Abstract
We present a propositional and a first-order logic for reasoning about higher-order upper and lower probabilities. We provide sound and complete axiomatizations for the logics and we prove decidability in the propositional case. Furthermore, we show that the introduced logics generalize some existing probability logics.
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Notes
The notation is motivated by the logic \({\mathsf {LUPP}}\) from Savić et al. (2017b), where LUP stands for “lower and upper probability”, while the second P indicates that the logic is propositional. We add I to denote iteration of upper and lower operators.
Also, note that the notions conjunctive and disjunctive are relative in this specific context, since the type of connective is closely related to the type of inequality that an operator uses. For example, the definition \(U_{\ge s}^G\alpha \equiv \bigvee _{a \in G}U_{\ge s}^a\alpha \) is equivalent to the “conjunctive’ definition \(U_{\le s}^G\alpha \equiv \bigwedge _{a \in G}U_{\le s}^a\alpha \).
Here, for simplicity, we can assume that all the singletons are in \(\mathcal G\), in order to capture the operators indexed by individual agents.
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This work was supported by the SNSF Project 200021_165549 Justifications and non-classical reasoning, by the Serbian Ministry of Education and Science through Projects ON174026, III44006 and ON174008, and by ANR-11-LABX-0040-CIMI.
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Doder, D., Savić, N. & Ognjanović, Z. Multi-agent Logics for Reasoning About Higher-Order Upper and Lower Probabilities. J of Log Lang and Inf 29, 77–107 (2020). https://doi.org/10.1007/s10849-019-09301-7
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DOI: https://doi.org/10.1007/s10849-019-09301-7