Abstract: A hedge is a linguistic device used to modify the impact of an utterance. They are called stressers or depressers depending on whether they strengthen or soften such impact. Typical examples are adjectives as in: "they lost a terrible amount of money" (stresser) or "they lost an insignificant amount of money" (depresser), or adverbs, as in: "Messi is definitely a better player than Maradona ever was" (stresser) or "Messi is a slightly better player than Maradona ever was" (depresser). Clauses can act as hedges too, for example: "I think you should reinstall the operating system of your computer, I know what I’m talking about" (stresser) or "I’m not an expert, but I think you should reinstall the operating system of your computer" (depresser). Other clauses acting as hedges are those which directly refer to the truth of some sentence like "it is very true that", "it is quite true that", "it is more or less true that", "it is slightly true that", etc.

Any sentence with a hedge, of any grammatical category, can be translated into one using clauses of the latter kind. In this formulation they have been represented in fuzzy logic systems (in broad sense) as functions from the set of truth values (typically the real unit interval) into itself, that modify the meaning of a proposition by applying over the membership function of the fuzzy set underlying the proposition. More specifically, in the setting of mathematical fuzzy logic, Hájek proposes in a series of papers to understand them as truth functions of new unary connectives called truth-stressing or truth-depressing hedges, depending on whether they reinforce or weaken the meaning of the proposition they apply over. The intuitive mathematical interpretation of a truth-stressing (resp. depressing) hedge on a chain of truth-values is a subdiagonal (resp. superdiagonal) non-decreasing functionpreserving 0 and 1. The class of such functions will be called hedge functions from now on.

The starting point of this paper are the works of Hájek and Vychodil on the axiomatization of truth-stressing and depressing hedges as expansions of BL by new unary connectives. They show that their logics are chain-complete, but standard completeness is only proved for the expansions over Gödel logic. We propose weaker axiomatizations that have as main advantages to preserve the standard completeness properties of the original logic and that any subdiagonal (resp. superdiagonal) non-decreasing function on [0, 1] preserving 0 and 1 is a sound interpretation of the truth stresser (resp. depresser) connectives. Hence we really recover all truth value functions ussense.