Fuzziness and probability: a discussion of Gaines5 axioms

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Abstract
Uncertain inference can be modelled in various ways. Probability and fuzzy logics are special cases of a more general result involving intervals of possibilities. It is argued that uncertainty analysis is unlike engineering science because it is difficult to test under controlled conditions. It is therefore important to expose its basic assumptions if engineers are to choose sensibly between alternative techniques. The minimal statements of the assumptions are the mathematical axioms, and it is the interpretation of these axioms which makes the inference model. Gaines' axioms, based on a mathematical lattice, for fuzzy and probability logics are discussed and explained with examples. Gaines demonstrated that single alternative axioms distinguish the two: probability logic rests on the law of the excluded middle whilst fuzzy logic relaxes that assumption but makes a total dependence assumption. Probability theory is applicable if the underlying structure of the problem is well understood. However, if unknown factors are important then the law of the excluded middle is too constraining and may be replaced by alternatives. The total dependence assumption of fuzzy logic is one such alternative which leads to bound values which involve least change.

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