4 15 00 ; 321 5 16 45 321 http://abelardfletkeioacjp/person/takemura/class2html 1 1 11 1 1 1 vocabulary (propositional connectives):,,, (quantifires): (individual variables): x, y, z, (individual constatns): a, b, c, d, e, n (n-ary predicate symbols): P ( 1,, n ), Q( 1,, n ), 1 1 P ( 1,, n ) n t 1,, t n P (t 1,, t n ) atomic formula 2 ( ) 3 ( B) ( B) ( B) 4 x ( x) 5 x ( x) U( 1, 2, 3 ) 3 Q( 1, 2 ) 2 (( y( xu(a, x, y))) ( z( Q(c, z)))) 1
U( xq(x)) Q(a) 1 P ( 1 ) R( 1 ) 1 Q( 1, 2 ) 2 1 ( x(p (x) ( y(q(x, y) ( z( R(z))))))) 2 ( z(( y( xq(x, y))) R(z))) 3 ( x(p (x) ( ( yq(a, y) ( zr(z)))))) x, x 1 ( ) 1 2 x x ( ) ( x) ( x) (( y( xu(a, x, y))) ( z( Q(c, z)))) y xu(a, x, y) z Q(c, z) 2 1 ( x(p (x) ( y(q(x, y) ( z( R(z))))))) 2 ( z(( y( xp (x, y))) R(z))) 3 ( x(p (x) ( ( yq(a, b, y) ( zr(z)))))) 2
12 N( ) = ( 1, 2 ) 1 2 = ( 1, 2 ) 1 = 2 x(n(x) (2 x = 16)) 2 16 2 x 3 x(n(x) (x x = 16)) y(n(y) (2 y = 16)) x(n(x) (2 x = 16)) 2 16 bound variable x y y(n(y) (2 y = z)) 2 z z free variable y(n(y) (2 y = x)) 2 x x z y(n(y) (2 y = z)) xp (x) Q(x) 3
P x Q x xp (x) Q(x) x x Q(x) x x(p (x) Q(x)) P (x) Q(x) x x x x x x x x scope xp (x) Q(x) x P (x) x(p (x) Q(x)) x P (x) Q(x) x x x x x x xp (x) Q(x) x x 1 scope of y {}}{ scope of x {}}{ y (( x (x = y)) (y = x)) x = y x x x = y y y = x y y y = x x 4 1 ( x(p (x) ( y(q(x, y) ( z( R(z))))))) 2 ( z(( y( xp (x, y))) R(z))) 3 ( x(p (x) ( ( yq(a, b, y) ( zr(z)))))) 2 x x x x( x(x = y)) yp (y) Q(x) x y 4
1 y z y ( xp (x, y) Q(y, x)) = z ( xp (x, z) Q(z, x)) 2 y x z y ( xp (x, y) Q(y, x)) z ( zp (z, z) Q(z, x)) 3 x w y x y ( xp (x, y) Q(y, x)) x ( wp (w, x) Q(x, x)) Remark 1 x ylove(x, y) x y y xlove(y, x) y xlove(x, y) a b c 1 a b c a b c 2 1 x ylove(x, y) y xlove(x, y) b 2 x ylove(x, y) b y xlove(x, y) a b c 5
13 Gentzen,,, I E B I B B B B I E B B B B E1 B B E2 I B B [] n B B I, n B B B B 6
I [] n I, n E I B B B E B I1 B B I2 E B B C B C B [] n C C [B] n C E, n E E E E R [ ] n R, n 7
1 ( provability ) proof figure conclusion B 1,, B n B 1,, B n provable B 1,, B n,,, 14 x x x + 1 y x y(x + 1 = y) x 2 2 + 1 y(2 + 1 = y) x [x] x x x t [x] x t [x := t] [t/x] 8
[x] y(x + 1 = y) y [x := 2] y(2 + 1 = y) [x] y(x + x = y) [x := 2] y(2 + 2 = y) x z w [x, z, w] [x, z] y(x + z = y) x z 2 x z y(x + 1 = y) x 2 x z y(z + 1 = y) y x y(y + 1 = y) 15 x P ( ) a xp (x) P (a) 9
x x t [x := t] x [x := t] x x x x [x] 2 1 x 2 x t x [x] [x := t] x t [x := t] [x := t] 3 () x, y x y(x < y) y(x < y) x y(x < y) y(y < y) 5 1 x(p (x) (Q(x) R(x))), P (d) Q(d) 2 P (d), x(p (x) (P (x) Q(x))) Q(d) 3 x(p (x) Q(x)), x(p (x) R(x)), P (d) Q(d) R(d) 4 x(p (x) (Q(x) R(x))), x(p (x) Q(x)) R(d) 5 x(p (x) Q(x)), Q(c) P (c) 6 x y(p (x) (P (y) Q(y)) Q(d) 10
16 I x I I [x] [x] [x] x x [x] x I 4 P ( ) Q( ) R( ) 1 x(p (x) Q(x)) 2 x(q(x) R(x)) x(p (x) R(x)) 1 x x P (x) 2 1 x x x(p (x) Q(x)) P (x) Q(x) Q(x) P (x) 3 2 x x x(q(x) R(x)) Q(x) R(x) x(p (x) Q(x)) P (x) Q(x) Q(x) R(x) P (x) 11
4 x x x x(q(x) R(x)) Q(x) R(x) x(p (x) Q(x)) P (x) Q(x) [ P (x) ] 1 Q(x) R(x) P (x) R(x) I, 1 5 x x x x x(q(x) R(x)) Q(x) R(x) x(p (x) Q(x)) P (x) Q(x) [ P (x) ] 1 Q(x) R(x) P (x) R(x) I, 1 x(p (x) R(x)) I 3 I P (x) R(x) xr(x) I R(x) I xr(x) R(x) P (x) x I [P (x)] 1 R(x) xr(x) I P (x) xr(x) I, 1 x x 1 12
x R(x) x x P (x) R(x) x 5 I 0 [x = 0] 1 x(x = 0) I x = 0 x(x = 0) 1 y(y = 0 x(x = 0)) I 0 = 0 x(x = 0) 6 ( I) x(p (x) Q(x)) xp (x) xq(x) 6 1 x yp (x, y) y xp (x, y) x(p (x) Q(x)) x(p (x) Q(x)) P (x) Q(x) P (x) Q(x) E E P (x) xp (x) I Q(x) xq(x) I I xp (x) xq(x) 2 x(p (x) Q(x)), x(q(x) R(x)) x(p (x) R(x)) 3 xp (x) x P (x) 4 x((p (x) Q(x)) R(x)) x(p (x) (Q(x) R(x)) 5 x( P (x) (P (x) Q(x))) 6 x(p (x) Q(x)), x(q(x) R(x)) x((p (x) Q(x)) R(x)) 7 1 2 8 1 2 13
9 1 2 10 1 2 11 1 2 2 7 2 7 12 x(p (x) Q(x)) xp (x) xq(x) 13 x(p (x) Q(x)) xp (x) xq(x) 14 xp (x) xq(x) x(p (x) Q(x)) 15 xp (x) xq(x) x(p (x) Q(x)) 16 x(p (x) Q(x)) xp (x) xq(x) 17 xp (x) x( P (x)) x 18 x( P (x)) xp (x) x 19 xp (x) x( P (x)) x 20 x( P (x)) xp (x) x 21 xp (x) x( P (x)) x 22 x( P (x)) xp (x) x 14
17 I x 2 3 2 2 3 2 P ( ) 3 a P (a) xp (x) I x [x := t] [x := t] [x := t] x I 7 x yl(x, y) x yl(x, y) 8 xp (x) x P (x) x yl(x, y) yl(a, y) x yl(x, y) I [ P (x)] 2 [ x P (x)] 1 x P (x) I E P (x) R, 2 xp (x) xp (x) I E x P (x) R, 1 15
Remark 2 I x x z(z = z) z = z x(z = x) I z = z z = x[z/x] 7 1 xp (x) xp (x) 2 x(p (x) (Q(x) R(x))), x(q(x) S(x)) x(r(x) S(x)) 3 x(p (x) (P (x) Q(x))) Q(x) 4 x(p (x) (Q(x) R(x)), x(p (x) S(x)), xp (x) x((q(x) R(x)) S(x)) 5 xp (x) xq(x) x y(p (x) Q(y)) 6 1 2 7 1 2 8 1 7 2 2 7 2 16
18 E x [x := t] E x x C [x := t] C 1 C x 2 x [x := t] [x := t] C [ [x := t] ] n x C C E, n 9 1 xp (x) 2 x(p (x) R(x)) R(x) 1 1 a P (a) a a 2 2 a a a x(p (x) R(x)) P (a) R(a) 17
3 1 a x(p (x) R(x)) P (a) R(a) R(a) P (a) 4 x(p (x) R(x)) P (a) R(a) R(a) xr(x) I P (a) 5 xr(x) a a a a P (a) a a xr(x) a P (a) xp (x) E xp (x) x(p (x) R(x)) P (a) R(a) [ P (a) ] 1 R(a) xr(x) I E, 1 xr(x) 3 E 1 xp (x) x(p (x) R(x)) P (a) R(a) R(a) R(a) E P (a) R(a) a a E (i) xp (x) (ii) P (a) (iii) a P (a) a C (iv) a P (a) C 18
10 x P (x) xp (x) x P (x) P (a) [P (a)] 1 [ xp (x)] 2 E, 1 xp (x) I, 2 E 8 1 x yp (x, y) y xp (x, y) 2 x((x) B(x)) (x) B(x) 3 1 2 4 1 2 5 1 2 6 1 2 7 1 2 8 xp (x) xq(x) x(p (x) Q(x)) 9 x(p (x) Q(x)) xp (x) xq(x) 10 x(p (x) Q(x)) xp (x) xq(x) 11 xp (x) x(p (x) ) x 12 x(p (x) ) xp (x) x 19
13 x(p (x) Q(x)) xp (x) xq(x) 14 xp (x) xq(x) x(p (x) Q(x)) 15 x(p (x) Q(x)) xp (x) xq(x) 16 xp (x) x P (x) 17 x P (x) xp (x) 18 xp (x) x P (x) 19 x P (x) xp (x) 19 I B B B B I E B B B B E1 B B E2 I B B [] n B B I, n B B B B 20
I [] n I, n E I B B B E B I1 B B I2 E B B C B C B [] n C C [B] n C E, n I [x] [x] x x [x] x I x t [x := t] x [x := t] x y y [x := y] 21
I [x := t] x [x := t] x I E x [x := t] C 1 C x 2 x [x := t] [x := t] C [ [x := t] ] n x C C E, n E E E E R [ ] n R, n 22