Subtitle a Realistic ( Non - Associative ) Logic and a Possible

When we know the subjective probabilities (degrees of belief) pt and pi of two statements S\ and S2, and we have no information about the relationship between these statements, then the probability of S\ kS2 can take any value from the interval [maxfpi + pi — l,Q),min(pi,p2)]. If we must select a single number from this interval, the natural idea is to take its midpoint. The corresponding "and" operation P\hpj'= (l/2)(max(/;i + pi — 1,0) + min(j>\,p2)) is not associative. However, since the largest possible non-associativity degree \(a&tb)&cc — a&c(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. © 2002 Elsevier Science Inc. All rights reserved. Corresponding author. E-mail addresses: [email protected] (R. Trejo), vladik(«Jcs.utep.edu (V. Kreinovich), [email protected] (I.R. Goodman). 0888-613X/02/S see front matter © 2002 Elsevier Science Inc. All rights reserved. PII:S0888-613X(01)00065-2