Maximality in finite-valued Lukasiewicz logics defined by order filters

摘要

In this paper we consider the logics L-n(i) obtained from the (n + 1)-valued Lukasiewicz logics Ln+1 by taking the order filter generated by i 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 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 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 L-n(i) and CPL obtained by adding to L-n(i) a kind of graded explosion rule. Finally, using these results, we show that the logics L-n(i) with n prime and i 1/2 are ideal paraconsistent logics.