We present dPASP, a novel declarative probabilistic logic programming framework that allows for the specification of discrete probabilistic models by neural predicates, relational logic constraints, and interval-valued probabilistic choices. This expressive combination facilitates the construction of models that combine low-level perception (images, texts, etc) and common-sense reasoning, thus providing an excellent tool for neurosymbolic reasoning. To support all such features, we discuss several semantics for probabilistic logic programs that allow one to express nondeterminism, non-monotonic reasoning, contradiction, and (vague) probabilistic knowledge. We also discuss how gradient-based learning can be performed with neural predicates and probabilistic choices under selected semantics. To showcase the possibilities offered by the framework, we present case studies that exploit different semantics and constructs.