By fuzzification we mean a procedure which assigns a fuzzy logicto an arbitrary crisp logic. In this paper we describe twoessentially different fuzzification procedures. In order to give aformal description of the notion of fuzziness of a formula, we extendthe language of the prepositional logic by a family of unaryprepositional operators, inductively applicable to the formulae. Theset of all formulae provable in an arbitrary prepositional crisplogic is extended by the fuzziness measure axioms concerning thoseoperators. A logical system obtained in such a way, main possible toexpress a fuzziness measure of truthfulness of any formula in contextof the observed logic, can be considered as a kind of polymodallogic. This example includes a description of the Kripke-typepossible worlds semantics covering the considered logical systems,being followed by the corresponding completeness results. The secondexample presents quite a syntactic concept of fuzzification,including some of the usual proof-theoretical results such as thecut-elimination theorem.
展开▼
