Towards more realistic (e.g., non-associative) ``and'' - and ``or''-operations in fuzzy logic (Q558521)

Human experts do not use all the numbers from the interval \([0,1]\) to describe their possible degrees of belief. They use a few words like ``very probable'', ``mildly probable'', etc. Each one of such words is a granule covering an entire sub-interval of values. Since the largest possible non-associativity degree \(| (x\& y)\& z-x\& (y\& z)| \) is equal to \(1/9\), this non-associativity is negligible if the corresponding realistic granular degree of belief has granules of width \(\geq 1/9\). One can fit no more than 9 granules of such width in the interval \([0,1]\). This may explain why humans are most comfortable with \(\leq 9\) items to choose from -- the famous ``7 plus minus 2'' law. Starting from this point of view some mathematical preliminary results are discussed.