A logical treatment of possibilistic conditioning

• Authors: Enrico Marchioni
• Venue: ECSQARU'05 Proceedings of the 8th European conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty
• Year: 2005

The notion of conditional possibility derived from marginal possibility measures has received different treatments. However, as shown by Bouchon-Meunier et al., conditional possibility can be introduced as a primitive notion generalizing simple possibility measures. In this paper, following an approach already adopted by the author w.r.t. conditional probability, we build up the fuzzy modal logic FCΠ, relying on the fuzzy logic FC∏, so as to reason about coherent conditional possibilities and necessities. First we apply a modal operator ⋄ over conditional events ϕ∣χ to obtain modal formulas of the type (ϕ|χ)⋄ whose reading is “ϕ∣χ is possible”. Then we define the truth-value of the modal formulas as corresponding to a conditional possibility measure. The logic FCΠ is shown to be strongly complete for finite theories w.r.t. to the class of the introduced conditional possibility Kripke structures. Then, we show that any rational assessment of conditional possibilities is coherent iff a suitable defined theory over FCΠ is consistent. We also prove compactness for rational coherent assessments of conditional possibilities. Finally, we derive the notion of generalized conditional necessity from that of generalized conditional possibility, and we show how to represent them introducing the logic GFCΠ.