Abstract
We discuss two-layered logics formalising reasoning with paraconsistent probabilities that combine the Łukasiewicz [0, 1]-valued logic with Baaz \(\triangle \) operator and the Belnap–Dunn logic. The first logic 
(introduced in [7]) formalises a ‘two-valued’ approach where each event \(\phi \) has independent positive and negative measures that stand for, respectively, the likelihoods of \(\phi \) and \(\lnot \phi \). The second logic 
that we introduce here corresponds to ‘four-valued’ probabilities. There, \(\phi \) is equipped with four measures standing for pure belief, pure disbelief, conflict and uncertainty of an agent in \(\phi \).
We construct faithful embeddings of 
and 
into one another and axiomatise 
using a Hilbert-style calculus. We also establish the decidability of both logics and provide complexity evaluations for them using an expansion of the constraint tableaux calculus for 
.
The research of Marta Bílková was supported by the grant 22-01137S of the Czech Science Foundation. The research of Sabine Frittella and Daniil Kozhemiachenko was funded by the grant ANR JCJC 2019, project PRELAP (ANR-19-CE48-0006). This research is part of the MOSAIC project financed by the European Union’s Marie Skłodowska-Curie grant No. 101007627.
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Notes
- 1.
In this paper, when dealing with the classical probability measures we will assume that they are finitely additive.
- 2.
- 3.
Note that \(\lnot \) does not occur in \((\alpha ^*)^-\) and thus we care only about \(e_1\) and \(v^+\). Furthermore, while n is the number of \(\phi _i\)’s, we can add superfluous modal atoms or variables to make it also the number of variables.
can be found in, e.g. [
is also given in [References
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Bílková, M., Frittella, S., Kozhemiachenko, D., Majer, O. (2023). Two-Layered Logics for Paraconsistent Probabilities. In: Hansen, H.H., Scedrov, A., de Queiroz, R.J. (eds) Logic, Language, Information, and Computation. WoLLIC 2023. Lecture Notes in Computer Science, vol 13923. Springer, Cham. https://doi.org/10.1007/978-3-031-39784-4_7
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