In this paper we consider the logics $\mathsf{L}_n^i$ obtained from the $(n+1)$-valued {\L}ukasiewicz logics {\L}$_{n+1}$ by taking the order filter generated by $i/n$ as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem which provides sufficient conditions for maximality between logics. As a consequence of this theorem it is shown that $\mathsf{L}_n^i$ is maximal w.r.t.\ {\sf CPL} whenever $n$ is prime. Concerning strong maximality (that is, maximality w.r.t. rules instead of only axioms), we provide algebraic arguments in order to show that the logics $\mathsf{L}_n^i$ are not strongly maximal w.r.t.\ {\sf CPL}, even for $n$ prime. Indeed, in such case, we show there is just one extension between $\mathsf{L}_n^i$ and {\sf CPL} obtained by adding to $\mathsf{L}_n^i$ a kind of graded explosion rule. Finally, using these results, we show that the logics $\mathsf{L}_n^i$ with $n$ prime and $i/n \< 1/2$ are ideal paraconsistent logics.

}, doi = {10.1093/logcom/exy032}, url = {https://doi.org/10.1093/logcom/exy032}, author = {Marcelo Coniglio and Francesc Esteva and Joan Gispert and Llu{\'\i}s Godo} }