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1. Both 1.a and 1.b are satisfied
1.a
ϕ
ϕ
∧
ψ
.
∧
ψ
1.b
ϕ
∧¬
ψ
ϕ
∧¬
ψ
.
2.
ϕ
¬
ψ
.
It is proved in [7] that the important implicational quantifiers (e.g
⇒
p,Base
!
p,α,Base
of lower critical implication) are inter-
esting implicational quantifiers. The similar theorems are proved for
Σ
-double
implicational quantifier and for
Σ
-equivalence quantifier in [11], they are pre-
sentedalsoin[12].
If
of founded implication and
⇒
⇔
∗
is the interesting
Σ
-double implicational quantifier then the deduc-
tion rule
⇔
∗
ψ
ϕ
⇔
∗
ψ
is correct if and only if at least one of the conditions 1 or 2 are satisfied:
ϕ
(
ϕ
∧
ψ
)and(
ϕ
∧¬
ψ
)
ϕ
∧
ψ
)
1. Both (
ϕ
∧
ψ
)
∨
(
¬
(
ϕ
∧¬
ψ
)
∨
(
¬
ϕ
∧
ψ
)
2.
ϕ
¬
ψ
or
ψ
¬
ϕ
⇔
∗
is interesting if it is a-
dependent, (
b
+
c
)-dependent and if it is also
⇔
∗
(0
,
0
,
0) = 0. The 4ft-
quantifier
≈
is (
b
+
c
)
-dependent
if there are non-negative integers
a
,
b
,
c
,
d
,
b
,
c
such that
The
Σ
- double implicational quantifier
=
b
+
c
and
(
a,b
,c
,d
).
b
+
c
≈
(
a,b,c,d
)
=
≈
It is proved in [11] that the important
Σ
- double implicational quantifiers
(e.g
!
p,α,Base
of lower critical
double implication) are interesting
Σ
- double implicational quantifiers.
If
⇔
p,Base
of founded double implication and
⇔
≡
∗
is the interesting
Σ
-equivalence quantifier, then deduction rule
≡
∗
ψ
ϕ
≡
∗
ψ
ϕ
is correct if and only if (
ϕ
∧
ψ
∨¬
ϕ
∧¬
ψ
)
(
ϕ
∧
ψ
∨¬
ϕ
∧¬
ψ
).
The
Σ
-equivalence quantifier
≡
∗
is interesting if it is (
a
+
d
)-dependent
and if
≡
∗
(0
,b,c,
0) = 0 for
b
+
c>
0. The definition of the fact that the
4ft-quantifier
is (
a
+
d
)-dependent is analogous to the definition that it is
(
b
+
c
)-dependent. It is proved in [11] that the important
Σ
- equivalence
quantifiers (e.g
≡
p,Base
of founded equivalence and
≡
≈
!
p,α,Base
of lower critical
equivalence are interesting
Σ
- equivalence quantifiers.
7 Definability in Classical Predicate Calculi
We have shown in Sect. 5 that the association rules we deal with are formulas
we can get from formulas of classical monadic predicate calculus by adding
4ft-quantifiers. Thus a natural question arises what association rules can be
