Generalized Logical Operations among Conditional Events

TL;DR

This paper extends the concepts of conjunction and disjunction to multiple conditional events within a coherence-based framework, exploring their properties, probabilistic assessments, and implications for nonmonotonic reasoning.

Contribution

It introduces a generalized approach to logical operations among multiple conditional events, establishing their properties and applications in probabilistic inference.

Findings

Conjunctions and disjunctions satisfy De Morgan's Laws, associativity, and commutativity.

Provides bounds and coherence conditions for probabilistic assessments of compound conditionals.

Characterizes p-consistency and p-entailment using conjunctions, with applications to inference rules.

Abstract

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of $n$ conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan's Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; in particular we examine the Fr\'echet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction $\mathcal{C}_{n+1}$ of $n+1$ conditional events to the family $\{\mathcal{C}_{n},E_{n+1}|H_{n+1}\}$. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid.