Logical Operations among Conditional Events: theoretical aspects and applications

Abstract

We generalize the notions of conjunction and disjunction of two conditional events to the case of $n$ conditional events. These notions are defined, in the setting of coherence, by means of suitable conditional random quantities with values in the interval $[0,1]$. We also define the notion of negation, by verifying De Morgan's Laws. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals and we show that some well known properties which are satisfied by conjunctions and disjunctions of unconditional events are also satisfied by conjunctions and disjunction of conditional events. We also examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. We consider the relation between the conjunction and other different definitions of conjunction among conditional events. In particular, we consider the notion of quasi-conjunction which is largely studied in non monotonic reasoning. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to the inference rules emph{And}, emph{Cut}, emph{Cautious Monotonicity}, and emph{Or} of System P. Then, we examine some non p-valid inference rules (emph{transitivity} and an example from Boole) by also illustrating two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid. We introduce a notion of iterated conditional and we characterize p-entailment by showing that a (p-consistent) family $F={A|H,B|K}$ p-entails $E|H$ if and only if $(E|H) | C(F)=1$, where $C(F)$ is the conjunction of the conditional events in $F$.