There are two natural and well-studied approaches to temporal ontology and reasoning: point-based and interval-based. Usually, interval-based temporal reasoning deals with points as a particular case of duration-less intervals. A recent result by Balbiani, Goranko, and Sciavicco presented an explicit two-sorted point-interval temporal framework in which time instants (points) and time periods (intervals) are considered on a par, allowing the perspective to shift between these within the formal discourse. We consider here two-sorted first-order languages based on the same principle, and therefore including relations, as first studied by Reich, among others, between points, between intervals, and inter-sort. We give complete classifications of its sub-languages in terms of relative expressive power, thus determining how many, and which, are the intrinsically different extensions of two-sorted first-order logic with one or more such relations. This approach roots out the classical problem of whether or not points should be included in a interval-based semantics. In this Part II, we deal with the cases of all dense and the case of all unbounded linearly ordered sets.

中文翻译：

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线性阶上点和区间的综合一阶理论（第二部分）

时间本体论和推理有两种经过充分研究的自然方法：基于点和基于区间。通常，基于间隔的时间推理将点视为无持续时间间隔的特殊情况。Balbiani、Goranko 和 Sciavicco 最近的一项结果提出了一个明确的二分类点间隔时间框架，其中时间瞬间（点）和时间段（间隔）被视为同等，允许视角在正式话语。我们在这里考虑基于相同原理的二阶一阶语言，因此包括 Reich 首次研究的关系，其中包括点之间、区间之间和排序间的关系。我们根据相对表达能力对其子语言进行完整分类，从而确定有多少，哪些，是具有一个或多个此类关系的二分类一阶逻辑的本质上不同的扩展。这种方法根除了点是否应该包含在基于区间的语义中的经典问题。在第二部分中，我们处理所有稠密集的情况和所有无界线性有序集的情况。