论二值原则——部分逻辑悖论产生的根源

On the Principle of Binary— The Roots of Some Logical Paradoxes

通过对三值逻辑方案遭遇强化型说谎者悖论反击而导致解悖失败的案例分析表明,只要是研究演绎推理及其规律的逻辑系统,包括所有严谨合理的经典与数理的以及那些扩展的和异常的逻辑系统都要遵守的一个基本规则,那就是强化型二值原则。金岳霖先生说:"二值原则不过是对命题的值引用二分法的结果"。因此,同一律和二分法是比二值原则还要更基本的逻辑设定。依据分离公理和戴德金分割,可以确定二分法的操作步骤。在对二分法操作步骤的逻辑分析中,如果按照"逻辑必然的实质就是穷尽可能"的要求,会发现通过母项内部的一个分界点来划分,并不能百分之百地确保得到的都是真正的二分。在两种特殊情况下,两个子项之间会出现一个遗漏元素或一个重叠元素,使二子项的关系成为上反对关系或下反对关系,而不是矛盾关系,这样就导致了二值原则普适性的失效。将这个发现应用到对理发师悖论的分析中,就明白理发师正是那一个重叠元素。更深入的考察,还会发现只要具有统一特征"矛盾等价式"的逻辑悖论,其结构就与理发师悖论相同。这些悖论的统一特征恰恰就是那一个重叠元素的特征,从而给出了说谎者悖论、集合悖论、格雷林悖论、理查德悖论等同一类型逻辑悖论产生根源的一个合理解释。

The case analysis of the failure of the paradox of defeat against the three-valued logic solution by counterattacking the intensified liar paradox shows that as long as it is a logical system to study deductive reasoning and its laws,including all rigorous and reasonable classical and mathematical,one of the basic rules to be followed by extended and anomalous logic systems is the reinforced binary principle.Mr.Yuelin Jin said:“The principle of binary is nothing more than a dichotomy of the value of the proposition.”Therefore,the same law and dichotomy are more basic logical settings than the binary principle.According to the separation of axioms and the Dedikine division,one can determine the dichotomy of the steps.In the logical analysis of dichotomous steps,if we follow the requirement that“the logical necessity is as far as possible”,we can find that dividing by a demarcation point within the parent item can not ensure 100%of the true dichotomy.In two special cases,a missing element or an overlapping element appears between two subsets,making the relationship between the two subsets an up-opposed or a down-opposed one,rather than a contradictory one,which leads to the failure of the universality of the binary principle.Applying this finding to the analysis of the hairdresser’s paradox,it is clear that the barber is exactly the one overlapping element.A closer examination will also find that as long as there is a unified paradoxical paradox of“paradoxical equivalence”,its structure is the same as the hairdresser’s paradox.The uniform feature of these paradoxes is precisely the characteristics of that overlapping element and thus,giving a reasonable justification for the same type of logic paradox:the liar paradox,the collective paradox,the Greylin paradox and the Richard Paradox Explanation.