This article introduces a new theory of intention representation which is based on a structure called a Dynamic Intention Structure (DIS). The theory of DISs was motivated by the problem of how to properly represent incompletely specified intentions and their evolution. Since the plans and intentions of collaborating agents are most often elaborated incrementally and jointly, elaboration processes naturally involve agreements among agents on the identity of appropriate agents, objects and properties that figure into their joint plans. The paper builds on ideas from dynamic logic to present a solution to the representation and evolution of agent intentions involving reference to incompletely specified and, possibly, mutually dependent intentions, as well as the objects referenced within those intentions. It provides a first order semantics for the resulting logic. A companion paper extends further the logical form of DISs and explores the problem of logical consequence and intention revision.