New applications of relational event algebra to fuzzy quantification and probabilistic reasoning

There have been a number of previous successful efforts that show how fuzzy logic concepts have homomorphic-like stochastic correspondences, utilizing one-point coverages of appropriately constructed random sets. Independent of this and fuzzy logic considerations in general, boolean relational event algebra (BREA) has been introduced within a stochastic setting for representing prescribed compositional functions of event probabilities by single compounded event probabilities. In the special case of the functions being restricted to division corresponding to pairs of nested sets, BREA reduced to boolean conditional event algebra (BCEA). BCEA has been successfully applied to issues involving comparing, contrasting and combining rules of inference, especially for those having differing antecedents. In this paper we show how, in a new way, not only BCEA, but also more generally, RCEA, can be applied to provide homomorphic-like connections between fuzzy logic quantifiers and classical logic relations applied to random sets. This also leads to an improved consistency criterion for these connections. Finally, when the above is specialized to BCEA, a novel extension of crisp boolean conditional events is obtained, compatible with the above improved consistency criterion.