Involutive basic substructural core fuzzy logics: Involutive mianorm-based logics

Abstract

This paper deals with the standard completeness of involutive non-associative, non-commutative, substructural fuzzy logics and their axiomatic extensions. First, fuzzy systems based on involutively residuated mianorms (binary monotonic identity aggregation operations on the real unit interval [0,1]), their corresponding algebraic structures, and their algebraic completeness results are discussed. Next, completeness with respect to algebras whose lattice reduct is [0,1], known as standard completeness, is established for these systems via a construction in the style of Jenei–Montagna. These standard completeness results resolve a problem left open by Cintula, Horčík, and Noguera in the recent Handbook of Mathematical Fuzzy Logic and Review of Symbolic Logic. Finally, we briefly consider the similarities and differences between constructions of the author and Wang's Jenei–Montagna-style.