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expresses the properties of some objects. The concept of satisfiability is used to
define decision logic language, that is:
If S is understandable, an object
x
∈U satisfies formula θ in S=(U, A), denoted
as
x|
=
S
θ (written as
x
|=θ for short), if and only if the following conditions are
satisfied:
(1) If and only if f(
a
,
x
)=
v
,
x
|=(
a
,
v
) ;
(2) If and only if
x
|=θ,
x
|=∼θ;
(3) If and only if
x
|=θ or
x
|=ψ,
x
|=θ∨ψ;
(4) If and only if
x
|=θ and
x
|=ψ,
x
|=θ∧ψ;
As corollaries of above conditions, we can get:
(1) If and only if
x
|=∼θ∨ψ,
x
|=θ→ψ;
|=ψ≡θ.
(3) If θ is a formula, set |θ|S is defined as: |θ|S={x∈U, x|=Sθ},
and it is called as the meaning of formular θ in S . The independent
variable with the meaning is a formula of language, and its value is a
subset of objects in system.
The following proposition explains the meaning of formulas.
Proposition 11.1
(2) If and only if
x
|=θ→ψ and
x
|=ψ→θ,
x
(1)
|(a, v)|
S
={x
∈
U: a(x)=v};
(2)
|
∼θ
|
S
=-|
θ
|
S
;
(3)
|
θ∨ψ
|
S
=|
θ
|
S
∪
|
ψ
|
S
;
(4)
|
θ∧ψ
|
S
=|
θ
|
S
∩
|
ψ
|
S
;
(5)
|
θ→ψ
|
S
=-|
θ
|
S
∪
|
ψ
|
S
;
(6)
|
θ≡ψ
|
S
=(|
θ
|
S
∩
|
ψ
|
S
)
∪
(-|
θ
|
S
∪
-|
ψ
|
S
).
From the proposition, it is easy to know that the meaning of formulas is the
object set expressed by formula θ, or we can say, it is the description of object set
|θ|S with knowledge representation language.
In the logic, we also use the concept “true”. If and only if |θ|S=U, that is, all
objects in universe satisfy formula θ, the formula is called as “true” in knowledge
representation system S.
If and only if
|
θ
|
S
=|
ψ
|
S
, formula θ and ψ are equivalent in
S
.
