Possibilistic conditioning framed in fuzzy logics

Abstract

The notion of conditional possibility derived from marginal possibility measures has received different treatments. 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 Rational Pavelka Logic RPL, 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 suitably defined theory over FCΠ is consistent. We also prove compactness for rational coherent assessments of conditional possibilities. We derive the notion of generalized conditional necessity from the notion of generalized conditional possibility, and we show and discuss how to represent those concepts introducing some logics generalizing FCΠ. Finally we show how to frame qualitative comparative relations in this logical framework.