.


:




:

































 

 

 

 





, KA , ; KA , ; S4A , .

, ( q q) F = (W, R) , R . , 4 ( q q) F = (W, R), R . D( q àq) (W, R) , Dom R = W (.. "tÎW$uÎW (tRu)). , . , . : 1, 0, &, Ú, , à ( ). - : Ø( & Ø).

: , p, p, , . , Ø. . , & à. . :

Ø([ p] & [Øp]);

Ø([ ] & [Ø ])

Ø([ ] & àà[Ø]);

à à Ø(à[] & [Ø à]) Ø(à[] & à[Øà]);

à à Ø(à[ ] & [Ø à]).

( pp) .

. 1973 , . .

F = (W, R) , h, h¢: P P(W) . h£h¢, h(p) £ h¢(p) p Î P, . h£h¢, h h¢, h¢ h.

. , F = (W, R) , h £ h¢ , t Î W . , (W, R, h), t |= A, (W, R, h¢), t |= A.

. . :

(W, R, h), t |= Þ t Î h(p) Þ t Î h¢(p) Þ (W, R, h¢), t |= .

= 1, = 0, . & È . Ø . ÿ ( à), , . (W, R, h), t |= . u Î W, tRu, (W, R, h),
t |= . , , (W, R, h¢), t |= . , (W, R, h¢),
t |= , .

.

. *() , :

1) Î *(), * ;

2) 1* = 1; 0* = 0;

3) (Ø)* = Ø(*);

4) ( & )* = * & *; ( Ú )* = * Ú *;

5) ( )*() = ("y(xRy A*(y)));

6) (à)*() = ($y(xRy & A*(y))).

R . :

( )*() = Ø( & Ø)*() = Ø( *() & Ø*()) = Ø("y(xRy p*(y)) & Ø*());

( (p Ú àq))*(x) = "y(xRy (pÚ àq)*(y)) = "y(xRy (p*(y) Ú $z(yRz & q*(z)))).

( ). . , . .

. , K .

. . :

= à(àq & à([Øc] & à[ p])) & àà[ p],

. p, p, q Øc & à. , Ø F = (W, R). (W, R, h) tÎW , M, t |= A. :

M, t |= à(àq & à(Øc & à[ p])) & àà[ p].

, , , R t. ti : M, ti |= Bi, Bi , , , Ø.

R(x, y) xRy. , R(x, y) = 1, (x, y) Î R.

, *(t). *(t) t1, , t6, :

R(t, t1), R(t1, t2), R(t1, t3), R(t3, t4), R(t, t5), R(t5, t6); M, t2 |= q; M, t3 |= Ø; M, t4| = p; M, t4 |= p.

, h, , , Ø, ( t3). h, . , , t , .

t4 , , 2 t4, . , , :

$y(R(t4, y) & R(x, y)).

, R(t6, x). q , = t2. , p q , , t.

. :

, :

p*(x) = $y(R(t4, y) & R(y, x)) Ú R(t6, x)

. q x, x = t2. , q*(x) = (x = t2).

A*(t):

1) $ti ;

2) *() $y(R(t4, y) & R(y, x)) Ú R(t6, x);

3) *(y) $z(R(t4, z) & R(z, y)) Ú R(t6, y);

4) *(z), *(t), *(t4) ..;

5) q*() x = t2, q*(y) y = t2 ..;

6) ( )* 1.

: a(t). :

1) (W, R, h), t |= A t Î W h;

2) (W, R, h), t |= A t Î W t1, , t6;

3) (W, R), t |= a(t) t Î W ( ).

Ø (W, R) , (W, R) |= Ø$ta(t). , a(t), (W, R).

, Ø Ø$ta(t) (W, R), Ø .

, Ø . ,

Ø = ( ) = Ø([ ] & [Ø])

Ø([ ] & [Ø]), "x R(x, x).

: Ø, .

: , Ø.

:

1) .

2) , , t; (, t1, t2, ), R , .

3) , .

4) *(t) :

*(t) $s, s , à;

1;

*(), q*(y), , ;

, .

a(t), , (W, R) t .

, Ø , (W, R) Ø$ta(t) ( , R).

, .

. *() : , , , ,
t, t1, t2, . :

*() = (x = t) Ú R(t, x) Ú $x1(R(t, x1) & R(x1, x)) Ú

Ú $x1$x2$xm(R(t, x1) & R(x1, x2) & & R(xm, x)).

2

Øa(t) Ø = ( ). & Ø. 1 *(t) = *(t) & Ø*(t). *(t) 1, . *(t) *() = R(t, x), *()=1 & Ø*(t)=ØR(t, t). : Øa(t) =
= R(t, t). , (W, R) |= Ø$ta(t). , (W,R) |= "tR(t, t).

3

Ø = ( ). , = & Ø . *(t) = *(t) & Ø *(t). *(t) = 1 *() = R(t, x). Ø *(t) :

Ø"x(R(t, x) *(x)) = Ø"x"y(R(t, x) (R(x, y) *(y))).

, a(t) = Ø"x"y(R(t, x) & R(x, y) R(t, y)). (W, R) |= Øa(t)

(W, R) |= "t"x"y(R(t, x) & R(x, y) R(t, y)).

4

Ø àØ. Ø(Ø & Ø àØ). Ø([Ø] & à[ p]). = [Ø] & à[ p]. :

*(t) = Øp*(t) & $t1(R(t, t1) & p*(t1)).

$t1 :

*(t) = $t1(Øp*(t) & R(t, t1) & p*(t1)).

p*(t1) : *() =R(t1, x). , *(t) = $t1(ØR(t1, t) & R(t, t1)). Ø (W, R) ,

(W, R) |= Ø$t$t1(ØR(t1, t) & R(t, t1)).

, R :

"t"t1(R(t, t1) R(t1, t)).

5

, . *(t) , = p Ú q:

A = à(àq & à([Ø( p Ú q)] & à[ p])) & àà[ p].

:

*(t) = R(t, t1) & R(t1, t2) & R(t1, t3) & R(t3, t4) & R(t, t5) & R(t5, t6) & q*(t2) & Øc*(t3) & p*(t4) & p*(t6).

: p*(t4) & p*(t6) = 1. ,

*() = $y(R(t4, y) & R(y, x)) Ú R(t6, x);

q*(x) = (x = t2).

*(t3) = p*(t3) Ú q*(t3) *(t), :

*(t) = R(t, t1) & R(t1, t2) & R(t1, t3) & R(t3, t4) & R(t, t5) & R(t5, t6) & Ø( p*(t3) Ú q*(t3));

a(t) = $x1$x2$x3$x4$x5$x6(R(t, t1) & R(t1, t2) & R(t1, t3) & R(t3, t4) & R(t, t5) & R(t5, t6) & Ø("x(R(t3, x) p*(x)) Ú (t3 = t2)) = $x1$x2$x3$x4$x5$x6(R(t, t1) & R(t1, t2) & R(t1, t3) & R(t3, t4) & R(t, t5) & R(t5, t6) & Ø("x(R(t3, x) $y(R(t4, y) & R(y, x)) Ú R(t6, x)) Ú (t3 = t2))

6

Ø(àà[ p] & à[ q] & [Ø( p Ú q)] (W, R), . :

= àà[ p] & à[ q] & [Ø( p Ú q)];

a(t) = $t1$t2$t3R(t, t1) & R(t1, t2) & p*(t2) & R(t, t3) & q*(t3) & (Ø( p*(t) Ú q*(t))).

:

p*(t2) = "x(R(t2, x) p*(x)),

q*(t3) = "x(R(t3, x) q*(x)),

:

p*(x) = R(t2, x); q*(x) = R(t3, x).

a(t) , p*(t2) = 1 , q*(t3) = 1, :

a(t) = $t1$t2$t3R(t, t1) & R(t1, t2) & R(t, t3) & Ø("x(R(t, x) R(t2, x)) Ú R(t3, x)).

(W, R) |= Ø$ta(t) 1- :

"t"t1"t2"t3(ØR(t, t1) Ú ØR(t1, t2) Ú ØR(t, t3) Ú "x(R(t, x) R(t2, x)) Ú R(t3, x).


, :

1) à (: $u(wRu));

2) à à (: wRu & wRu vRu);

3) à (: wRv & wRu v = u);

4) (: wRv $u(wRu & uRv));

5) à à (: wRv & wRx $u(vRu & xRu)).

[F] [P] . à <F> = Ø[F]Ø
<> = Ø[]Ø. M = (W, R, h) h: P P(W). , [F]A [P]A :

M, t |= [F]A, M, u |= A u Î W, , R(t, u);





:


: 2016-11-12; !; : 296 |


:

:

, .
==> ...

885 - | 793 -


© 2015-2024 lektsii.org - -

: 0.072 .