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
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/
, , ,
,
/
(x:=a)
Module #1 - Logic
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Module #1 - Logic
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(x,y)= O x y
x y (x,y)
Module #1 - Logic
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Module #1 - Logic
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(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
P(x,y)=y
Module #1 - Logic
P(x,y)=y
Module #1 - Logic
P(x,y)=y
Module #1 - Logic
P(x,y)=y
Module #1 - Logic
P(x,y)=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
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...
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)
;
, , ...
, !
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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
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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
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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
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1. x ((x) (x))
2. x ((x) (x))
3. x ((x) (x))
4. x ((x) (x)) ,
Module #1 - Logic
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4. x ((x) (x)) , , x ((x) (x)) , a, (a) (a) .
, (a) (a) (a) (a)
, (4)
Module #1 - Logic
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: ..={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
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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
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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]
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