In the first part of this paper, I show how one natural methodological approach to linguistic problems leads directly to a framework which emphasizes the generalization of the problem of composition to a number of dimensions. In the second part, I introduce a general notion of compositional function, which provides an appropriate abstract setting in which to discuss the problem of multi-dimensional composition. In this general setting, we may think of a grammar G,much as Montague did (1974, Chapter 7, hereafter "UG'), as the closure of a set of compositional functions over a set of postulated basic expressions (each endowed with properties in the various dimensions under consideration). There are then a number of ways of conceiving of languages associated with G, depending on whether we wish to consider every analysis generated by G,some special class of analyses generated by G (such as those associated with a designated symbol "S'),the phonological structures definable in the structure G,and so on.