Encoding de Finetti's coherence within Lukasiewicz logic and MV-algebras

TL;DR

This paper explores the algebraic and proof-theoretical foundations of a probabilistic logic based on Lukasiewicz logic, establishing soundness, completeness, and analyzing the unification problem within this framework.

Contribution

It introduces a translation from probabilistic logic to Lukasiewicz logic, proving key properties and extending unification theory to probabilistic reasoning.

Findings

Proof-theoretical properties for FP(L,L) established

A class of algebras for probabilistic logic introduced

Probabilistic unification shown to be of nullary type

Abstract

The present paper investigates proof-theoretical and algebraic properties for the probability logic FP(L,L), meant for reasoning on the uncertainty of Lukasiewicz events. Methodologically speaking, we will consider a translation function between formulas of FP(L,L) to the propositional language of Lukasiewicz logic that allows us to apply the latter and the well-developed theory of MV-algebras directly to probabilistic reasoning. More precisely, leveraging on such translation map, we will show proof-theoretical properties for FP(L,L) and introduce a class of algebras with respect to which FP(L,L) will be proved to be locally sound and complete. Finally, we will apply these previous results to investigate what we called "probabilistic unification problem". In this respect, we will prove that Ghilardi's algebraic view on unification can be extended to our case and, on par with the Lukasiewicz propositional case, we show that probabilistic unification is of nullary type.