The differential semantics of Lukasiewicz syntactic consequence

Abstract

The classical condition "$\phi$ is a semantic consequence of $\Theta$" in infinite-valued propositional \L ukasiewicz logic \L$_\infty$ is refined using enriched valuations that take into account the effect on $\phi$ 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 $\phi$ 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 $\vdash$.