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It is important that the function
Tb
⇒
∗
makes it possible to use a simple
test of inequality instead of a rather complex computation. For example we
can use inequality
b<Tb
⇒
!
p,α,s
(
a
) instead of the condition
a
+
b
(
a
+
b
)!
i
!(
a
+
b
i
)!
p
i
(1
p
)
a
+
b−i
−
≤
α
∧
a
≥
s
−
i
=
a
!
p,α,s
of lower critical implication, see [12]. An other form of
the table of critical frequencies for implicational quantifier is defined in [2].
Note that if we apply a corresponding data mining procedure then the
computation of the function
Tb
⇒
!
p,α,s
for quantifier
⇒
usually requires much less effort than
the evaluation of all conditions
a
+
b
i
=
a
(
a
+
b
)!
i
!(
a
+
b−i
)!
p
i
(1
p
)
a
+
b−i
s
.It
depends on the cardinality of the set of relevant questions to be automatically
generated and verified, for more details see namely [12].
Letusremarkthatitcanbe
Tb
⇒
∗
(
a
)=
−
≤
α
∧
a
≥
∞
. A trivial example gives the
T
T
quantifier
⇒
defined such that
⇒
(
a,b
) = 1 for each couple
a,b
.Then
it is
Tb
⇒
T
(
a
)=
for each
a
.
The partial table of maximal
b
and table of maximal
b
are called
tables of
critical frequencies
. Further tables of critical frequencies for
Σ
-double impli-
cational quantifiers and for
Σ
-equivalence quantifiers are defined in [8].
∞
3 Classical Definability and TCF
3.1 Association Rules and Observational Calculi
Monadic observational predicate calculi (MOPC for short) are defined and
studied in [2] as a special case of observational calculi. MOPC can be under-
stood as a modification of classical predicate calculus such that only finite
models (i.e. data structures in which the formulas are interpreted) are admit-
ted and more quantifiers than
∀
and
∃
are used. These new quantifiers are
called
generalized quantifiers
. The 4ft quantifiers are special case of generalized
quantifiers.
Classical monadic predicate calculus (CMOPC for short) is a MOPC with
only classical quantifiers. In other words it is a classical predicate calculus with
finite models. The formulas (
∀
x
)
P
1
(
x
)and(
∃
x
)(
∃
y
)((
x
=
y
)
∧
P
1
(
x
)
∧¬
P
2
(
y
))
are examples of formulas of CMOPC.
If we add the 4ft-quantifiers to CMOPC we get MOPC the formulas of
which correspond to association rules. Formulas (
⇒
p,Base
x
)(
P
1
(
x
)
,P
2
(
x
)and
(
P
4
(
x
)) are examples of such association
rules. Values of these formulas can be defined in Tarski style see [2]. We sup-
pose that the formulas are evaluated in
⇔
p,Base
x
)(
P
1
(
x
)
∨
P
3
(
x
)
,P
2
(
x
)
∧
- data matrices (i.e. finite data
structures), see example in Fig. 1 where predicates
P
1
,...,P
n
are interpreted
by columns - functions
f
1
,...,f
n
respectively.
{
0,1
}
