Fuzzy decision implications

The aim of this paper is to provide the semantical and syntactical characteristics of fuzzy decision implications, an expression concerning two systems of fuzzy-sets. This Pavelka-style fuzzy logic starts from a fuzzy set of fuzzy decision implications and makes deductions from partially true decision implications. Following this idea, in semantical aspect, we present some basic results concerning completeness and more importantly, introduce the notion of “unite closure”, which is in fact important not only in semantical aspect but also in syntactical aspect. Besides, we derive three deduction rules, namely (F-Transformation), (F-Add) and (F-Sh↑) for syntactical aspect of fuzzy decision implications, which are proved to be sound and complete with respect to semantical aspect. The result ensures that one can obtain a closed fuzzy set of fuzzy decision implications by the semantical way or by the syntactical way, i.e., by the three deduction rules.