A mathematical theory is proposed and exemplified, which covers an extended class of black boxes. Every kind of stimulus and response is pictured by a channel connecting the box with its environment. The input-output relation is given by a postulate schema according to which the response is, in general, a nonlinear functional of the input. Several examples are worked out: the perfectly transmitting box, the damping box, and the amplifying box. The theory is shown to be (a) an extension of the S-matrix theory and the accompanying channel picture as developed in microphysics; (b) abstract and applicable to any problem involving the transactions of a system (physical, biological, social, etc.) with its milieu; (c) superficial, because unconcerned with either the structure of the box or the nature of the stimuli and responses. The motive for building the theory was to show the capabilities and limitations of the phenomenological approach.