Maximality in finite-valued Łukasiewicz logics defined by order filters
Marcelo E. Coniglio, Francesc Esteva, Joan Gispert, Lluı́s Godo · Journal of Logic and Computation · 2018
In this paper we consider the logics |$\mathsf{L}_n^i$| obtained from the |$(n+1)$|-valued Łukasiewicz logics Ł|$_{n+1}$| 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 that 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. CPL whenever |$n$| is prime. Concerning strong maximality (i.e. 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. CPL, even for |$n$| prime. Indeed, in such case, we show that there is just one extension between |$\mathsf{L}_n^i$| and 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.