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/sub 1/) and p(S/sub 2/) in two statements S/sub 1/ and S/sub 2/, then the possible values of p(S/sub 1/&S/sub 2/) form an interval p=[max(p/sub 1/+p/sub 2/-1,0), min(p/sub 1/,p/sub 2/)]. 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(S/sub 1/&S/sub 2/) 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 /spl ges/1/9, we will not notice this associativity. This may explain the famous 7/spl plusmn/2 law, according to which we use no more than 9 granules.