Structural completeness in fuzzy logics. (English) Zbl 1190.03027

Cintula, Petr; Metcalfe, George

Notre Dame J. Formal Logic 50, No. 2, 153-182 (2009).

Structural completeness means that each admissible, i.e., theoremhood-preserving, inference rule is derivable. The authors study this property for a range of well-known t-norm-based mathematical fuzzy logics.
They give general methods to establish this property. Among other interesting results, they prove structural completeness for the product logic and the cancellative hoop logic, prove a suitable weakening of this property for the strict monoidal t-norm logic, and show that the logic of Wajsberg hoops (a fragment of the Łukasiewicz logic) as well as the logic of basic hoops (a fragment of the basic fuzzy logic) miss this structural completeness.

Reviewer: Siegfried J. Gottwald (Leipzig)

MSC:

03B52	Fuzzy logic; logic of vagueness
03B22	Abstract deductive systems
03B47	Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03B50	Many-valued logic

Keywords:

structural completeness; mathematical fuzzy logics; substructural logics; residuated lattices