Databases Reference
In-Depth Information
Tabl e 4 .
4ft table 4ft(
P
1
,
P
2
,
M
)
M P
2
¬P
2
P
1
a
b
r
¬P
1
c
d
k
n
→
∗
we will observe the associ-
ation rule
P
1
→
∗
P
2
that we will understood as the formula
(
To study the definability of the quantifier
→
∗
,x
)(
P
1
(
x
)
,P
2
(
x
))
of an observational predicate calculus (see Sect. 5).
To show that this formula is definable in classical predicate calculus
we have to find a formula
Φ
consisting of some of symbols: predicates
P
1
,
P
2
, logical connectives
∧
,
∨
,
¬
, classical quantifiers
∀
,
∃
, equality = (and of
course inequality
=) and variables
x
1
,
x
2
...
that is logically equivalent to
→
∗
,x
)(
P
1
(
x
)
,P
2
(
x
).
The fact that the formula
Φ
is logically equivalent to (
(
→
∗
,x
)(
P
1
(
x
)
,P
2
(
x
)
means that
Φ
is true in data matrix
M
if and only if
a,b
∈
3
,
5
×
0
,
1
o
r
a,b
∈
6
,
∞
)
×
0
,
3
where frequencies
a
,
b
are given by 4ft table 4ft(
P
1
,
P
2
,
M
) see Table 4. We will
construct
Φ
such that
Φ
=
Φ
1
∨
Φ
2
,
Φ
1
is equivalent to
a,b
∈
3
,
5
×
0
,
1
and
Φ
2
is equivalent to
a,b
∈
6
,
∞
)
×
0
,
3
. We will use the formulas
κ
a
(
x
)=
P
1
(
x
)
∧
P
2
(
x
)and
κ
b
(
x
)=
P
1
(
x
)
∧¬
P
2
(
x
)
k
says
“there are at least
k
mutually different objects”. It is defined using the classical
quantifier
k
and the quantifiers
∃
where
k
is a natural number. The quantifier
∃
∃
and the predicate of equality. An example of its application is the
3
κ
a
(
x
) saying “there are at least three mutually different objects
satisfying”
κ
a
(
x
) that is defined this way:
(
∃
formula
∃
3
x
)
κ
a
(
x
)=(
∃x
1
∃x
2
∃x
3
)
κ
a
(
x
)
∧
(
x
1
=
x
2
∧ x
1
=
x
3
∧ x
2
=
x
3
).
The formula
Φ
1
equivalent to
a,b
∈
3
,
5
×
0
,
1
canbedefinedas
3
x
)
κ
a
(
x
)
6
x
)
κ
a
(
x
))
2
κ
b
(
x
))
Φ
1
=(
∃
∧¬
((
∃
∧¬
((
∃
The formula
Φ
2
equivalent to
a,b
∈
6
,
∞
)
×
0
,
3
can be defined as
6
x
)
κ
a
(
x
)
4
κ
b
(
x
)).
Φ
2
=(
∃
∧¬
((
∃
The formula
Φ
=
Φ
1
Φ
2
defined this way consists of symbols: predicates
P
1
,
P
2
, logical connectives
∨
∧
,
∨
,
¬
, classical quantifier
∃
, inequality
=and
→
∗
,x
)(
P
1
(
x
)
,P
2
(
x
). It
of suitable variables and it is logically equivalent to (
→
∗
is classically definable. We have also seen that
the formula
Φ
is constructed on the basis of the table of maximal b
Tb
⇒
∗
shows that the quantifier
of
⇒
∗
. The table
Tb
⇒
∗
has two steps that are used in the construction of
Φ
.
