Maximality in finite-valued Lukasiewicz logics defined by order filters

Marcelo E. Coniglio; Francesc Esteva; Joan Gispert; Lluis Godo

paper

Maximality in finite-valued Lukasiewicz logics defined by order filters

arXiv:1803.09815

Abstract

In this paper we consider the logics $L_{n}^{i}$ obtained from the (n+1)-valued Lukasiewicz 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 analysed. We present a very general theorem which 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 between the logics $L_{n}^{i}$ (that is, maximality w.r.t. rules instead of 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 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/n < 1/2 are ideal paraconsistent logics.