数学专业英语 2-11 A

1、2.11 数理逻辑入门数理逻辑入门Elementary Mathematical Logic成员：成员：数理逻辑:又称符号逻辑、理论逻辑。它既是数学的一个分支也是逻辑学de一个分支。是用数学方法研究逻辑或形式逻辑学科。其研究对象是对证和计算这两个直观概念进行符号以后的形式系统。数理逻辑是数学基础de一个不可缺少的组部分数理逻辑包括: “命题演算”和“谓词演算”。数理逻辑就是精确化、数学化的形式逻辑。它是现代计算机技术的基础。新的时代将是数学大发展的时代，而数理逻辑在其中将会起到很关键的作用。逻辑是探索、阐述和确立有效推理原则的学科，最早由古希腊学者亚里士多德创建的。用数学的方法研究关于推理、证

2、明等问题的学科就叫做数理逻辑。也叫做符号逻辑New Words:assert 断言，主张断言，主张 predicate 谓词谓词 conjunction 合取合取 quantifier 量词量词connective 连词连词 quantification 量词化量词化 disjunction 析取析取 statement 语句语句exclusive 执行执行 universe 通集，底集通集，底集 propositional 命题命题 Statements involving variables, such as “x3”, “x+y=3”, “x+y=z” are often found i

3、n mathematical assertion and in computer programs.11-A Predicates包含变量的语句，比如包含变量的语句，比如“x3”, “x+y=3”, “x+y=z” 常出现在数常出现在数学论断和计算机程序中。学论断和计算机程序中。These statements are neither true nor false when the values of the variables are not specified. In this section we will discuss the ways that propositions can b

4、e produced from such statements.若未给语句中的所有变量赋值，则不能判定该语句是真是假，若未给语句中的所有变量赋值，则不能判定该语句是真是假，本节要讨论由这种语句生成命题的方法。本节要讨论由这种语句生成命题的方法。 The statement “x is greater than 3” has two parts. The first part, the variables, is the subject of the statement.语句语句“x大于大于3”分成两部分，第一部分，变量，是语分成两部分，第一部分，变量，是语句的主语。句的主语。The secon

5、d part-the predicate, “is greater than 3”-refers to a property that the subject of the statement can have.第二部分，谓语，第二部分，谓语，“大于大于3”，指的是语句主语具有的，指的是语句主语具有的性质。性质。We can denote the statement “x is greater than 3” by P(x), where P denotes the predicate “is greater than 3” and x is the variable.把语句把语句“x大于大于

6、3”记为记为P(x), 其中其中P表示谓词表示谓词“大于大于3”，而而x是变量。是变量。The statement P(x) is also said to be the value of the propositional function P at x. Once a value has been assigned to the variable x, the statements P(x) becomes a proposition and has a truth value.语句语句P(x)也称为命题函数也称为命题函数P在在x点处的值。一旦赋予变点处的值。一旦赋予变量量x一个值，语句一

7、个值，语句P(x)就成为一个命题就成为一个命题,有了真假值。有了真假值。EXAMPE1. Let P(x) denote the statement “x3”. What are the truth values of P(4) and P(2)?例1 让P(x)表示该语句 x 3”。P(4)和P(2) 是真的?Solution:The statement P(4) is obtained by setting x=4 in the statement “x3”. Hence, P(4) , which is the statement”43”, is true. However, P(2),

8、which is the statement “23” , is false. 解:语句P(4)设置了x = 4在语句中表示“x 3”。因此,P(4),这是“4 3”语句是真的然而,P(2)是2 3,是假的。We can also have statements that involve more than one variable. For instance, consider the statement”x=y+3”. We can denote this statement by Q(x,y) , where x and y are variables and Q is the pred

9、icate. When values are assigned to the variables x and y , the statement Q(x,y) has a truth value.我们也可以有涉及多个变量de语句.例如,考虑该语句”x=y+3”。我们可以表示这种说法Q(x,y),当x和y是变量及Q是谓语,当值是指定的变量x和y,语句(x,y)有一个真值,EXAMPLE2.Let Q(x, y) denote the statement “x=y+3”. What are the truth values of the propositions Q(1,2) and Q(3,0)

10、?例2:让(x,y)表示该语句“ x = y + 3” Q(1,2)及Q(3,0)哪个 是真的命题 To obtain Q(1,2), set x=1 and y=2 in the statement Q(x,y). Hence, Q(1,2) is the statement “1=2+3”, which is false. The statement Q(3,0) is the proposition “3=0+3”, which is true.Q(1,2),语句Q(x,y)中设x = 1,y = 2。因此,语句Q(1,2 )中”1 = 2 + 3”,这是假的。语句Q(3,0)是命题“3

11、 = 0 + 3”,这是真的。Similarly,we can let R(x,y,z) denote the statement “x+y=z” . when values are assigned to the variables x, y, and z, this statement has a truth value.同样地,我们可以让R(x,y,z)表示语句 x + y = z”.当值被指定到变量x,y,和z,该语句是真值。EXAMPLEX3. What are the truth values of the propositions R(1,2,3) and R(0,0,1)?例3

12、:R(1、2、3)和R(0,0,- 1) 哪个是真的命题?Solution:The proposition R(1,2,3) is obtained by setting x=1, y=2, and z=3 in the statement R(x , y, z) . We see thatR(1,2,3) is the statement “1+2=3”, which is true. Also note that R(0,0,1) , which is the statement “0+0=1” is false. 解：命题解：命题R(1、2、3)在语句在语句R(x,y,z)中得到中得到x

13、 = 1,y= 2,Z = 3表示。我们看到表示。我们看到,R(1、2、3)语句语句“1 + 2 = 3” ,这是真这是真的。还注意到的。还注意到,R(0,0,- 1),语句语句“0 + 0 = 1”是假的。是假的。In general , a statement involving the n variables x,x2,xn can be denoted by P(x1,x2,xn).一般说来一般说来,一个语句中包含一个语句中包含n变量变量x,x2,xn 可以用可以用P(x1,x2,xn)表示表示A statement of the form P(x1,x2,xn) is the val

14、ue of the propositional function P at the n-tupe (x1,x2,xn), and P is also called a predicate.一个语句的形式一个语句的形式P(x1,x2,xn)是命题函数是命题函数P在在n的数组的数组(x1,x2,xn)下的值，下的值，P也可称为谓语也可称为谓语Propositional functions occur in computer programs , as the following example demonstrates命题功能发生在计算机程序当中命题功能发生在计算机程序当中,下面用例子来演示下面用

15、例子来演示EXAMPLE4. Consider the statement if x 0 then x:=x+1.When this statement is encountered in a program.,the value of the variable x at that point in the execution of the program is inserted into P(x) , which is “x0”例例4： 考虑考虑ru下语句下语句: 如果如果 x 0 且且 x:=x+1当这条语句中遇到一个程序当这条语句中遇到一个程序,变量的值变量的值x此时在程序的此时在程序的执行过程中插入执行过程中插入P(x),这是这是“x 0 如果如果P(x) P(x) 对这个对这个x x的值是真的的值是真的, ,赋值语句赋值语句x:= x + 1x:= x + 1是是执行执行, ,所以所以x x的值增加的值增加1 1。如果。如果P(x) P(x) 对于这个对于这个x x的值是的值是错误的错误的, ,赋值语句不执行赋值语句不执行, ,所以所以x x的值是没有改变的值是没有改变dede。I