Consequences of the minimum specificity principle on conditioning and on independence in possibility theory

The consequences of Dubois and Prade’s minimum specificity principle are shown under a continuous t-norm T , when dealing with conditioning and independence in possibility theory. The minimum specificity principle singles out a particular sub-class of T -conditional possibilities (referred to as T -DP-conditional possibilities) adhering to a suitable axiomatic definition that relies on the residuum of T . Such a sub-class differentiates from the larger class of T -conditional possibilities in terms of non-closure with respect to pointwise limits, non-connectedness of extension sets, and Kolmogorov-like representation. We then switch to coherence for a partial assessment in both frameworks, highlighting that, under T -DP-conditioning, coherence of the global assessment cannot be characterized in terms of coherence on every finite sub-family. Finally, both for T - conditioning and T -DP-conditioning, we introduce an independence notion that implies logical independence and we investigate the differences due to minimum specificity.