Combining Linear Orders with Modalities for Possible Histories

Two main areas of temporal logics are those of linear time and of branching time. Linear orders, though, play a crucial role also in logics for branching time. Prior’s Ockhamist and Peircean semantical rules for branching time, in fact, involve quantification over histories in tree-like structures, where histories are maximal linearly ordered sets of moments. Moreover, this quantification can be viewed as the result of the application of a modal operator. This means that the language and semantics for branching time can be obtained as the combination of languages and semantics for linear time with a modality for possible histories. In this paper, we study various degrees of combining linear time and modal operators and semantics, and we discuss the problem of transferring logical properties from linear to branching time