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The basic idea of Interval Mathematics is that ordinary set-theoretical intervals IR/provide a consistent support for numerical computing. The set-theoretical form of Interval Analysis has produced a large amount of work since its initiation in the late 1950s,[1, 60, 61]. There are some later papers devoted to the structural analysis of the method or its completion [13, 14, 49, 66, 72, 87, 96, 97], but with no fundamental departure from its initial set-theoretical foundations. This book presents a new interval theory, the Modal Interval Analysis (MIA), as a structural, algebraic, and logical completion of the classical intervals. The starting point of MIA is quite simple: to define a modal interval attaching a quantifier to a classical interval, and to introduce the basic relation of inclusion between modal intervals by means of the inclusion between the sets of predicates they accept. So a modal interval consists in a classical interval, which …