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

TL;DR

This paper extends He1jek's axiomatic system for first-order logic based on BL-chains to achieve strong completeness for the predicate logic of continuous t-norms by adding new axioms and rules.

Contribution

It introduces new axioms and an infinitary rule to extend the existing system, establishing strong completeness for continuous t-norms.

Findings

Achieves strong completeness with respect to continuous t-norms

Extends He1jek's system with new axioms and rules

Provides a more robust logical framework for fuzzy logic

Abstract

The axiomatic system introduced by H\'ajek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom $(\forall x \phi)^2 \leftrightarrow \forall x \phi^2$ and the infinitary rule \[ \frac{\phi \vee (\alpha \to \beta^n):n \in \mathbb{N}}{\phi \vee (\alpha \to \alpha \& \beta)} \] to achieve strong completeness with respect to continuous t-norms.