An original approach to fuzzification of grounding sets is introduced. This model is considered for the case of grounding knowledge, belief and possibility extensions of conjunctions. Artificial cognitive agents that carry out the grounding are assumed to observe their external worlds and store results of observations in the so-called base profiles. Two levels of fuzzification are introduced to an original model for grounding: the first one deals with fuzzification of atom observations in particular base profiles and the second deals with fuzzification of grounding sets by introducing a fuzzy membership of base profiles to grounding sets. Both levels of fuzzification are applied to make the original mechanism of grounding more context sensitive.