A Realistic ( Non - Associative ) Logic Anda Possible Explanations of 7 2 LawRa
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چکیده
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 j(a& b) & c ? a & (b & c)j 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 minus 2" law). We also show that the use of interval computations can simplify the (rather complicated) proofs.
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A Realistic ( Non - Associative ) Logic Anda Possible Explanations of 7 2
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 \...
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