We observe that some disjunctions admit interpretations richer than the boolean interpretation by communicating intended orderings between disjuncts. Such disjunctions, which we denote as ordered disjunctions, can be encoded by disjunctions with parentheses of the form A (or B). We present a linguistic analysis of disjunctions with parentheses, and categorize their ordered interpretations into three types: preference, likelihood, and appropriateness of wording. We argue that the source of the orderings encoded by disjunctions with parentheses is pragmatic effects of parentheses weakening information inside while emphasizing information outside, derived from the basic function of parentheses to mark parentheticals. We study reasoning involving ordered disjunction, showing that ordered disjunction do not satisfy the commutative law, and it gives rise to novel types of free choice inferences and ignorance inferences. These phenomena are captured in our proposed model based on the framework of Bilateral State-based Modal Logic (BSML).