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In-Depth Information
ɂ
ɂ
2
∨
…
∨
ɂ
k
, where each
ɂ
i
(1
±
i
±
k) is an objective formula with the form
1
∨
B
ϕ
i
,1
∧
…
∧
B
ϕ
i,mi
∧
¬
B
ϕ
i
,1
∧
…
∧
¬
B
ϕ
i,ni
∧
ii
.
be a countable set of propositional letters. Let 2
L
be the set of all the
functions from the elements of
Let
L
to {0, 1}, i.e., 2
L
is the set of all the assignments
L
. Let W be a subset of 2
L
and
be an element of 2
L
. Then, the truth-relation
of
L
w
for any formula of the logic
ќ
B
or the logic
O
ќ
can be defined
according to the following definitions.
Definition 2.20
W,
w
For any formula
ɂ
of the logic
ќ
B
, the truth-relation W,w
ż
ɂ
is
defined inductively as follows:
(1) For any propositional letter pW,w
p iff w(p) = 1;
iff W, w
|
;
(2) W,w
¬
(3) W,w
(
∧
ϕ
) iff W,w
and W,w
ϕ
;
(4) W,w
B
iff W,w'
for every w' ∈ W.
Definition 2.21
For any formula
ɂ
of the logic O
ќ
, W,w
ż
O
ќ
iff W,w
ż
B
ϕ
and for every w', if W,w'
Therefore, the rule for
O
is in fact a very simple modification of the rule for
B
.
This can also be seen by rewriting both rules as follows:
W,w
B
iff w'∈W
¼
W,w'
for every w';
ż
ϕ
then w'
∈
W.
W,w
O
iff w'∈W
⇔
W,w'
for every w'.
The modal operator
O
is closely related to stable expansion. To a certain
extent, the operator
O
can be used to describe stable expansions, as shown by the
following theorem and corollary.
