A realistic (non-associative) logic and a possible explanations of 7±2 law Original Research Article

When we know the subjective probabilities (degrees of belief) p1 and p2 of two statements S1 and S2, and we have no information about the relationship between these statements, then the probability of S1 & S2 can take any value from the interval [max(p1+p2−1,0),min(p1,p2)]. If we must select a single number from this interval, the natural idea is to take its midpoint. The corresponding “and” operation p1 & p2 =def (1/2)(max(p1+p2−1,0)+min(p1,p2)) is not associative. However, since the largest possible non-associativity degree |(a & b) & c−a & (b & c)| is equal to 1/9, this non-associativity is negligible if the realistic “granular” degree of belief have granules of width ⩾1/9. This may explain why humans are most comfortable with ⩽9 items to choose from (the famous “7 plus or minus 2” law). We also show that the use of interval computations can simplify the (rather complicated) proofs.