Strong completeness for the predicate logic of the continuous t-norms

Abstract

The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains.In this study, we extend this system with the axiom $(orall x phi)^2 leftrightarrow orall x phi^2$ and the infinitary rule[rac{phi ee (alpha o eta^n):n in mathbb{N}}{phi ee (alpha o alpha & eta)}]to achieve strong completeness with respect to continuous t-norms.Abstract 2:We study the S5-modal expansion of the logic based on the L ukasiewicz t-norm. We exhibit a finitary propositional calculus and show that it is finitely strongly complete with respect to this logic. This propositional calculus is then expanded with an infinitary rule to achieve strong completeness. These results are derived from properties of monadic MV-algebras: functional representations of simple and finitely subdirectly irreducible algebras, as well as the finite embeddability property. We also show similar completeness theorems for the extension of the logic based on models with bounded universe.