A note on the notion of truth in fuzzy logic

Abstract In fuzzy predicate logic, assignment of truth values may be partial, i.e. the truth value of a formula in an interpretation may be undefined (due to lack of some infinite suprema or infima in the underlying structure of truth values). A logic is supersound if each provable formula ϕ is true (has truth value 1) in each interpretation in which the truth value of ϕ is defined. It is shown that among the logics given by continuous t -norms, Godel logic is the only one that is supersound; all others are (sound but) not supersound. This supports the view that the usual restriction of semantics to safe interpretations (in which the truth assignments is total) is very natural.