Editing Fuzzy set (section)

== Fuzzy logic ==
{{main|Fuzzy logic}}

As an extension of the case of [[multi-valued logic]], valuations (<math>\mu : \mathit{V}_o \to \mathit{W}</math>) of [[propositional variable]]s (<math>\mathit{V}_o</math>) into a set of membership degrees (<math>\mathit{W}</math>) can be thought of as [[membership function (mathematics)|membership functions]] mapping [[Predicate (mathematical logic)|predicates]] into fuzzy sets (or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzy [[premise]]s from which graded conclusions may be drawn.<ref>[[Siegfried Gottwald]], 2001. ''A Treatise on Many-Valued Logics''. Baldock, Hertfordshire, England: Research Studies Press Ltd., {{ISBN|978-0-86380-262-1}}</ref>

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the [[engineering]] fields of [[automation|automated]] control and [[knowledge engineering]], and which encompasses many topics involving fuzzy sets and "approximated reasoning."<ref>{{cite journal | doi=10.1016/0020-0255(75)90036-5 | title=The concept of a linguistic variable and its application to approximate reasoning—I | date=1975 | last1=Zadeh | first1=L.A. | journal=Information Sciences | volume=8 | issue=3 | pages=199–249 }}</ref>

Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at [[fuzzy logic]].