The quest for the basic fuzzy logic

The quest for basic fuzzy logic was initiated by Petr Hájek when he proposed his basic fuzzy logic BL, complete with respect to the semantics given by all continuous t-norms. Later weaker systems, such as MTL, UL or $$mathrm{psMTL}^r$$, complete with respect to broader (but still meaningful for fuzzy logics) semantics, have been introduced and disputed the throne of the basic fuzzy logic. We contribute to the quest with our own proposal of a basic fuzzy logic. Indeed, we put forth a very weak logic called $$mathrm {SL}^ell $$, introduced and studied in earlier works of the authors, and propose it as a base of a new framework which allows to work in a uniform way with both propositional and first-order fuzzy logics.