BL (logic)

Basic fuzzy Logic (or shortly BL), the logic of the continuous t-norms, is one of t-norm fuzzy logics. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic of all left-continuous t-norms MTL.
Syntax.
Language.
The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives:
The following are the most common defined logical connectives:
Well-formed formulae of BL are defined as usual in propositional logics. In order to save parentheses, it is common to use the following order of precedence:
Axioms.
A Hilbert-style deduction system for BL has been introduced by Petr Hájek (1998). Its single derivation rule is modus ponens:
The following are its axiom schemata:
The axioms (BL2) and (BL3) of the original axiomatic system were shown to be redundant (Chvalovský, 2012) and (Cintula, 2005). All the other axioms were shown to be independent (Chvalovský, 2012).
Semantics.
Like in other propositional t-norm fuzzy logics, algebraic semantics is predominantly used for BL, with three main classes of algebras with respect to which the logic is complete: