The Complexity of Fuzzy Logic

Lukasiewicz logic is a “fuzzy” logic in which truth value can be real numbers in the unit interval. There are connectives for min, max, addition and complement (1-x). The “value” of a closed formula in a fuzzy (relational model) is defined in the natural way. A formula is called valid iff it has value 1 in every fuzzy model. We show that the set of valid formulas in Lukasiewicz predicate logic is a complete Pi^0_2 set. We also show that if we restrict our attention to the classical language (min, max, complement) then the classically valid formulas are exactly those formulas whose fuzzy value is 1/2.