The differential semantics of Lukasiewicz syntactic consequence

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

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