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 $(\forall x\varphi {)}^2\leftrightarrow \forall x{\varphi }^2$ and the infinitary rule \[ \frac{ϕ\vee (α\to β^n):n \in \mathbb{N}}{ϕ\vee (α\to α\& β)} \] to achieve strong completeness with respect to continuous t-norms.