The last decade has seen an enormous development in infinite-valued systems and in particular in such systems which have become known as mathematical fuzzy logics.
The paper discusses the mathematical background for the interest in such systems of mathematical fuzzy logics, as well as the most important ones of them. It concentrates on the propositional cases, and mentions the first-order systems more superficially. The main ideas, however, become clear already in this restricted setting.
"Mathematical fuzzy logics." Bull. Symbolic Logic 14 (2) 210 - 239, June 2008. https://doi.org/10.2178/bsl/1208442828