禁止使用自指代命题——说谎者悖论的排除和哥德尔定理的讨论

The Prohibition of Self-reference & Substitution Propositions——The Elimination of the Liar Paradox and Discussion of Godel's Incompleteness Theorem

应把“自指代命题”从“自指命题”中区分出来,前者违反同一律,作代换还可能违反矛盾律,因此禁止使用自指代命题。对内容不明的自指命题作真假对错的评判,不可能给出确定的结论,但也不会出现矛盾。说谎者悖论是一个佯悖。它被称为悖论,是因为推理者混淆了思维的层次,构造了自指代命题并进行代换才导致矛盾。哥德尔不完全性定理所构造的自指代命题的可证性存在矛盾的双重标准,定理的证法中共用了矛盾的双重标准,其结论值得商榷。结论中的“不可判定”命题,现在有三种不同的错误解释:是非不可分辨的命题(三值)、是非可分辨但不确定的命题(二值)、除自指代命题之外的是非都不可证的其他命题,它们都不是哥德尔的证法所支持的结论。它导致“真理丧失说”和“数学丧失了确定性”缺乏依据。

No certain conclusion or contradiction will be reached when we try to judge whether a self-reference proposition with unclear content is true or false. Based on this, the present paper eliminates the liar paradox and points out that it is a pseudo paradox. The cause for the mistake is that people confuse different levels of thinking, construct the self-reference and substitution propositions, and get in contradiction by substitution. Hence, the self-reference and substitution propositions should be prohibited. Moreover, by analysis of the Socrates-Plato paradox, the paper points out that it is a fallacy, which includes a self-contradiction. The paper also analyzes ＂the undecidable propositions＂ in the conclusion of Godel＇s incompleteness theorem, and finds that they are often mistakenly explained in three ways： propositions whose truth are undistinguishable; propositions whose truth are uncertain; and any other propositions than the self-reference and substitution propositions whose truth are unprovable. Thus, the conclusions of ＂loss of truth＂or ＂loss of certainty in Math＇cannot be reached.