Strong conjunction (binary). 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 (
nullary — a
propositional constant); or are common alternative signs and zero a common alternative name for the propositional constant (as the constants bottom and zero of substructural logics coincide in MTL).
The following are the most common defined logical connectives:
Weak conjunction (binary), also called lattice conjunction (as it is always realized by the
lattice operation of
meet in algebraic semantics). Unlike
MTL and weaker substructural logics, weak conjunction is definable in BL as
(Weak) disjunction (binary), also called lattice disjunction (as it is always realized by the
lattice operation of
join in algebraic semantics), defined as
Top (nullary), also called one and denoted by or (as the constants top and zero of substructural logics coincide in MTL), defined as
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).
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 [0, 1] with the usual order; they are uniquely determined by the function that interprets strong conjunction, which can be any continuous
t-norm.
Bibliography
Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.
Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic20: 177–212.
Cintula P., 2005, "Short note: On the redundancy of axiom (A3) in BL and MTL". Soft Computing9: 942.