This paper discusses Probabilistic Logic Programming (PLP) under Distribution Semantics, a leading approach to practical reasoning under uncertainty. A key advantage of distribution semantics is its ease of implementation in Prolog or Python libraries, with two well-maintained implementations available: ProbLog and cplint/PITA. However, current formulations of distribution semantics rely on point probabilities, making it difficult to express epistemic uncertainty, such as that encountered in hierarchical classification in computer vision models. Belief functions, as non-additive capacities, generalize probability measures and address epistemic uncertainty through interval probabilities. This paper presents interval-based capacity logic programs (CLPs), extending distribution semantics to include belief functions, and describes the properties of this new framework that make it suitable for practical applications.