It is a common thought that one can distinguish between certain fore- and background assumptions in inference, notably in non-demonstrative inference. What can such fore-v-background structures consist it? In particular, how do they relate to consciousness? According to a ‘Boring View’, such structures can be captured by specifying, for the various grounds or assumptions of the inference, whether they are phenomenally conscious, or access conscious, or else how easily available they are to such consciousness. According to an ‘Interesting View’, there are fore-v-background structures over and above such classifications. The chapter points to reasons for thinking that an Interesting View at least merits exploration. Some recent contributions to such a view are discussed, as are with some pertinent remarks in Husserl, and in psychological work on the role of schema or gist representations in memory. It is proposed that background assumptions can figure in consciousness by being condensed into a consciously, though inattentively, entertained notion of their overall thematic gist, where this thematic gist gives the drift of a possible elucidation of how or why such-and-such salient grounds mean that so-and-so conclusion holds.