The differential semantics of Lukasiewicz syntactic consequence

TL;DR

This paper introduces a refined semantic consequence relation in infinite-valued Łukasiewicz logic using differential properties of truth functions, establishing its equivalence with syntactic consequence.

Contribution

It defines a new differential-based semantic consequence relation in Łukasiewicz logic that aligns with syntactic derivability, enhancing the understanding of stability under perturbations.

Findings

The differential semantics coincides with syntactic consequence.

Enriched valuations capture stability effects on truth-values.

New notion of consequence refines classical semantics.

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 $\vdash$.