Uncertain random logic and uncertain random entailment

Probabilistic logic and uncertain logic, as two branches of the multi-valued logic, have been proposed for dealing with knowledge with random factors and with human uncertainty, respectively. As a generalization, this paper proposes an uncertain random logic to deal with complex knowledge containing random factors and human uncertainty simultaneously, and derives a formula to calculate the truth value of an uncertain random proposition. As an inverse problem of the uncertain random logic, this paper also proposes an uncertain random entailment model which calculates the truth value of a function of some uncertain random propositions based on the truth values of some other functions of these uncertain random propositions. As a byproduct, a probabilistic entailment model is also discussed.