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中南大学学报(社会科学版)
ZHONGNAN DAXUE XUEBAO(SHEHUI KEXUE BAN)

2004年02月第10卷第1期
   
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文章编号:1672-3104(2004)01-0016-05
 
数理逻辑中的归纳定义和归纳证明
 
孙明湘,沈旭明
 
(中南大学哲学系,湖南长沙,410083)
 
摘  要: 运用数学归纳法能证明一个表示逻辑定理的全称命题的真实性,即通过证明一集合对象具有某性质,从而证明该集合所有对象具有该性质,其原因在于用归纳法证明的集必须首先是一个用归纳定义给出的归纳集,它是与自然数集相同的最小归纳集,它具有封闭性,即:如果该集合的初始元有某性质,并且有一生成函数使得在初始元基础上,可不断生成新的元,如果这些生成元也有该性质,那么由生成元运用生成函数所生成的其他生成元,也有该性质,于是可断定,该集合中所有元都有该性质。归纳集所具有的这种封闭性质,就是数学归纳法原理。它是一前件真而后件不能假的蕴涵命题,因此归纳证明实际是通过证明它的前件(奠基和归纳两步)真,从而证明后件(归纳命题)必然真的演绎证明。
 
关键词: 归纳集;归纳定义;封闭性;归纳证明
 
 
The definition of induction and proving of induction in mathematical logic
 
SUN Ming-xiang,SHENG Xu-ming
 
(Department of Philosophy, Central South University, Changsha 410083, China)
 
Abstract: Why can the truthfulness of general proposition be proved to be truth in limited process by using of the mathematical induction. That is, why can we to prove all objects in aggregation with some natures through proving one or some objects with these natures in aggregation . The reason is that the aggregation proved by induction must be the inductive aggregation that generated from the definition of induction at first ,it is one of the smallest inductive aggregation like natural number aggregation ,it is close. Namely :if primary elements with some natures in the aggregation, and a function can generate new elements with these natures constantly on the basis of the primary elements, other generative elements with such natures can be generated from the generative elements by using of a generative function. It can be concluded that all the elements in this aggregation possess the natures. The clothing of the inductive aggregation is the principle of mathematical induction. It is a implicative proposition that former proposition is true and latter proposition cannot be false. So inductive proof is deductive proposition through proving truthfulness of former proposition (two process of foundation and induction )to be proved that the later proposition (inductive proposition)is truth certainly.
 
Key words: inductive aggregation; definition of proposition; closing; inductive proof
 
 
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