1、1* College of Computer Science & Technology, BUPT命题逻辑的推理n 数理逻辑的主要任务是提供一套推理规则。从给定的前提出发，推导出一个结论来。n 前提是一些已知的命题公式，结论是从前提出发 ,应用推理规则推出的命题公式。n 而这推理过程称为演绎或形式证明 . 2* College of Computer Science & Technology, BUPT关于重言蕴含 n 定义：n A、 C是两个公式，如果 A C是重言式则称 A重言蕴含C， 或称 A能逻辑地推出 C 记作 A Cn 说明：n 与 是有区别的， 是联结词， A C仍然是公式，而
2、是公式关系符。n 描述了两个公式的关系，只能说 A C式成立或不成立。n 公式 A C， 当且仅当 A真 C假时才为假，因此 A C要成立的充要条件是对一切赋值如果使 A为真，必须使 C也为真。 3* College of Computer Science & Technology, BUPT重言蕴含 n 定义：设 A1、 A2、 、 Am， C是命题公式，如果（ A1、 A2 、 、 Am） C是重言式，则称 C是前提集合 A1、 A2、 、 Am 的有效结论，或称由 A1、 A2 、 、 Am 逻辑地推论出 C。n 由上面 的定义，上面可以记作 A1、 A2 Am C。 4* Colleg
3、e of Computer Science & Technology, BUPTRules of Inferencen Many of the tautologies are rules of inference. They have the formn P1 P2.Pn Qn wheren Pi are called the hypotheses（ 前提）n andn Q is the conclusion（ 结论） .n Q logically follows from P1 , P2, ,Pn5* College of Computer Science & Technology, BUP
4、TRules of Inference -推理规则n As a rule of inference they take the symbolic form:P1P2.PnQn where means therefore or it follows that.6* College of Computer Science & Technology, BUPTNoten To “prove the theorem“ means to show that the implication is a tautology. n not trying to show that Q (the conclusio
5、n) is true, but only that Q will be true if all the Pi are true. n mathematical proofs often begin with the statement “suppose that P1 , P2, . . . , and Pn are true“ and conclude with the statement “therefore, Q is true.n The proof does not show that Q is true, but simply shows that Q has to be true
6、 if the Pi are all true.7* College of Computer Science & Technology, BUPTRules of Inferencen Arguments based on tautologies represent universally correct methods of reasoning. n Their validity depends only on the form of the statements involved and not on the truth values of the variables they conta
7、in. Such arguments are called rules of inference.8* College of Computer Science & Technology, BUPTRules of Inferencen The various steps in a mathematical proof of a theorem must follow from the use of various rules of inference.n A mathematical proof of a theorem must begin with the hypotheses, proc
8、eed through various steps, each justified by some rule of inference, and arrive at the conclusion.9* College of Computer Science & Technology, BUPTmodus ponens(假言推理规则）n The tautology P (PQ) Q becomesPPQQn This means that whenever P is true and PQ is true we can conclude logically that Q is true.10* College of Computer Science & Technology, BUPTFamous rules of inferencen PP Q Addition（ 析取引入规则）n P QP Simplification（ 合取消去规则）n QP QP Modus Tollens （ 拒取式）n P QQ RP R Hypothetical syllogism（ 假言三段论）
