An even more realistic (non-associative) logic and its relation to psychology of human reasoning

If we know the degrees of certainty (subjective probabilities) p(S ₁) and p(S₂) in two statements S₁ and S ₂, then the possible values of p(S₁&S₂ ) form an interval p=[max(p₁+p₂-1,0), min(p ₁,p₂)]. As a numerical estimate, it is natural to use a mid-point p of this interval; this mid-point is a mathematical expectation of p(S₁&S₂) over a uniform (second-order) distribution on all possible probability distributions. This mid-point operation & is not associative. We show that the upper bound on the difference a&(b&c)-(a&b)&c is 1/9, so if the size of the corresponding granules is ⩾1/9, we will not notice this associativity. This may explain the famous 7±2 law, according to which we use no more than 9 granules