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离散  数学. 计算机学院. 冯伟森. Email : [email protected] 2014年8月27日星期三. 习题课二. 第二章. 一、基本概念 全总个体域(全论域)、全称量词、存在量词、特性谓词、指导(作用)变元、辖域(作用域)、约束变元、自由变元、约束变元的改名规则、自由变元的代入规则、常量符号、变量符号、函数符号、谓词符号、谓词公式、公式的解释、永真公式 ( 重言式 ) 、永假公式 ( 矛盾式,不可满足公式)、.

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3610317

[email protected]

2014827


3610317


3610317

() (


3610317

()SkolemUS()ES()UG()EG()


3610317

  • ()


3610317

  • USESUGEG


3610317

  • 1

  • A(x):x

  • B(x):x

  • (x)(B(x)A(x))(x)(A(x)B(x))

  • (x)(A(x)B(x))

  • abab

  • A(x):x

  • F(xy):xy

  • Gx,y):

  • ()()A()A()

  • (F()G,))


3610317

A(x):

B(x):x

F(xy):

B(x)A(x)F(x)

(x)A(x)F(x)B

A(x):

(x)

(x)

(x)A(x)(x)(x)


3610317

  • A(x):

  • F(xy):

  • P

  • Q:

  • a:

  • (x)A(x)F(xa)PQ


3610317

  • 2

  • 00x-10x-1x+10

  • A(x):

  • xA(x)A(x)A()

  • A(x A(x)

  • (Why?)


3610317

A(x):B(x):x:

(x) A(x) B(x) A() B()

F

A(x):B(x):x

(x) B(x) A(x)

F


3610317

  • A(x):

  • B(x):x

  • C(x):x

  • D(x)

  • (x,y)

T


3610317

1

0

0

D

p(x):x;

s(x):x;

EQUAL(x,y):x=y

(x)(y)[EQUAL(y,s(x))

(z)[EQUAL(z,s(x))EQUAL(y,z)]]


3610317

0

(x)[EQUAL(0,s(x))

(x)[EQUAL(x,0)

(y)[EQUAL(y,p(x))(z)[EQUAL(z,p(x))

EQUAL(y,z)]]]


3610317

2


3610317

2(1)

P(x)x

Q(x)x

R(x)x

S(x)x

(x)(P(x)(Q(x)R(x))

(x)(P(x)(R(x)S(x)))

(x)(P(x)S(x)),

(x)(P(x)S(x))

(x)(P(x)Q(x))


3610317

2(2)

(1) (x)(P(x)S(x)) P

(2) (x)(P(x)S(x)) T,(1),E

(3) P(c)S(c) ES,(2)

(4) P(c) T,(3),I

(5) (x)(P(x)(Q(x)R(x)) P

(6) P(c)Q(c)R(c) US,(5)

(7) Q(c)R(c) T,(4),(6),I

(8)(x)(P(x)(R(x)S(x))) P

(9) P(c)(R(c)S(c)) US8

(10) R(c)S(c) T,(4),(9),I


3610317

2(3)

(11)(R(c)S(c))(S(c)R(c)) T,(10),E

(12) R(c)S(c) T,(12),I

(13)S(c) T,(3),I

(14) R(c) T,(12),(13),I

(15) Q(c) T,(7),(14),I

(16) P(c)Q(c) T,(4),(15),I

(17) (x)(P(x)Q(x)) ES,(16)


3610317

3

S(x)xT(x)x

P(x)xL(x,y)xy

(x)(S(x)(y)(T(y)L(x,y)))

(x)(y)((S(x)P(y))L(x,y))

(x)(T(x)P(x))

(x)(S(x)(y)(T(y)L(x,y)))

(x)(y)((S(x)P(y))L(x,y))

(x)(T(x)P(x))


3610317

  • (x)(S(x)(y)(T(y)L(x,y)))P

  • S(c)(y)(T(y)L(c,y))ES,1)

  • S(c)T,2),I

  • (y)(T(y)L(c,y))T,2),I

  • T(x)L(c,x)US,4)

  • (x)(y)((S(x)P(y))L(x,y))P

  • (y)((S(c)P(y))L(c,y))US,6)

  • (S(c)P(x))L(c,x)US,7)

  • S(c)(P(x)L(c,x))T,8),E

  • P(x)L(c,x)T,3),8),E

  • L(c,x)P(x)T,10),E

  • T(x)P(x)T,5),11),E

  • (x)(T(x)P(x))US,12)


3610317

4

(x)(P(x)Q(x)) (x)((y)(P(y)R(x,y))(y)(Q(y)R(x,y)))

1.

  • (x)((y)(P(y)R(x,y))

    (y)(Q(y)R(x,y)))P()

  • (x)((y)(P(y)R(x,y))

    (y)(Q(y)R(x,y)))T,1),E

  • ((y)(P(y)R(c,y))

    (y)(Q(y)R(c,y)))ES,2)

  • (y)(P(y)R(c,y))

    (y)(Q(y)R(c,y))T,3),E

  • (y)(P(y)R(c,y))T,4),I


3610317

4(1)

  • P(b)R(c,b)ES,5)

  • P(b)T,6),I

  • (x)(P(x)Q(x)) P

  • P(b)Q(b)US,8)

  • Q(b)T,7),9),I

  • R(c,b)T,6),I

  • (Q(b)R(c,b) T,10),11)I

  • (y)(Q(y)R(c,y))T,4),E

  • (y)(Q(y)R(c,y))T,13),E

  • (Q(b)R(c,b))US,14)

  • (Q(b)R(c,b))(Q(b)R(c,b))US,14)


3610317

4(2)

x

2.CP

  • (y)(P(y)R(x,y)) P()

  • P(f(x))R(x,f(x))ES1)

  • P(f(x))T,2)

  • (x)(P(x)Q(x))P

  • P(f(x))Q(f(x))US,4)

  • Q(f(x))T,3),5),I

  • R(x,f(x))T,2)

  • Q(f(x))R(x,f(x))T,6),7),I

  • (y)(Q(y)R(x,y))EG,8)

  • (y)(P(y)R(x,y))(y)(Q(y)R(x,y)) CP,1),9)

  • (x)((y)(P(y)R(x,y))(y)(Q(y)R(x,y)))

    • EG,10)


3610317

5

Q(x)xR(x)x N(x)x C(x)x

(x)(Q(x)R(x), (x)(N(x)R(x))

(x)(C(x) R(x))

(x)(C(x)(Q(x)N(x))


3610317

(x)(Q(x)R(x))P

Q(x)R(x)US(1)

(x)(N(x)R(x))P

N(x)R(x)US(3)

(x)(C(x)R(x))P

C(x)R(x)US(5)

R(x)C(x)T(6)E


3610317

Q(x)C(x)T(2)(7)I

N(x)C(x)T(4)(7)I

(Q(x)C(x))(N(x)C(x))

T(8)(9)E

C(x)(Q(x)N(x))T(10)EI

(x)(C(x)(Q(x)N(x))UG(11)


3610317

6

P(x)xQ(x)x R(x)x S(x)x

(x)(P(x)Q(x)R(x))),

(x)(P(x)(S(x)Q(x)))

(x)(P(x)S(x))(x)(P(x)S(x))

(x)(P(x)R(x))


3610317

(x)(P(x)S(x))(x)(P(x)S(x))P

(x)(P(x)S(x))TE

(x)(P(x)S(x))TE

P(c) S(c)ES

P(c) TE

S(c) TE

(x)(P(x)Q(x)R(x)))P

P(c)Q(c)R(c))US


3610317

Q(c)R(c) TI

(x)(P(x)(S(x)Q(x)))P

11P(c)(S(c)Q(c))US

S(c)Q(c)T11I

Q(c)S(c)T12E

Q(c)T13EI

R(c)TE

P(c)R(c)T15E

(x)(P(x)R(x))EG16


3610317

1

  • 1

  • 2a[N]n


3610317

  • 3

  • a[N]

  • a[N]2n

    4

    C


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