Quasi Conjunction, Quasi Disjunction, T-norms and T-conorms: Probabilistic Aspects

TL;DR

This paper explores probabilistic bounds and inference rules related to quasi conjunction and disjunction of conditional events, connecting them with t-norms and t-conorms within a coherence-based framework, advancing nonmonotonic reasoning theory.

Contribution

It introduces a probabilistic analysis of quasi conjunction and disjunction, linking them to well-known t-norms and t-conorms, and develops new inference rules in a coherence-based setting.

Findings

Identifies bounds coinciding with known t-norms and t-conorms.

Defines Quasi And and Quasi Or rules for conditional events.

Analyzes probabilistic bounds and logical dependencies among events.

Abstract

We make a probabilistic analysis related to some inference rules which play an important role in nonmonotonic reasoning. In a coherence-based setting, we study the extensions of a probability assessment defined on $n$ conditional events to their quasi conjunction, and by exploiting duality, to their quasi disjunction. The lower and upper bounds coincide with some well known t-norms and t-conorms: minimum, product, Lukasiewicz, and Hamacher t-norms and their dual t-conorms. On this basis we obtain Quasi And and Quasi Or rules. These are rules for which any finite family of conditional events p-entails the associated quasi conjunction and quasi disjunction. We examine some cases of logical dependencies, and we study the relations among coherence, inclusion for conditional events, and p-entailment. We also consider the Or rule, where quasi conjunction and quasi disjunction of premises coincide with the conclusion. We analyze further aspects of quasi conjunction and quasi disjunction, by computing probabilistic bounds on premises from bounds on conclusions. Finally, we consider biconditional events, and we introduce the notion of an $n$-conditional event. Then we give a probabilistic interpretation for a generalized Loop rule. In an appendix we provide explicit expressions for the Hamacher t-norm and t-conorm in the unitary hypercube.