Are truth degrees about truth?

We often explain the meaning of formulae of predicate fuzzy logics by using truth degrees [M76]. In response to this, Petr Hajek insisted that our conception of truth must be revised: fuzzy truth values are based on “the comparative conception of truth”, formulae may be more or less true [H99]. In this paper, we question the necessity of such revision by introducing an alternative way, which doesn’t need such revision, to explain the meaning of formulae in fuzzy logics as in Charles Parson’s article. One of the main problems of truth degree theory is that it can’t explain the meaning of formulae in fuzzy logics, e.g. BL∀, which are defined in proof-theoretic ways. Actually fuzzy truth values [0, 1] are not complete for them, so it isn’t suitable to interpret such logics in terms of fuzzy truth values. Francesco Paoli gave a revision of truth degrees to represent “a really fuzzy ” approach [Pl03]. In his theory, truth degrees are not linearly ordered: they form a lattice. However, his lattice seems to be an abstraction of algebraic representations of proof theoretic structures, rather than the truth values. So we should think again what the role of truth degrees is. To explain this, it is helpful to introduce Michael Dummett’s analysis of 3-valued logi