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(2) If P∈T, then
B
P ∈ T.
(3) If P
∉
Tthen ¬
B
P ∈ T.
No further conditions could be drawn by an ideally rational agent in such a state;
therefore, the state of belief characterized by such a theory is also described
by Moore as stable autoepistemic theories. If a stable autoepistemic theory T
is consistent, it will satisfy the following two consitions:
(4) If
B
P ∈ T, then P∈T.
(5) If ¬
B
P ∈ Tthen P
∉
T.
was proposed and studied by Moore. This
logic is built up a countable set of propositional letters, the logical connectives ¬
and
∧
, and a modal connective
B
.
An autoepistemic logic named
ќ
B
2.8.2
OOOO
ќ
Logic
Based on the autoepistemic logic
ќ
B
, Levesque introduced another modal
connective
O
and built the logic
O
ќ
. Therefore, there are two modal operators,
B
and
O
, where
B
ϕ
is read as “
ϕ
is believed” and
O
ϕ
is read as “
ϕ
is all that is
believed” (Levesque, 1990). Formulas of
ќ
B
and
O
ϕ
are formed as usual as that
of ordinary logic. The objective formulas are those without any
B
and
O
operators; the subjective formulas are those where all nonlogical symbols occur
within the scope of a
B
or
O
. Formulas without
O
operators are called basic.
Be similar to that of classical propositional logic, any formula of the
autoepistemic logic
B
ϕ
can be transformed into a (disjunctive or conjunctive)
normal form.
Theorem 2.6
(Theorem on Moore disjunctive normal form) Any formula
ɂ
∈
ќ
B
can be logical equivalently transformed into a formula of the form
