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

Abstract This paper is an extended version of Bílková et al. ((2023b). Logic, Language, Information, and Computation. WoLLIC 2023 , Lecture Notes in Computer Science, vol. 13923, Cham, Springer Nature Switzerland, 101–117.). We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Łukasiewicz $[0,1]$ -valued logic with Baaz $\triangle$ operator and the Belnap–Dunn logic. We consider two probabilistic logics – $\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ (introduced by Bílková et al. 2023d. Annals of Pure and Applied Logic , 103338.) and $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ (from Bílková et al. 2023b. Logic, Language, Information, and Computation. WoLLIC 2023 , Lecture Notes in Computer Science, vol. 13923, Cham, Springer Nature Switzerland, 101–117.) – that present two perspectives on the probabilities in the Belnap–Dunn logic. In $\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ , every event $\phi$ has independent positive and negative measures that denote the likelihoods of $\phi$ and $\neg \phi$ , respectively. In $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ , 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 $\phi$ . In addition to that, we discuss two logics for the paraconsistent reasoning with belief and plausibility functions from Bílková et al. ((2023d). Annals of Pure and Applied Logic , 103338.) – $\mathsf {Bel}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ and $\mathsf {Bel}^{\mathsf {N}{\mathsf {\unicode {x0141}}}}$ . Both these logics equip events with two measures (positive and negative) with their main difference being that in $\mathsf {Bel}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ , the negative measure of $\phi$ is defined as the belief in $\neg \phi$ while in $\mathsf {Bel}^{\mathsf {N}{\mathsf {\unicode {x0141}}}}$ , it is treated independently as the plausibility of $\neg \phi$ . We provide a sound and complete Hilbert-style axiomatisation of $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ and establish faithful translations between it and $\mathsf {Pr}^{\mathsf {\unicode {x0141}}^2}_\triangle$ . We also show that the validity problem in all the logics is $\mathsf {coNP}$ -complete.