Two-layered logics for probabilities and belief functions over Belnap--Dunn logic

TL;DR

We address reasoning about uncertainty in Belnap–Dunn logic using two-layered formalisms. The outer layer uses Łukasiewicz-based arithmetic to reason about probabilities and belief/plausibility defined on BD events, with two probabilistic perspectives: $\pm$-probabilities and $4$-valued probabilities, and two belief frameworks: $\mathsf{Bel}^{\hbox{\sf L}^2}_\triangle$ and $\mathsf{Bel}^{\mathsf{N}{\hbox{\sf L}}}$; the paper provides sound and complete Hilbert axiomatisations for the $4$-valued probabilities, faithful embeddings between the $\pm$ and $4$ probabilistic logics, and NP/CoNP complexity results for satisfiability and validity across the probabilistic and belief-logics. It develops BD-based belief/plausibility formalisms and shows how these can be embedded into modal probabilistic BD logics, yielding a unified, decidable framework for paraconsistent uncertainty. These contributions advance a modular, two-layered approach to combining non-classical logic with quantitative uncertainty, with precise completeness and complexity results that support automated reasoning in BD settings.

Abstract

This paper is an extended version of an earlier submission to WoLLIC 2023. We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Lukasiewicz $[0,1]$-valued logic with Baaz $\triangle$ operator and the Belnap--Dunn logic. We consider two probabilistic logics that present two perspectives on the probabilities in the Belnap--Dunn logic: $\pm$-probabilities and $\mathbf{4}$-probabilities. In the first case, every event $φ$ has independent positive and negative measures that denote the likelihoods of $φ$ and $\negφ$, respectively. In the second case, the measures of the events are treated as partitions of the sample into four exhaustive and mutually exclusive parts corresponding to pure belief, pure disbelief, conflict and uncertainty of an agent in $φ$. In addition to that, we discuss two logics for the paraconsistent reasoning with belief and plausibility functions. They equip events with two measures (positive and negative) with their main difference being whether the negative measure of $φ$ is defined as the belief in $\negφ$ or treated independently as the plausibility of $\negφ$. We provide a sound and complete Hilbert-style axiomatisation of the logic of $\mathbf{4}$-probabilities and establish faithful translations between it and the logic of $\pm$-probabilities. We also show that the satisfiability problem in all logics is $\mathsf{NP}$-complete.

Figures (3)

• Figure 1: A counterexample to \ref{['equ:exportimport']}: $\mu(\{u_1\})=\mu(\{u_2\})=\frac{1}{3}$, $\mu(W)=1$, and $\mu(\varnothing)=0$. All variables have the same values exemplified by $p$.
• Figure 2: The values of all variables coincide with the values of $p$ state-wise. $\mu(X)=\frac{1}{2}$ for every $X\subseteq W$; $\pi(\varnothing)=\pi(\{w'_1\})=0$, $\pi(W')=1$, $\pi(X')=\frac{1}{2}$ otherwise.
• Figure 3: $\mu(\{w_1\})=\frac{1}{3}$, $\mu(\{w_2\})=\frac{1}{2}$, $\mu(\{w_3\})=\frac{1}{6}$.

Theorems & Definitions (84)

• Definition 2: Set semantics of $\mathsf{BD}$
• Remark 3
• Definition 4: $\mathsf{BD}$-models with $\pm$-probabilities
• Definition 6: $\mathsf{BD}$-models with $\mathbf{4}$-probabilities
• Example 9: Paraconsistent vs. classical probabilities
• Definition 10
• Definition 11: ${\hbox{\sf\L}}_\triangle$
• Definition 12: ${\hbox{\sf\L}}^2_{(\triangle,\rightarrow)}$
• Remark 14
• Definition 15: $\mathbf{4}\mathsf{Pr}^{{\hbox{\sf\L}}_\triangle}$: language and semantics