Cuetos and Mitchell (1988) observed that in constructions in which a relative clause can attach to one of two possible sites, English speakers prefer the more recent attachment site, but Spanish speakers prefer the least recent attachment site, in violation of the proposed universal principle Late Closure (Recency Preference), which favors attachments to the most recent sites. Based on this evidence, Cuetos and Mitchell concluded that Late Closure is not a universal principle of the human sentence processing mechanism. In this paper, we provide new evidence from Spanish and English self-paced reading experiments on relative clause attachment ambiguities that involve three possible attachment sites. The results of our experiments suggest that a principle like Late Closure is in fact universally operative in the human parser, but that it is modulated by at least one other factor in the processing of relative clause attachment ambiguities. We propose that the second factor involved in the processing of these and related constructions is the principle of Predicate Proximity, according to which attachments are preferred to be as structurally close to the head of a predicate phrase as possible, and we further consider the origins and predictions of the theory combining these two factors.