05 DM PredicateLogic 0803 01

Module #1 - Logic

3/8/2013 11

HY118-

Kees van

Deemter,

University of Aberdeen

, 08/03/2013

. e-mail: [email protected]

Module #1 - Logic

3/8/2013 22

Module #1 - Logic

3/8/2013 33

/

, , ,

,

/

(x:=a)

Module #1 - Logic

3/8/2013 44

Module #1 - Logic

3/8/2013 55

(x,y)= O x y

x y (x,y)

Module #1 - Logic

3/8/2013 66

Module #1 - Logic

3/8/2013 77

(x,y)= O x y

y x (x,y)

Module #1 - Logic

3/8/2013 88

B(x,y)= x y,

:

Module #1 - Logic

3/8/2013 99

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1010

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1111

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1212

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1313

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1414

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1515

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1616

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1717

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

3/8/2013 1818

x(y (x,y))= ,

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

x(y (x,y))=

y(x (x,y))=

Module #1 - Logic

P(x,y)=y

Module #1 - Logic

3/8/2013 (c)2001-2004, Michael

P. Frank

26

..

( );

. ,

http://en.wikipedia.org/wiki/Empty_domain . :

x .

x .

..

(Free Logic):

http://en.wikipedia.org/wiki/Free_logic

,

.

Topic #3 Predicate Logic

Module #1 - Logic

3/8/2013 2727

...

P=

D .

Q(x) = x

P : x Q(x)

;

Module #1 - Logic

3/8/2013 2828

...

P=

D .

Q(x) = x

P : x Q(x)

;

, , ...

, !

Module #1 - Logic

3/8/2013 2929

o ;

P=

D ( )

Q(x) = x

(x) = x

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

;

, , ...

, !

Module #1 - Logic

3/8/2013 (c)2001-2004, Michael

P. Frank

30

..

1. x(y R(x,y))

2. y(x R(x,y))

3. x(y R(x,y))

3 2 .

2 , 1

:

3 2

2 1

Topic #3 Predicate Logic

Module #1 - Logic

3/8/2013 3131

, ..

, ..,

x>0 P(x)

x , P(x).

;

Module #1 - Logic

3/8/2013 3232

, ..

, ..,

x>0 P(x)

x , P(x).

;

x (x>0 P(x))

Module #1 - Logic

3/8/2013 3333

x>0 P(x) :

x ,

P(x).

x>0 P(x) ;

=???

Module #1 - Logic

3/8/2013 3434

x>0 P(x) :

x ,

P(x).

x>0 P(x) ;

=x (x>0 P(x))

Module #1 - Logic

3/8/2013 3535

x>0 P(x) x , P(x).

=x (x>0 P(x)) x>0 P(x)

x , P(x).

=x (x>0 P(x))

Module #1 - Logic

3/8/2013 3636

x (x>0 P(x)) ; ; ; x , P(x).

x (x>0 P(x)) ; ; ;

x x>0 P(x)

... x

Module #1 - Logic

3/8/2013 3737

. (D). 118.

(x) / x /

(x) / x /

1. x ((x) (x))

2. x ((x) (x))

3. x ((x) (x))

4. x ((x) (x))

Module #1 - Logic

3/8/2013 3838

1. x ((x) (x))

2. x ((x) (x))

3. x ((x) (x))

4. x ((x) (x)) ,

Module #1 - Logic

3/8/2013 3939

4. x ((x) (x)) , , x ((x) (x)) , a, (a) (a) .

, (a) (a) (a) (a)

, (4)

Module #1 - Logic

3/8/2013 4040

: ..={a, b, c,} x P(x) P(a) P(b) P(c) x P(x) P(a) P(b) P(c)

, :x P(x) (x P(x)) x P(x) x P(x) (x P(x)) x P(x)

;

Module #1 - Logic

3/8/2013 4141

x P(x) x P(x)

P(x) x,

x P(x) x P(x)

,

Module #1 - Logic

3/8/2013 4242

:

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

x P(x) x P(x)

, ( )

Module #1 - Logic

3/8/2013 4343

x y P(x,y) y x P(x,y)

x y P(x,y) y x P(x,y)

Module #1 - Logic

3/8/2013 4444

:

xyz P(x,y,z) . x y z P(x,y,z)

xyz P(x,y,z) . x y z P(x,y,z)

Module #1 - Logic

3/8/2013 4545

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

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

Module #1 - Logic

3/8/2013 4646

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

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

;

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

?

Module #1 - Logic

3/8/2013 4747

;

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

!!!

(. )

P(x): x 24

Q(x): x 4

Module #1 - Logic

3/8/2013 4848

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

P(x): x 24

Q(x): x 4

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

24/4 4/3 [FALSE]

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

24/4

4/3 [ TRUE]

05 DM PredicateLogic 0803 01

Documents

Transcript of 05 DM PredicateLogic 0803 01