Bulletin of the Section of Logic
Volume 18/2 (1989), pp. 44–49
reedition 2006 [original edition, pp. 44–50]
Jan Fijalkowski
ON THEORIES OF NON-MONOTONIC
CONSEQUENCE OPERATIONS
In this paper paper we examine some algebraic properties of nonmonotonic consequence operations. The idea of introducing such operations goes back to Gabbay (cf. Gabbay [1985]). They were also examined
by Makinson and Wójcicki (cf. Makinson [198.], Wójcicki [198.]). Because
of pure algebraic character of this paper and to avoid misunderstandings,
we prefer to speak about “cumulation” or “closure” operators instead of
“non-monotonic consequence” and “consequence” operations.
Let X be an arbitrary set and F an arbitrary family of subsets of X.
It is well known, that existence of a closure operator C on X satisfying
the condition {A : C(A) = A} = F is equivalent to the fact that F closed
under arbitrary intersections. If we take a cumulation, instead of closure,
operator, the situation is more complicated. In this paper we consider a
finite set X (the case which seems the most important for applications) and
we show that if F satisfies the condition X ∈ F then there exist at least two
cumulation operators C 1 and C 2 on X, such that {A : C i (A) = A} = F
for i = 1, 2. We also show, that this observation is not always true, if F is
infinite.
Given a set X, Su(X) denotes the set of all subsets of X.
If we are given a set X, a mapping C : Su(X) → Su(X) is called a
closure operator on X, if for every A, B ⊆ X, it satisfies:
T 1 A ⊆ C(A)
T 2 C(C(A)) = C(A)
T 3 A ⊆ B implies C(A) ⊆ C(B)
A mapping C : Su(X) → Su(X) is called a cumulation operator (cf.
Makinson [198.]) on X if, for every A, B ⊆ X, it satisfies T 1 and
On theories of non-monotonic consequence operations
45
M A ⊆ C(B) implies C(A ∪ B) = C(B)
We can easily verify, that every closure operator on an arbitrary set
X is a cumulation operator on X.
It is easy to prove, that every cumulation operator on X satisfies T 1
and
M ∗ B ⊆ A ⊆ C(B) implies C(A) = C(B), for A, B ⊆ X.
We say, that a set R of subsets of given set X is closed under intersection, if for A, B ∈ R, A ∩ B ∈ R.
Given a closure or cumulation operator C on X, a subset A of X is
called a theory if C(A) = A. The set of all theories (for the given set X
and operator C) is denoted by T hC (X).
It is well known that, if C is a closure operator on X, then the set
T hC (X) is closed under (arbitrary) intersection. We have
Remark 1. Let C be a cumulation operator on given set X such that
T hC (X) is closed under intersection. Then C is a closure operator on X.
Proof. Let A, B ∈ Su(X) and assume, that A ⊆ B. It follows now from
T 1, that A ⊆ C(A) ∩ C(B), hence A ⊆ C(A) ∩ C(B) ⊆ C(A). Since C
satisfies M , it satisfies M ∗ , therefore C(C(A) ∩ C(B)) = C(A). Now we
will have C(A) ∩ C(B) = C(A), since T hC (X) is closed under intersection.
Thus C(A) ⊆ C(B). 2
Now we will mention
Remark 2. Let X be a set, and T h(X) the T
set of subsets of X, such
that X ∈ T h(X) and for every set R ⊆ T h(X), T ∈R T ∈ R (i.e. T h(X)
is closed under arbitrary intersection). It is well known that then exists a
mapping C : Su(X) → T h(X), such that for every T ∈ T h(X), C(T ) = T ,
and C is a closure operator on X.
We will show now, that if T h(X) satisfying the condition X ∈ T h(X), is
not closed under arbitrary intersection, it could be impossible to define the
mapping C : Su(X) → T h(X), such that C is a cumulation operator on
X and T hC (X) = T h(X).
Example 1. Let X be a set of real numbers R, and define T h(X) =
{[0, 1], . . . , [0, 1/n], . . .} ∪ {R}. Take A = {0}. It is easy to see, that
46
Jan Fijalkowski
for every T ∈ T h(X) we can find T ∗ ∈ T h(X), such that A ⊂ T ∗ ⊂ T .
Assume, that C : Su(X) → T h(X) is a cumulation operator on X such that
T hC (X) = T h(X) and T = C(A). Of course C satisfies M ∗ . From M ∗ it
follows that C(T ∗ ) = T 6= T ∗ , contrary to assumption, that C(T ∗ ) = T ∗ .
2
It is well known, that given a set X and a closure operator C on X, this
operator is determined by the set T hC (X). In the other words, if C1 , C2
are closure operators on X, such that T hC1 (X) = T hC2 (X), the for every
A ∈ Su(X), C1 (A) = C2 (A).
This remark does not hold for cumulation operators. Indeed, we have
Example 2. Take X = {a, b}, a 6= b. Define C 1 , C 2 as follows
C 1 (X) = C 2 (X) = X
C 1 ({a}) = C 2 ({a}) = {a}
C 1 ({b}) = C 2 ({b}) = {b}
C 1 (∅) = {a}, C 2 (∅) = {b}
We can easily verify, that C 1 , C 2 are well defined cumulation operators
on X and T hC 1 (X) = T hC 2 (X). Of course, neither C 1 nor C 2 satisfies
T 3. Let us notice, that the set T hC 1 (X) = T hC 2 (X) is not closed under
intersection. 2.
This result can be generalized in the following form:
Theorem. Let X be an arbitrary finite set. By T h(X) we denote fixed
set of subsets of X which is not closed under intersection and such that
X ∈ T h(X). Then there exists at least two different mappings C 1 , C 2 :
Su(X) → T h(X) such that C 1 , C 2 are cumulation operators on X and
T hC 1 (X) = T hC 2 (X) = T h(X).
Proof. Let n be the number of elements of X. We proceed by induction
on n.
First step. Take n = 2. This case was already considered (see Example
2).
Second Step. Suppose that n > 2 and the assertion holds for every set
with number of elements less then n and every set T h(X) ⊆ Su(X) which
is not closed under intersection and satisfies the condition X ∈ T h(X).
Let T be a set of maximal (with ⊆ as partial order) elements of the
On theories of non-monotonic consequence operations
47
set T h(X)\{X}. From the assumption it follows that T is nonempty.
Now we have to consider two cases.
Case 1. There exists T ∗ ∈ T , such that the set Su(T ∗ ) ∩ T h(X) is not
closed under intersection.
Case 2. All the sets of the form Su(T ) ∩ T h(X) where T ∈ T are closed
under intersection.
Consider the first case. Let T = {T0 , . . . , Tr−1 }, Tr = X.
A0 , . . . , Ar be sets defined as follows:
Let
A0 = Su(T0 )
...
S
Ak = Su(Tk )\ i<k Ai
...
S
Ar = Su(Tr )\ i<r Ai
The sets A0 , . . . , Ar−1 are nonempty, since Tk ∈ Ak for 0 ≤ k < r.
Indeed, let’s assume that there exists 0 ≤ k < rSsuch that Tk 6∈ Ak .
Since Tk ∈ Su(Tk ), the set Tk must be the element of i<k Ai , hence Tk ∈
Ak0 for some k0 < k, and therefore Tk ⊆ Tk0 , contrary to the definition of
the set T . It is easy to see, that the set Ar is also nonempty.
From the definition of A0 , . . . , Ar it follows, that for every A ∈ Su(X)
there exists exactly one set Ak , 0 ≤ k ≤ r such that A ∈ Ak .
The sets A1 , . . . , Ar have the following property:
(∗) for every A, B ∈ Su(X), if A ⊆ Tk and B ∈ Ak then A ∪ B ∈ Ak .
Indeed, let’s assume that A ⊆ Tk , B ∈ Ak , but A ∪ B 6∈ Ak . Since
B ∈ Ak we have B ⊆ Tk , hence A ∪ B ⊆ Tk . Therefore if A ∪ B 6∈ Ak then
there exists i0 < S
k such that A ∪ B ∈ S
Ai0 . Hence B ⊆ A ∪ B ⊆ Ti0 . On the
other hand B 6∈ i<k Ai , hence B 6∈ i<i0 Ai , therefore B ∈ Ai0 , contrary
to assumption that B ∈ Ak .
Now we can define (according to the assumption of this step of the
proof), two different mappings C01 , C02 : Su(T0 ) → Su(T0 ) ∩ T h(X) satisfying the assertion of the theorem for X = T0 .
For 0 < k < r we define the mapping Ck : Su(Tk ) → Su(Tk ) ∩ T h(X)
according to the Remark 2 if Su(Tk ) ∩ T h(X) is closed under intersection,
otherwise according to the assumption of this step of the proof.
48
Jan Fijalkowski
From A0 = Su(X) it follows that for A ∈ A0 we have C0i (A) ∈ A0 for
i = 1, 2.
We want to show that for A ∈ Ak , 0 < k < r, Ck (A) ∈ Ak . Indeed, it
follows from the definition of the mapping Ck that Ck (A) ⊆ Tk , and that
A ⊆ Ck (A). Therefore Ck (A) ∪ A = Ck (A) ∈ Ak what follows from (∗).
Now we can define the mappings C 1 , C 2 : Su(X) → T h(X) as follows:
C 1 (A) = C01 (A), C 2 (A) = C02 (A)
C 1 (A) = Ck (A) = C 2 (A)
C 1 (A) = X = C 2 (A)
for A ∈ A0
for A ∈ Ak , 0 < k < r
for A ∈ Ar .
It follows immediately from this definition that C 1 and C 2 are different
mappings satisfying T 1 such that T hC 1 (X) = T hC 2 (X) = T h(X). Now
we will show that they satisfy M . Indeed, take B ∈ Ak , 0 < k < r and
A ⊆ C i (B), i = 1, 2. Since C i (B) = Ck (B) we have A ⊆ Tk , so from
(∗) it follows that A ∪ B ∈ Ak , hence C i (A ∪ B) = Ck (A ∪ B) ∈ Ak .
From the assumption that the theorem is true for every k < n it follows
now immediately that C i (A ∪ B) = C i (B) for i = 1, 2. We use the same
arguments to show that for B ∈ A0 , A ⊆ C i (B) we have C i (A∪B) = C i (B)
for i = 1, 2. It follows from the definition of C i (B) that the same is true
for B ∈ Ar , A ⊆ C i (B). So M is satisfied by C 1 , C 2 and they are desired
mappings.
Consider now the case 2. Since T h(X) is not closed under intersection,
there exists a set A0 6∈ T h(X) such that A0 is an intersection of some
elements of T h(X). Let Te1 and Te2 be two different minimal (with ⊆ as a
partial order) elements of the set {T ∈ T h(X) : A0 ⊆ T }. The existence
of such elements follows from the choice of A0 , and finitariness of T h(X).
Now we can find two sets T (1) , T (2) ∈ T such that Te1 ⊆ T (1) , Te2 ⊆ T (2) ,
and Te1 , Te2 6∈ Su(T (1) )∩Su(T (2) ), otherwise Su(T (1) )∩T h(X) or Su(T (2) )∩
T h(X) will not be closed under intersection.
Now we can take T0 = T (1) and define as above the mapping C 1 :
Su(X) → T h(X) such that C 1 is a cumulation operator on X, such that
T hC 1 (X) = T h(X).
In the same manner, taking T0 = T (2) , we can define the mapping C 2 .
We can easily notice, that C 1 (A0 ) = Te1 6= Te2 = C 2 (A0 ), so C 1 and C 2 are
different and therefore the theorem is proved. 2.
On theories of non-monotonic consequence operations
49
The Theorem and the Remark 2 show that the definition of cumulation
operator C on a finite set X satisfying the condition T hC (X) = T h(X)
(where T h(X) is an arbitrary subset of Su(X) satisfying the condition
X ∈ T h(X)) is always possible.
On the other hand, we have showed that this operator is determined
by the set T h(X) if and only if T h(X) is closed under intersection. A
slight modification of the Example 1 (we can add for instance [−1, 0] to the
set T h(X)) shows that even if the set T h(X) is not closed under arbitrary
intersection, sometimes there exists only one cumulation operator on X,
such that T hC (X) = T h(X).
The problem of finding the properties of the set T h(X) equivalent to
the existence (or uniqueness) of the cumulation operator C on X such that
T hC (X) = T h(X) remains still open, when both Su(X) and T h(X) are
infinite.
References
[1985] Dov Gabbay, Theoretical foundations for non monotonic reasoning in expert systems, [in:] K. R. Apt, ed., Logics and Models of
Concurent Systems, Berlin, Springer Verlag.
[198.] David Makinson, General Theory of Cumulative Inference,
Studia Logica, to appear.
[198.] Ryszard Wójcicki, Heuristic Rules of Inference in non-monotonic
arguments, Studia Logica, to appear.
Polish Academy of Sciences
Institute of Philosophy and Sociology
Section of Logic
8 marca, 90–365 Lódź, Poland