Primitive Intervals versus Point-Based Intervals: Rivals or Allies?

The notion of time is a very interesting and exciting subject both in science and everyday life. One of the fundamental questions is: what is time composed of? While the traditional time structure is based on a set of points, a notion that has been prevalently adopted in classical physics and mathematics, it has also been noticed that intervals have been widely adopted for expression of commonsense temporal knowledge, especially in the domain of artificial intelligence. However, there has been a long-standing debate on whether intervals should be treated as primitive or not, leading to two different approaches to the treatment of intervals. In the first, intervals are modelled as derived objects constructed from points, e.g. sets of points, or pairs of points. In the second, intervals are taken as primitive themselves. This article provides a critical examination of these two approaches. By means of proposing a definition of intervals in terms of points and types, we shall demonstrate that, while the two different approaches have been viewed as rivals in the literature, they are actually reducible to logically equivalent expressions under some requisite interpretations, and therefore they can also be viewed as allies.