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

Abstract

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