BL (logic)

Syntax

Language

• Implication
• Strong conjunction . The sign & is a more traditional notation for strong conjunction in the literature on fuzzy logic, while the notation follows the tradition of substructural logics.
• Bottom ; or are common alternative signs and zero a common alternative name for the propositional constant.

• Weak conjunction , also called lattice conjunction. Unlike MTL and weaker substructural logics, weak conjunction is definable in BL as
• Negation , defined as
• Equivalence , defined as
• disjunction , also called lattice disjunction, defined as
• Top , also called one and denoted by or , defined as

• Unary connectives
• Binary connectives other than implication and equivalence
• Implication and equivalence

  Axioms

Semantics

• General semantics, formed of all BL-algebras — that is, all algebras for which the logic is sound
• Linear semantics, formed of all linear BL-algebras — that is, all BL-algebras whose lattice order is linear
• Standard semantics, formed of all standard BL-algebras — that is, all BL-algebras whose lattice reduct is the real unit interval with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any continuous t-norm