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.
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