Semantics
Algebraic semantics is predominantly used for propositional t-norm fuzzy logics, with three main classes of algebras with respect to which a t-norm fuzzy logic is complete:
- General semantics, formed of all -algebras — that is, all algebras for which the logic is sound.
- Linear semantics, formed of all linear -algebras — that is, all -algebras whose lattice order is linear.
- Standard semantics, formed of all standard -algebras — that is, all -algebras whose lattice reduct is the real unit interval with the usual order. In standard -algebras, the interpretation of strong conjunction is a left-continuous t-norm and the interpretation of most propositional connectives is determined by the t-norm (hence the names t-norm-based logics and t-norm -algebras, which is also used for -algebras on the lattice ). In t-norm logics with additional connectives, however, the real-valued interpretation of the additional connectives may be restricted by further conditions for the t-norm algebra to be called standard: for example, in standard -algebras of the logic with involution, the interpretation of the additional involutive negation is required to be the standard involution rather than other involutions which can also interpret over t-norm -algebras. In general, therefore, the definition of standard t-norm algebras has to be explicitly given for t-norm logics with additional connectives.
Read more about this topic: T-norm Fuzzy Logics