Formalization of Probabilities with Non-linearly Ordered Ranges

Abstract

This chapter is devoted to logical formalization of reasoning about probabilities with “non-standard” ranges, e.g., ranges that are not the unit interval of real numbers. The main part of the chapter considers probabilities whose ranges are fields of p-adic numbers Qp for arbitrary prime number p. We describe the probability logic LDQp which allows statements in which probabilities are estimated by p-adic balls. We provide an example to illustrate how LDQp can be used to syntactically express statements about interference of waves in the double-slit experiment. Some variants of LDQp that formalize reasoning about p-adic conditional probabilities are also presented. Next, we consider the field of complex numbers as another unordered field and present two logics for reasoning about complex valued probability. Finally, we discuss formal systems for reasoning about probabilities whose ranges are partially ordered monoids.