Structural and functional quantization of vagueness

This paper proposes two methods of quantizing vague categories in terms of quantities whose identities have been either widely recognized in the fuzzy set theory literature or established by Bayesian Decision Theory. The methods are called structural quantization and functional quantization respectively. The idea behind structural quantization is that for a 'comparative' vague category like good-scores, there exists a preference ordering among the possible scores and how good a particular score is depends on how large the portion of examinees whose scores are lower than that score is. Functional quantization adopts the conclusion of Bayesian Decision Theory that a rational agent's belief can be represented by a probability. To quantize the term good-score, the method first assesses the two probabilities that respectively represent a rational agent's beliefs about John's score before and after learning that "John has got a good-score". It then derives a membership function by comparing those two probabilities. The equivalence of the two quantization methods is hypothesized. The logic connectives 'not', 'and' and 'or' are discussed in the setting of structural quantization, while aggregation of information is addressed in the setting of functional quantization.