Basic Para-consistent and Para-complete Logic and some of its derivatives

Abstract

The operator “classical negation”, symbolized “~”, is characterized from the semantic point of view by the following equivalence:

A accepted ~A not accepted

This equivalence states that an enunciate is accepted if and only if its negation is not accepted. 4 conditional enunciates can be read:

A accepted Þ~A not accepted
~A accepted ÞA not accepted
A not accepted Þ~A accepted
~A not accepted ÞA accepted

The first two enunciates are equivalent and both prohibit: that an enunciate and its negation be both accepted, therefore, it is forbidden for an enunciate to be compatible with its negation; the two last ones are equivalent and prohibit an enunciate and its negation to be both not accepted, therefore, the indeterminations are prohibited regarding the negation. Classic negation prohibits compatibility of an enunciate withits negation and the indeterminations regarding the negation. The Basic Para-consistent and Para complete Logic system LBPco, shown in this work, is a generalization of classical logic, in this one there is an operator known as “weak negation”, which bears the characteristic of not prohibiting compatibility of an enunciate with its negation, nor the indeterminations regarding the negation. The Basic Para-consistent Logic LBPc and Basic Para-complete logic are particular cases of LBPco, in the first one indeterminations are prohibited and compatibility is permitted, in the second one, compatibility is prohibited and indeterminations are permitted. When rounding, regarding the connectives implication conjunction and disjunction, the performance of the new operator to the one of classic negation, the Positive Paraconsistent and Para-complete logic LPPco, Positive Para-consistent Logic LPPc, and Positive Para-complete LPPco logic are obtained. When rounding, regarding the strong and weak negation connectives, the performance of the new operator to the one of classic negation, the Para-consistent and Para-complete LPco, Para-consistent LPc, and Paracomplete LPo logic are obtained. When permitting, to the basic systems, the indeterminations and the compatibilities only to atomic enunciates, the Basic Para-consistent and Para-complete weak at an atomic level LBPcoDA, Basic Para-consistent logic at atomic level LBPcDA and Basic Para-complete logic weak at atomic level LBPoDA are obtained; this same restriction to the stronger systems gives origin to the Para-consistent and Para-complete logic at atomic level LPcoA, Para-consistent logic at atomic level LPcA and Para-complete logic at atomic level LPoA. All the systems are shown axiomatically and are semantically characterized using a powerful tool of visual inference called Semantic Forcing Trees.