Propositional Calculus with Multiple Negations
Oscar Ramírez
TL;DR
The paper develops Propositional Calculus with Multiple Negations $\textbf{CPN}_{n}$, a paraconsistent extension of classical propositional logic that introduces $n$ weak negations to control inconsistencies across multiple possible worlds. It builds a Hilbert-style deductive system with a multi-argument implication $\longrightarrow_{(n)}$ and derived $n$-connectives, together with axioms $A_1$--$A_7$ and the rule $MP_n$, establishing deduction and reductio theorems analogous to classical logic. Semantically, it introduces a multiverse framework where each negation corresponds to a distinct world, with world-local truth values and world-specific contradictions, and proves a Soundness Theorem alongside a Completeness Theorem, while outlining which CPC results fail under weak negations. The work also analyzes the interplay among different $\textbf{CPN}_{n}$ levels, showing that truth-values can be contingent on chains and their complements, thereby enabling controlled paraconsistent reasoning without explosion in a multi-world setting.
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.