Abstract
Classical logic can deal with uncertainty caused by incomplete information and, beside the standard interpretation-based semantics, also has a semantics in terms of modalities expressing “known to be true”, “known to be false”, where binary-valued possibility theory and necessity measures are instrumental. When uncertainty is caused by inconsistency, the standard semantics of classical logic collapses, but the other one survives provided that we replace necessity measures by more general set functions. This is the topic of this paper where we show that various inconsistency-tolerant inferences for propositional bases, some of which defined at the syntactic level only, have a semantics in terms of binary-valued capacities.
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Notes
- 1.
BUT we cannot express the situation when \(E\not \subseteq [p]\) (i.e. \(K\not \models _E p\)) in the language \(\mathcal {L}\). Likewise, we do not have that \(K\models _E p\vee q\) if and only if \(K\models _E p\) or \(K\models _Eq\) (only the if part holds). To fully capture the epistemic semantics, the syntax needs a modality \(\Box \) in front of \(\mathcal {L}\)-formulas so as to properly express the statement \(E \subseteq [p]\) as \(N(p) = 1\) [1].
- 2.
In [14], it is called QC logic, where QC stands for qualitative capacities.
- 3.
\(N_k\) is the necessity measure associated with \(MC_k\).
- 4.
The convention for representing epistemic states is in agreement with possibility theory, but it is opposite to Dunn-Belnap’s for whom the (now conjunctive) set {0, 1} represents \(\textbf{C}\) (at the same time true and false) and the empty set represents \(\textbf{U}\) understood as the lack of information.
- 5.
Remember that values \(\textbf{U}\) and \(\textbf{C}\) play strictly equivalent roles for the truth ordering in \(\mathbb {V}_4\).
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Dubois, D., Prade, H. (2022). A Capacity-Based Semantics for Inconsistency-Tolerant Inferences. In: Dupin de Saint-Cyr, F., Öztürk-Escoffier, M., Potyka, N. (eds) Scalable Uncertainty Management. SUM 2022. Lecture Notes in Computer Science(), vol 13562. Springer, Cham. https://doi.org/10.1007/978-3-031-18843-5_8
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