Logic and Paradox

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.