The widely prevailing view that standard bivalent logic is the only possible sound logical system, imposed by metaphysical necessity, has been shattered by the development of multivalent logics during the 20th century. It is now clear that standard bivalent logic is merely the minimal representative of a wide variety of viable logics with any number of truth values. These viable logics can be subdivided into families. In this article, the Kleene family and the PPCn family are subjected to special examination, as they appear to be most relevant for the study of the logical properties of human language.