Gradual Generalized Modus Ponens

Gradual relationship between premises and conclusions is often an underlying property of fuzzy rules. In this paper, we propose to integrate the gradual hypothesis, sometimes called monotonicity, to Generalized Modus Ponens (GMP). To achieve this objective, we defined the Gradual Generalized Modus Ponens (GGMP), based on a partioning of the universe of discourse. Moreover, we prove, for this formulation, some major preservation properties, such as for the convexity, the continuity and the normality. Finally, we show that the ordering of different fuzzy observations is conserved for the associated conclusions.