A paraconsistent 3-valued logic related to Godel logic G3

Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values

“…Formulas can be assigned  ,  or both  and  in the former;  ,  or neither  nor  in the latter (cf. [29], [31]). The aim of this paper is to provide an overdetermined BD-semantics for all natural implicative expansions of Kleene's strong 3-valued matrix with two designated values.…”

“…The present paper pursues previous work by the authors. In [31] and in [29], both an o-semantics and a u-semantics is provided for Łukasiewicz's 3-valued logic Ł3 and the 3-valued logic G3 Ł , respectively 5 ; in [28] the logic G3 is endowed with a u-semantics. These results connect 3-valued logics with relevance logics; actually, with the most basic relevance logic, Anderson and Belnap's FDE and the BD-semantics devised for it.…”

Compositional Meaning in Logic

“…The logic PE3 is further shown to be strongly maximally paraconsistent in its language, and endowed with a sound and complete analytic cut-free sequent system. This result marks an important and clear-cut distinction between truthfunctional meaning in logic as provided by logical matrices, and more general approaches adopting generalized logical matrices (as used to define orderpreserving logics) or non-deterministic matrices, for which forms of threevalued self-extensional paraconsistency in richer languages, including implications, are known to exist [61,4].…”

Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix II. Only one designated value

“…[17] and the references therein). On the other hand, Lt5 is the logic G3 1 Ł , related to Gödelian 3-valued logic G3, studied in [15].…”

“…We have dubbed this second variant "overdetermined (BD-semantics)", the first one being named "underdetermined (BD-semantics)"; cf. [15], [17]. )…”