BL (logic)
In mathematical logic, basic fuzzy logic, the logic of the continuous t-norms, is one of the t-norm [fuzzy logics]. It belongs to the broader class of substructural logics, or logics of residuated lattices; it extends the logic MTL of all left-continuous t-norms.
Syntax
Language
The language of the propositional logic BL consists of countably many propositional variables and the following primitive logical connectives: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.The following are the most common defined logical connectives:Weak conjunction, also called lattice conjunction. Unlike MTL and weaker substructural logics, weak conjunction is definable in BL asNegation, defined asEquivalence, defined as disjunction, also called lattice disjunction, defined asTop, also called one and denoted by or, defined as
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:
- Unary connectives
- Binary connectives other than implication and equivalence
- Implication and equivalence
Axioms
A Hilbert-style deduction system for BL has been introduced by Petr Hájek. Its single derivation rule is modus ponens:The following are its axiom schemata:
The axioms and of the original axiomatic system were shown to be redundant and. All the other axioms were shown to be independent.