Despite the progress being made on generally-fast algorithms for solving intractable problems, the demands of the real world require the capability to produce reasoners that are guaranteed to return answers, however preliminary, in a guaranteed amount of time. We have been exploring an approach to this problem involving limiting the reasoning context to a current focus of attention. Preliminary results using this approach have been encouraging. However, a semantical understanding of such tractable reasoning mechanisms is necessary in order to understand what confidence can be placed in them: what do they and do they not tell you? We are developing a semantic framework for tractable context-limited reasoning and experimenting with algorithms that implement this semantics.