A family of genuine and non-algebraisable C-systems

ABSTRACT In 2016, Béziau introduced the notion of genuine paraconsistent logic as logic that does not verify the principle of non-contradiction; as an important example, he presented the genuine paraconsistent logic in terms of three connectives , , and . In this paper, we show that is an axiomatic extension of through the introduction of a non-primitive deductive implication. Furthermore, we prove that is an algebraisable logic with Blok-Pigozzi's method. From the proof that is non-algebraisable logic, we are able to see that is not algebraisable logic and studying the borders of algebrisabilty, we can give an enumerable family of new genuine, paraconsistent and non-algebraisable logics, extensions of . Finally, we introduced n-valued ( ) and infinite-valued logic and show that they are genuine and non-algebraisable paraconsistent ones; in addition, we present semantics for this extensions of by means of Fidel's structures.