Logics with Probability Operators

Abstract

This chapter presents the logic LFOP₁ which may be suitable to formalize reasoning about degrees of beliefs. The aim is that this chapter serves as an illustration for syntax, semantics and the main proof techniques used elsewhere in the book. The logic enriches first order calculus with probabilistic operators of the form $\mathrm P\geq _{s}$ with the intended meaning “probability is at least s”.We define a possible-world semantics with a finitely additive probability measure on sets of worlds. We provide an infinitary axiomatization which contains an infinitary rule with countable many premisses and one conclusion, related to the Archimedean property of real numbers. Other probability logics considered by the authors of this book are then presented, and an overview of related works of other authors is given.