Propositional Calculus with Multiple Negations

TL;DR

This paper introduces a generalized propositional calculus with multiple negations, enabling controlled weak inconsistencies while maintaining non-trivial logical behavior, bridging classical and paraconsistent logics.

Contribution

It proposes the Propositional Calculus with Multiple Negations, a novel system combining classical and paraconsistent features with multiple negations.

Findings

Allows controlled weak inconsistencies

Maintains non-triviality in the logic system

Bridges classical and paraconsistent logic

Abstract

One advantage of paraconsistent logic is that it can deal with inconsistencies without making the system trivial. However, unlike classical propositional calculus, its deductive system is limited, and the meaning of paraconsistent negation is still not clear. This article presents a logical system that brings together the strengths of both approaches. The Propositional Calculus with Multiple Negations $\left(\textbf{CPN}_{n}\right)$ is a generalization of classical propositional logic in which a finite number of negations (each weaker than the classical one but with similar behavior) are added. This makes it possible to introduce weak inconsistencies in a controlled way without leading to triviality.