Studies on reductio ad absurdum and its Application in Indian Propositional Logic

在印度的解释学派Mimamsa中，这种借助于反证法证明的方法被称为Arthapatti，不同于推理(Anumana)。如果我们将“A”、“B”和“C”定义为以下意义的命题：A=“凯特拉在他的房子里”；B=“凯特拉在外面”；C=“凯特拉活着”，如果我们预先假设凯特拉在他的房子里，或者只要他还活着，我们就可以把阿尔萨帕蒂假设凯特拉在外面的例子表述如下：((C⊃(A∨B))Λ(Λ，A))⊃B。这个公式可以通过自然演绎的方式正式证明。库马里拉说，要证明的结论隐含在证明它的理由中，这不是阿尔萨帕蒂的缺陷(阿尔萨帕蒂帕里切达K.29)。库马里拉将普遍命题中所表达的“屋内无活人”与“他在屋外的存在”之间的关系称为“不可分割的关系”(Avinabhavita)。可以肯定的是，在我们意识到凯特拉存在于外界之前，这种不可分割的关系就已经存在了。但我们并没有在前者的基础上意识到这一点。当我们意识到凯特拉在外面的存在时，这种不可分割的关系是相当假定的(阿尔萨帕蒂帕里切达K.30)。虽然库马里拉没有建立命题逻辑的公理方法，但他提出了一个独特的关于逻辑的问题。他坚持认为，必须有可能从另一个命题中推导出一个命题，在这个命题中隐含着一个命题，而不必求助于两个术语之间的普遍关系。凯特拉在外部的存在可以从其主体被限制在凯特拉的命题的组合中推断出来。因此，我们可以有把握地说，这种类型的阿萨帕蒂不能被简化为阿纳马纳。

The way of proof with the help of reductio ad absurdum was called arthapatti and distinguished from inference (anumana) in the hermeneutical school in India, Mimamsa. If we define "A" "B" and "C" as propositions of the following sense : A="Caitra is present in his house" ; B="Caitra is present in outside" ; C="Caitra is living", and if we presuppose that Caitra is present either in his house or in outside as far as he is living, we can formulate the example of arthapatti assuming that Caitra is present in outside as follows : ((C⊃(A∨B)) Λ (CΛ¬A )) ⊃B.This formula can be proven formally by means of natural deduction. Kumarila remarks that it is not a defect of an arthapatti that the conclusion to be proved is implied in the reason which proves it (Arthapattipariccheda k.29). Kumarila calls the relation between "absence of a living person in house" and "his presence in outside" expressed in the universal proposition "inseparable relation" (avinabhavita). It is sure that this inseparable relation is there before we realize Caitra's presence in outside. But we do not realize it on the basis of the former. The inseparable relation is rather assumed when we realize Caitra's presence in outside (Arthapattipariccheda k.30). Though Kumarila did not establish an axiomatic method of propositional logic, he brought up a unique question on Logic. He maintained that it must be possible to derive a proposition from another ones in which it is implied without recourse to the universal relation between two terms. Caitra's presence in outside can be deduced from a combination of propositions of which subject is restricted to Caitra. Therefore we can safely say that this type of arthapatti cannot be reduced to an anumana.