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Formal fuzzy logic
Fuzzy logic is a relatively new chapter of formal logic whose aim is to formalize the reasonings involving predicates that are vague in nature (as an example small, near, similar). An example of such kind of reasoning is
- If a tomato is red, then the tomato is ripe. Since this tomato is very red, this tomato is very ripe.
Further examples of reasonings involving vague predicates are in the item Paradoxes and fuzzy logic and in the section Fuzzy logic with no truth-functional semantics. The main tool for fuzzy logic is the notion of a fuzzy subset since a vague predicate is interpreted by a fuzzy subset. Notice that in literature the name "fuzzy logic" also denotes a large series of topics based on an informal usage of the notion of a fuzzy subset and which are usually devoted to applications.
As a matter of fact, fuzzy logic is an evolution and an enlargement of multi-valued logic since all the definitions and results in the literature on multi-valued logic are also considered in fuzzy logic. In particular, as in multi-valued logic, the starting point is a fixed valuation structure, i.e. a bounded lattice L equipped with suitable operations to interpret the logical connectives. The minimum 0 means 'False', the maximum 1 means 'True', the remaining elements are interpreted as intermediate truth values. The following is the main class of valuation structures (see Hájek 1998, Novák et al. 1999 and Gottwald 2005) corresponding to the connectives and .
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