Possibility of Conditionals and Conditional Possibilities: From the Triviality Result to Possibilistic Imaging

Lewis-Gärdenfors imaging is an updating procedure for probability functions that generalizes Bayesian conditionalization, allowing to approach the probability of conditionals and counterfactual formulas without incurring in Lewis well-known triviality result. Precisely, while the probability of a so called Stalnaker conditional (as formalizable in Lewis logic C2) was proved to be an imaged probability by Lewis in his celebrated paper from 1976, a variant of Gärdenfors generalized imaging (proposed by Dubois and Prade in 1994) has been recently proved to characterize the probability of Lewis counterfactuals, the latter refer to those conditionals of Lewis’s logic C1. The present contribution extends the analysis on Lewis’s triviality result, imaging and generalized imaging to cope with possibility and necessity measures. In particular, after showing that the triviality result also holds in the possibilistic framework, we introduce a way to define the possibility measure of conditional and counterfactual formula. Then we prove that the possibilistic version of Lewis-Gärdenfors imaging (that is inspired by a definition given again by Dubois and Prade in 1994) actually characterizes, as in the aforementioned cases, the possibility of Stalnaker conditionals and Lewis counterfactuals. Furthermore, we show that possibilistic imaging can also be described within the setting of Boolean algebras of conditionals and Lewis algebras. These are algebraic models for conditional and counterfactual formulas recently introduced by two of the present authors. On these structures one can (canonically) define a notion of possibility measure that turns out to be the conditional possibility and imaged possibility mentioned above, respectively, and hence it represents the possibility of conditional and counterfactual formulas.