Weakening-free, non-associative fuzzy logics: Micanorm-based logics

Abstract

This paper deals with standard completeness for weakening-free, non-associative, substructural fuzzy logics. First, fuzzy systems, which are based on micanorms (binary monotonic identity commutative aggregation operations on the real unit interval [0,1]), their corresponding algebraic structures, and algebraic completeness results are discussed. Next, completeness with respect to algebras whose lattice reduct is [0,1], so-called standard completeness, is established for these systems using construction in the style of Jenei–Montagna. In particular, standard completeness results for the involutive logics, which was a problem left open by Horčík in the recent Handbook of Mathematical Fuzzy Logic, are provided. Finally, we briefly consider the similarities and differences between constructions of the author and Wang's Jenei–Montagna-style.