说谎者悖论的隐蔽的假设

A Hidden Assumption in the Liar Paradox

说谎者悖论是最古老最有影响的一个悖论。关于说谎者悖论,主流的理论着重于对“自我指涉”和“真谓词”的分析。本文指出,自我指涉和真谓词虽然是说谎者悖论的明显的外部特征,却不是问题的要害。问题的要害是说谎者悖论里的一个隐蔽的假设。本文先通过考察悖论与反证法的关系给出一个有助于寻找隐蔽假设的一般原理,然后应用这个原理找出说谎者悖论里的隐蔽的假设,以解答说谎者悖论。提供有力证据的是一个新发现的“三卡悖论”,它证实该原理适用于说谎者悖论。按照这一原理,说谎者悖论与一个布尔方程无解的反证法有相同的假设,即“该方程有解”的假设,也即“存在变元的值满足该方程”的假设,只不过对说谎者悖论来说,这个假设是隐蔽的。于是,解答说谎者悖论的关键就是搞清楚这一假设所说的“变元”“值”“方程”“解”对说谎者悖论来说是什么意思。人们将惊讶这一假设在说谎者悖论里隐藏之深。同样的分析也给出了雅布罗(S.Yablo)悖论的解答。

The mainstream theories in the literature about the Liar paradox focus on selfreference and truth predicate.In this paper we point out that,while selfreference and truth predicate are evident features of the Liar paradox,they are not the crux of the problem.Rather,the crux is a hidden assumption in the Liar argument.We first give a general principle about hidden assumption through the relation between paradox and proof by contradiction,and then apply this principle to find the hidden assumption in the Liar argument.A strong evidence is a new“Threecards paradox”which confirms that the principle applies to the Liar paradox.Then,according to the principle,modulo translation,the Liar paradox has,hiddenly,the same assumption as a proof by contradiction of no solution of a Boolean equation-the existence of a solution of the equation,or the existence of a value of the variable that satisfies the equation.Thus,the key to understand the Liar paradox is to clarify the meanings of“variable”,“value”,“equation”and“solution”for paradoxes.One will be surprised to see how deeply this assumption is hidden in the Liar argument.The same analysis gives a solution to the Yablo’s paradox.