, 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$uÎW (tRu)). , . , . : 1, 0, &, Ú, , à ( ). - : Ø( & Ø).
: , p, p, , . , Ø. . , & à. . :
Ø([ p] & [Øp]);
Ø([ ] & [Ø ])
Ø([ ] & àà[Ø]);
à à Ø(à[] & [Ø à]) Ø(à[] & à[Øà]);
à à Ø(à[ ] & [Ø à]).
( pp) .
. 1973 , . .
F = (W, R) , h, h¢: P P(W) . h£h¢, h(p) £ h¢(p) p Î P, . h£h¢, h h¢, h¢ h.
. , F = (W, R) , h £ h¢ , t Î W . , (W, R, h), t |= A, (W, R, h¢), t |= A.
. . :
(W, R, h), t |= Þ t Î h(p) Þ t Î h¢(p) Þ (W, R, h¢), t |= .
= 1, = 0, . & È . Ø . ÿ ( à), , . (W, R, h), t |= . u Î W, tRu, (W, R, h),
t |= . , , (W, R, h¢), t |= . , (W, R, h¢),
t |= , .
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.
. *() , :
1) Î *(), * ;
2) 1* = 1; 0* = 0;
3) (Ø)* = Ø(*);
4) ( & )* = * & *; ( Ú )* = * Ú *;
5) ( )*() = ("y(xRy A*(y)));
6) (à)*() = ($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 , .
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t4 , , 2 t4, . , , :
$y(R(t4, y) & R(x, y)).
, R(t6, x). q , = t2. , p q , , t.
h¢ . :
, :
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).
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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$t3R(t, t1) & R(t1, t2) & R(t, t3) & Ø("x(R(t, x) R(t2, x)) Ú R(t3, x)).
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(W, R) |= Ø$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);