On the Paraconsistent Companions of Involutive Fuzzy Logics that Preserve Non-falsity

It is known that most systems of fuzzy logic trivialise in the

presence of contradictory information of the type {φ, ¬φ}, since with the

standard truth-preserving [0, 1]-valued semantics, there is no evaluation

assigning truth-degree 1 to both φ and ¬φ. 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 φ and ¬φ 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 Lukasiewicz, Nilpotent minimum and

G¨odel with involution logics.