models of comp(P) are models of P, so all the classes of models described are
contained in the class of 3-valued models of P. Since they all contain the class
of 2-valued Herbrand models of comp(P), it is enough to show that if Q is true
in all 2-valued Herbrand models of comp(P), it is true in all 3-valued models of
P.
The hypothesis implies that Q is true in lfp(T
p
). If
Q is 3
Xl
. . . 3x
m
(A
1
(x
1
, . . . ,χ
Γ
) Λ . . . Λ A
s
(x
1
, . . . , x
r
) )
then
A
1
(t
iy
. . . ,
t
r
) Λ · · · Λ A
s
(t
lf
. . . ,
t
r
)
is true in lfp(T
p
) for some ground terms t
p
...,t
r
. Since lfp(T
p
) = T
p
fco all the
A
s
(tj,...,t
r
) are in T
p
fn for some finite n. By induction
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