Abstract
This paper defends the use of possibility and necessity models based on the Logics of Formal Inconsistency, taking advantage of their expressivity in terms of the notions of consistency (\(\circ \)) and inconsistency (\(\bullet \)). The present proposal directly generalizes the approach of Besnard and Lang (Proceedings of 10th Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann, San Francisco, pp. 69–76 1994), whose main guidelines we borrow here. Some basic properties of possibility and necessity functions over the Logics of Formal Inconsistency are obtained and it is shown, by revisiting a paradigmatic example, how paraconsistent possibility and necessity reasoning can, in general, attain realistic models for artificial judgement. We will call such models credal calculi, emphasizing some of their appealing consequences.
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Notes
- 1.
There is another sense of trivialism, according to which everything is true. This should not be confused with deductive trivialism.
- 2.
This is independent from the fact that classical logic endorses the validity of the Principle of Non-Contradiction: (PNC) \( \quad \vdash \lnot (\alpha \wedge \lnot \alpha )\), see Carnielli et al. (2018).
- 3.
See discussions on the Derivability Adjustment Theorems in Carnielli et al. (2007).
- 4.
Shafer’s proposal as a theory of probable reasoning (and the Dempster-Shafer theory for that matter) is open to a number of objections, as discussed in Williams (1978). This aspect is not particularly relevant to our approach, since it is robust enough to correct distortions caused by a less precise belief function.
- 5.
We are supported by the following properties: (1) every real number is both an upper and a lower bound of an empty set, and (2) If \(A\subset B\) are sets of real numbers and sup exists, then \(sup A \le sup B\), see, e.g., Harzheim (2005).
- 6.
Formally, the fact that the consequence relation \(KB\,\mid \!\sim \,\phi \) is monotonic follows from results in Dubois and Prade (1991) but it can be checked that if \(KB\subset KB'\) and \(KB\,\mid \!\sim \,\phi \) it does not necessarily follow that \(KB'\,\mid \!\sim \,\phi \), in view of the role of the sup operator.
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Acknowledgements
Both authors acknowledge support from the National Council for Scientific and Technological Development (CNPq), Brazil under research grants 307376/2018-4 and 308077/2018-0. We are indebted to Jérôme Lang and Philippe Besnard for personal communications that encouraged this paper to appear, and to Lluís Godo for some sharp remarks on a previous version of this paper.
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Carnielli, W., Bueno-Soler, J. (2021). Credal Calculi, Evidence, and Consistency. In: Arieli, O., Zamansky, A. (eds) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Outstanding Contributions to Logic, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-71258-7_4
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