It is known that most systems of fuzzy logic trivialise in the presence of contradictory information of the type \(\{\varphi , \lnot \varphi \}\) , since with the standard truth-preserving [0, 1]-valued semantics, there is no evaluation assigning truth-degree 1 to both \(\varphi \) and \(\lnot \varphi \) . In this paper we consider an alternative semantics for some well-known fuzzy logics with an involutive negation (definable or primitive), where an evaluation validates a formula as soon as it gets a non-zero truth-value. This is a paraconsistent semantics, since both \(\varphi \) and \(\lnot \varphi \) can simultaneously be evaluated with a positive truth-degree without trivialising the reasoning, and it has been called non-falsity preserving semantics by Avron. In this paper we study the properties of this semantics and axiomatise it for the case of several systems of fuzzy logic, among them Łukasiewicz, Nilpotent minimum and Gödel with involution logics.