Paraconsistent fuzzy logic preserving non-falsity

We introduce proof systems and semantics for two paraconsistent extensions of the system T of Anderson and Belnap, and prove strong soundness, completeness, and decidability for both. The semantics of both systems is based on excluding just one element from the set of designated values. One of the systems has the variable sharing property, and so it is a relevant logic. The other is an extension of the first that may be viewed as a semi-relevant counterpart of Łukasiewicz Logic which preserves non-falsity rather than truth.