This project is an attempt to give an account of the logical operators in a variety of settings, over a broad range of sentential as well as nonsentential items, in a way that does not rely upon any reference to truth conditions, logical form, conditions of assertability, or conditions of a priori knowledge. Furthermore, it does not presuppose that the elements upon which the operators act are distinguished by any special syntactic or semantic features. In short, it is an attempt to explain the character of the logical operators without requiring that the items under consideration be “given” or regimented in some special way, other than that they can enter into certain implication relations with each other. The independence of our account from the thorny questions about the truth conditions of hypotheticals, conjunctions, disjunctions, negations, and the other logical operators can be traced to two sources. Our account of the logical operators is based upon the notion of an implication structure. Such a structure consists of a nonempty set together with a finitary relation over it, which we shall call an implication relation. As we shall see, implication relations include the usual syntactic and semantic kinds of examples that come to mind. However, implication relations, as we shall describe them, are not restricted to relations of deducibility or logical consequence. In fact, these relations are ubiquitous: Any nonempty set can be provided with an implication relation.