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A major problem in the consistent-histories approach to quantum theory iscontending with the potentially large number of consistent sets of historypropositions. One possibility is to find a scheme in which a unique set isselected in some way. However, in this paper we consider the alternativeapproach in which all consistent sets are kept, leading to a type of `manyworld-views' picture of the quantum theory. It is shown that a natural way ofhandling this situation is to employ the theory of varying sets (presheafs) onthe space $\B$ of all Boolean subalgebras of the orthoalgebra $\UP$ of historypropositions. This approach automatically includes the feature wherebyprobabilistic predictions are meaningful only in the context of a consistentset of history propositions. More strikingly, it leads to a picture in whichthe `truth values', or `semantic values' of such contextual predictions are notjust two-valued (\ie true and false) but instead lie in a larger logicalalgebra---a Heyting algebra---whose structure is determined by the space $\B$of Boolean subalgebras of $\UP$.Comment: 28 pages, LaTe

Topos theory and consistent histories: The internal logic of the set of all consistent sets