A General Interpretation of Conditioning and its Implication on Coherence

In this paper we introduce a relation between Boolean events that can be interpreted as a conditioning relation. The goal is to find “elementary” events (atoms) derived from a finite family of conditional events and for this purpose we adopt MV-algebra operations. Two different decompositions (whose elements are called dicotomic and tricotomic conditional atoms, respectively) are proposed and their interpretations are given. Finally, we examine them as a tool for checking the coherence of a partial conditional probability assessment