The differential semantics of Lukasiewicz syntactic consequence

Abstract

The classical condition "$\varphi$ is a semantic consequence of $\Theta$" in infinite-valued propositional Łukasiewicz logic Ł${}_{\infty }$ is refined using enriched valuations that take into account the effect on $\varphi$ of the stability of the truth-value of all $\theta \in \Theta$ under small perturbations (or, measurement errors) of the models of $\Theta$. The differential properties of the functions represented by $\varphi$ and by all $\theta \in \Theta$ naturally lead to a new notion of semantic consequence ${\models }_{\partial }$ that turns out to coincide with syntactic consequence $⊢$.