Introducing granularity-dependent quantitative distance and diameter measures in common-sense reasoning contexts

In this paper, I will present a method for constructing correlated series of granularity-dependent distance and diameter measures on the basis of a theory of qualitative spatial concepts. Each granularity-dependent measure function has as its range a discrete subset of R. As we proceed through a series of such functions, the distance between the values in the range will become smaller and the measurements returned by the functions will become correspondingly more precise. My method for constructing the series of functions is partially based on work in Kranz, Luce, Suppes, and Tversky, The Foundations of Measurement, Vol. 1. I use a result from that volume to prove that my series of distance and diameter measures converge to continuous, extensive distance and diameter measures. But it is the discrete measures in the series, not the continuous limit measures, that should be used in analyses of common-sense concepts. Unlike the continuous measure functions, arbitrary values for the discrete measure functions can, in most contexts, be determined through practical procedures. Moreover, the ability to move from one granularity-level to the next is appropriate for common-sense contexts in which the level of precision is typically kept at the minimum necessary to accomplish the task at hand.