An Integrated First-Order Theory of Points and Intervals over Linear Orders (Part II)

Abstract

There are two natural and well-studied approaches to temporal ontology andreasoning: point-based and interval-based. Usually, interval-based temporalreasoning deals with points as a particular case of duration-less intervals. Arecent result by Balbiani, Goranko, and Sciavicco presented an explicittwo-sorted point-interval temporal framework in which time instants (points)and time periods (intervals) are considered on a par, allowing the perspectiveto shift between these within the formal discourse. We consider here two-sortedfirst-order languages based on the same principle, and therefore includingrelations, as first studied by Reich, among others, between points, betweenintervals, and inter-sort. We give complete classifications of itssub-languages in terms of relative expressive power, thus determining how many,and which, are the intrinsically different extensions of two-sorted first-orderlogic with one or more such relations. This approach roots out the classicalproblem of whether or not points should be included in a interval-basedsemantics. In this Part II, we deal with the cases of all dense and the case ofall unbounded linearly ordered sets.Comment: This is Part II of the paper `An Integrated First-Order Theory of Points and Intervals over Linear Orders' arXiv:1805.08425v2. Therefore the introduction, preliminaries and conclusions of the two papers are the same. This version implements a few minor corrections and an update to the affiliation of the second autho