We describe here the theory behind the language comprehension program Wimp. Wimp understands by first finding paths between the open-class words in a sentence using a marker passing, or spreading-activation, technique. This paper is primarily concerned with the "meaning" (or interpretation) of such paths. We argue that they are best thought of as backbones of proofs that the terms (words) at either end of the paths exist in the story and show how viewing paths in this way naturally leads to the kinds of inferences which are normally thought to characterize "understanding." In a companion paper we show how this interpretation also accomplishes much of the work normally expected in the parsing of language (noun-phrase reference, word-sense disambiguation, etc) so we only briefly touch on this topic here. Wimp has been implemented and works on all of the examples herein.