A Realistic (non-associative) Logic and a Possible Explanations of 7 + 2 Law
Abstract
When the subjective probabilities (degrees of belief) p1 and p2 of two statements S1 and S2 are known, and there is 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 thus interval, the natural idea is to take its midpoint. The corresponding "and" operation p1 and p2(def) (1/2)(max(p1 + p2- 1,0) + min(p1, p2)) is not associative. However, since the largest possible nonassociativity 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 greater than or equal to 1/9. This may explain why humans are most comfortable with less than or equal to 9 items to choose from (the famous "7 plus or minus 2" law). This paper also shows that the use of interval computations can simplify the (rather complicated) proofs.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 2001
- Accession Number
- ADA506814
Entities
People
- I. R. Goodman
- Jesus Martinex
- Raul Trejo
- Reginaldo Gonzalez
- Vladik Kreinovich
Organizations
- Naval Information Warfare Systems Command