The Herbrand model H of a definite logic program P is an initial model among the class of all the models of P, interpreting P as an initial theory. Such a theory (program) proves (computes) only positive literals (atoms) in P, so it does not deal with negation. In this paper, we introduce isoinitial semantics for logic programs and show that it can provide a rich semantics for logic programs, which can deal with not just negation, but also incomplete information, parametricity and compositionality.