Maximality in finite-valued Lukasiewicz logics defined by order filters

TL;DR

This paper investigates the maximality properties of finite-valued Lukasiewicz logics defined by order filters, establishing conditions for maximality and analyzing their relationships with classical propositional logic.

Contribution

It provides a general theorem for maximality conditions, shows when these logics are maximal w.r.t. classical logic, and characterizes their paraconsistent nature for prime n.

Findings

$L_n^i$ is maximal w.r.t. CPL when n is prime.

$L_n^i$ are not strongly maximal w.r.t. CPL for prime n.

Certain $L_n^i$ are identified as ideal paraconsistent logics.

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.