Logic and Paradox

TL;DR

This paper proposes a redefinition of logic with four set-theoretic principles to eliminate paradoxes like the liar paradox and G"odel's incompleteness, aiming to establish a more solid foundation for human language and knowledge.

Contribution

It introduces a novel set-theoretic logic framework with four principles to resolve classical paradoxes and logical errors.

Findings

The new logic eliminates all classical paradoxes.

It provides a consistent foundation for human language.

The redefinition improves logical soundness and validity.

Abstract

This article discusses the logical errors in the liar paradox, G\"odel's incompleteness theorems, Russell's paradox, and the halting problem. In order to avoid these errors, a redefinition of logic has been presented, which is concluded as four principles in set-theoretic language, including 1. Don't talk about empty; 2. Elements of set should have identity; 3. Sets should have definitions; and 4. A total set should be defined. The new definition of logic can eliminate all paradoxes and invalid statements, thus becoming a solid foundation for human language and knowledge.