Fuzzy knowledge combination

Abstract

A general answer is given to what one should conclude from disagreeing experts. the answer is generalized further to incorporate the experts' credibility weights. the answer rests on a wide range of intuitively based epistemic axioms, scientific and philosophical conjectures, and formal mathematical relationships. A recurring theme is the making of Bellman - Zadeh fuzzy decisions, wherein a decision is the intersection of fuzzy goal and fuzzy constraint subsets of some space of alternatives. Another result is that measures of central tendency, such as the arithmetic mean, make poor knowledge combination operators. Formally, fuzzy knowledge combination operators are sought. the function space of knowledge combination operators ø: K″ → K is shrunk by imposing successive axioms. the final shrunken set is said to consist of admissible knowledge combination operators. Some of its mathematical properties are explored and a simple admissible operator is finally chosen. Knowledge sources Xi: S → K are mappings from epistemic stimuli or questions into a knowledge response set K. the uncertainty of the underlying epistemic situations is captured by the cardinality of K and by the fuzziness of its partial ordering. Admissible knowledge combination operators Aggregate knowledge responses in some desirable way. the arithmetic mean is not admissible. Nor in general is a probabilistic framework even definable in the abstract poset setting employed by this theory. the fuzzy knowledge combination theory is extended by associating general credibility weights with the knowledge sources. A new set of weighting axioms is required to satisfy certain intuitions and to satisfy the admissibility axioms. General weighting functions are obtained and thereby weighted admissible operators are obtained. the weighted mean still proves inadmissible. Appendix I contains a technical glossary and summary of the proposed fuzzy knowledge combination theory. Appendix II contains proofs of the probabilistic uncertainty theorems required for the uncertainty testbed used in the theory.