Considering the execution path and cut operators of a Prolog program can improve the precision of program analysis. Known semantics for Prolog either makes use of limited amount of path information and hence leads to less precise analysis or is goal dependent and therefore not suitable for goal independent program analysis. This paper deals with the problems by proposing a goal-independent denotational semantics for Prolog with cut, from which we can compute the set of partially computed answers associated to each program point that are obtained in the execution of any goal. With existing abstraction techniques this semantics can be abstracted into a finitely computable semantics that can serve as a basis for goal-independent Prolog program analysis.