Firing condition of fuzzy rules, generalised possibility and necessity measures

When inference is performed with the compositional rule of inference (CRI), fuzzy rules can be mathematically modelled by a pair of operators, the implication function and the modus ponens generating function, each pair being a possible model for the rule. Analysing the firing condition of fuzzy rules, that is the condition the hypothesis and the observation must verify to infer a non-trivial conclusion, it is possible to get a deeper understanding of the behaviour of the different models that can be used, and valuable semantic criteria can be obtained to select the best suitable model. The firing condition of any model can be expressed by means of a generalised possibility or necessity measure generated by its implication and modus ponens generating functions, measures that quantify the amount of uncertainty or possibilistic uncertainty of the conclusion. Two different types of possibility distributions can be identified as those concluded from different types of models, that is necessary and possible possibility distributions, corresponding to necessary and possible conclusions.