@article {56247, title = {Maximality in finite-valued {\L}ukasiewicz logics defined by order filters}, journal = {Journal of Logic and Computation}, volume = {29}, year = {2019}, pages = {125-156}, publisher = {OUP}, abstract = {

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} }