Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

Abstract

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa's school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the "core" of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.

Ideal Paraconsistent Logics

“…In [3] we have shown that LP is a strongly maximal paraconsistent logic (in the absolute sense of Definition 14), and in Example 4 we have shown that it is also maximal relative to classical logic. Nevertheless, LP is not strongly maximal relative to classical logic.…”

“…Example 6. Sette's logic P 1 [32] (and all of its fragments containing Sette's negation), the logic PAC [8,4], J 3 [18], and the 2 20 three-valued logics considered in [3] (including the 2 13 LFIs from [15]), are all ideal paraconsistent logics. 8 Note 5.…”

“…This is insufficient, of course. Thus, in [3] we have shown that the three-valued logic whose only connective is Sette's negation [32], is maximally paraconsistent and it is obviously contained in classical logic. Still, nobody would take it as an 'ideal' paraconsistent logic, because its language is not sufficiently expressive.…”

“…This includes all the 2 20 three-valued paraconsistent logics shown in [3] to be maximally paraconsistent, including the 2 13 three-valued logics of formal inconsistency (LFIs), shown in [15,28] to be maximal relative to classical logic. In Section 5 we provide a systematic way of constructing ideal logics with any finite number of truth-values.…”

Неклассические Модификации Многозначных Матриц Классической Логики. Часть II

“…В работе [14] доказано, что каждая матрица из 8Kb является мак-симально паранепротиворечивой в сильном смысле. Ниже я покажу, что это так и для T L 2 .…”