LBPcPo Paraconsistent and paracomplete system

Abstract

LBPcPo system is an extension of classical positive logic. It includes two operators of alternate negation, an incompatibility operator and acompleteness operator from which strong negation, alternate affirmationand alternate completeness operators are defined. The defining feature of this system is a valuation semantics that proves that with regards to an operator of alternate negation the system is paracomplete, with regards to the other which is paraconsistent and the operator of strong negation has all the features of classical negation. The LBVA system (Basic Logicfor Truth and Falsehood) is also introduced as a tool to deductively characterize definitions of truth and falsehoods stated by Aristotle. Results show that LBVA system is equivalent to LBPcPo system. The Aristotelian falsehood operator matches the paracomplete negation, and the Aristotelian truth operator matches the alternate affirmation.