The notion of restricted arrow has been introduced by Beall et al. to analyse restricted quantification, and is characterized as a particular relevant conditional by semantic means of a certain ternary relation in the so-called Routley-Meyer semantics for entailment. In this paper, the author presents a range of positive propositional substructural logics with the restricted arrow, and shows that they are also sound and complete with respect to suitably defined frameworks on the base of Routley-Meyer semantics. For the completeness proof, the usual one-step method by canonical model construction does not work. It is then carried out by a step-by-step method which has been applied for the completeness proof of some modal logics.