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

借助还原法和谬论的证明方式被称为arthapatti，区别于印度解释学学派Mimamsa的推理（anumana）。如果我们把“A” “B”和“C”定义为以下意义上的命题：A=“Caitra在他的房子里”；B=“Caitra在外面”；C=“Caitra在活着”，如果我们假设Caitra在他活着的时候要么在他的房子里要么在外面，那么我们可以将arthapatti的例子表述如下：（C、（A∨B）） Λ （CΛ¬A）)、B。这个公式可以用自然演绎法形式化地证明。Kumarila评论说，要证明的结论隐含在证明它的原因中，这不是arthapatti的缺陷（Arthapattipariccheda k.29）。鸠摩罗把“一个活着的人在家里的缺席”和“他在外面的存在”之间的关系称为普遍命题“不可分割的关系”（avinabhavita）。可以肯定的是，在我们意识到凯特拉的存在之前，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认识到这一点。当我们意识到Caitra在外部的存在时，这种不可分割的关系就被假设了（Arthapattipariccheda k.30）。鸠摩罗虽然没有建立命题逻辑的公理化方法，但他提出了一个独特的逻辑问题。他坚持认为，必须有可能从另一个命题推导出一个命题，在这个命题中，它是隐含的，而不依赖于两个项之间的普遍关系。Caitra的外在存在可以从命题的组合中推导出来，这些命题的主体被限制在Caitra之内。因此，我们可以有把握地说，这种类型的骨关节炎不能被简化为人类。

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.