Maximality in finite-valued Łukasiewicz logics defined by order filters

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.

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