Depto. de Ciencias de la Computación e Inteligencia Artificial. Facultad de. Ciencias. Universidad ... Depto. de Matemática Aplicada. Universidad Politécnica de ...
ON CONSEQUENCE IN APPROXIMATE REASONING J.L. Castro*, E. Trillas** and S. Cubillo*** * Depto. de Ciencias de la Computación e Inteligencia Artificial. Facultad de Ciencias. Universidad de Granada. 18071-Granada. Spain. ** Depto. de Inteligencia Artificial. Universidad Politécnica de Campus de Montegancedo, 28660, Boadilla del Monte, Madrid. Spain.
Madrid.
*** Depto. de Matemática Aplicada. Universidad Politécnica de Madrid. Campus de Montegancedo, 28660, Boadilla del Monte, Madrid. Spain. Abstract The purpose of
this paper is to explore the notion
of consequence in
Approximate Reasoning. While logical consequence has an standard (semantic and syntactic)
definition
in
Exact
Logic,
several
different
definitions
of
that
concept can be considered in the Approximate Reasoning framework. In this paper some definitions are compared with consequences derived from inference methods used in Artificial Intelligence. We will show that some inference methods are not covered by that definitions. Finally a more general concept of Approximate
Consequence
is
introduced
covering
a
wide
variety
of
such
inference methods.
§1. Introduction 1.1. Two ways of defining the notion of consequence in two-valued logic are of special importance: the so-called Tarski and Gentzen styles. Let F a non empty set. A consequence operator, in the standard sense of Tarski [12], is an operation C on the power set of F, satisfying: C1) INCLUSION: A ⊆ C(A), C2) MONOTONICITY: If A ⊆ B, then C(A) ⊆ C(B), C3) IDEMPOTENCE: C(C(A)) = C(A); for any A,B⊆F. C(A) represents the set of all consequences of A, and if
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b∈C(A), b is said to be a conclusion from premises A. Usually a fourth property is added for admiting only deductions on the basis of a finite number of premises: C4) COMPACTICITY: if b∈C(A), then b∈C(A’) for some finite subset A’⊆A. Then C is called a compact consequences operator. As is well known, this concept can be equivalently expressed through a relation 9 between subsets of F and elements of F, with the following conditions: G1) REFLEXIVITY:
A9 b, whenever b∈A,
G2) MONOTONICITY:A∪B 9 b, whenever A 9 b and for any B⊆F, A9 b, whenever A9 bi for all i∈I and A ∪ {bi}i∈I 9 b, Such a relation is called a consequence relation [6]. G3) CUT:
For any operator C satisfying C1)-C3), the relation 9 defined by A 9 b satisfies
G1)-G3).
if and only if
Conversely,
for
any
b∈C(A)
relation
9
satisfying
G1)-G3),
the
operator C defined by C(A) = { b : A 9 b } satisfies C1)-C3). The conditions G and C are not quite pairwise equivalent. For example, cut does not follow from idempotence alone, but needs help from both inclusion and monotony as well. The compacticity axiom is now translated in terms of 9 as follows: G4) COMPACTICITY: If A 9 b, then A’9 b for some finite subset A’⊆ A, being obvious that G4) is equivalent to C4). Presentations in terms of a relation 9 often focus on only the finitary part of the notion, giving the conditions: G1) REFLEXIVITY:
A9 b, whenever b∈A,
G2u) UNIT MONOTONICITY:A∪{c} 9 b, whenever A 9 b, for any c∈F, G3u) UNIT CUT:
A9 b, whenever A9 c and A ∪ {c} 9 b,
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or in terms of a operator C: A ⊆ C(A),
C1) INCLUSION:
C2u) UNIT MONOTONICITY C(A) ⊆ C(A ∪ {b}), C3u) UNIT IDEMPOTENCE: If b∈C(A) then C(A) = C(A ∪ {b}). The link between the finitistic and infinitistic versions is as follows; if C set
satisfies
the
infinitistic
of finitistic ones; and
conditions
then
it
satisfies
the
corresponding
if C satisfies the finitistic conditions and is
compact then C satisfies the infinitistic conditions. Similarly for 9. When authors speak of a "Gentzen-style" presentation, they mean one that is expressed in terms of 9 ,while a "Tarski-style" is one in terms of C. Finitistic cases are usually presented in Gentzen-style, while general cases are usually presented in Tarski-style. The soundness of the notion of consequence in two-valued logic is checked proving that all known monotonic two-valued logics are covered by that notion [see 13]. 1.2. On the other hand, in Chakraborty (1988), the possibility of introducing the notion of graded consequence in a Multiple-Valued Logic was explored. The starting
point
was
the
axiomatization
of
the
corresponding
notion
in
a
two-valued situation. This started from Tarski [12] and has taken a very concrete shape in Shoesmith and Smiley [11] who have extended the notion further to Multiple Conclusion Logic: A α-consequence relation 0 (in the sense of Chakraborty) is defined as a fuzzy relation 0 :P(F)xF [---------------L [0,1], satisfying the following conditions: G1’) α-REFLEXIVITY:
If b ∈ A then 0(A,b) ≥ α,
G2’) GRADED MONOTONICITY : If A’⊆ A then 0(A’,b) ≤ 0(A,b),
0(A,b) ≥ 0(A∪B,b) ∧ i n f 0(A,x), x∈B for any subsets A,B and any element b of F. 0(A,b) means the degree in which b G3’) GRADED CUT
:
is considered to be a consequence of A.
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A fuzzy relation R:P(F)xF [---------------L [0,1] is said to be compact if and only if it satisfies G4’) GRADED COMPACTICITY: R(A,b) ≤
su p R(A’,b). A’⊆ A A’ f i n i t e
For any α-consequence relation 0, the relation 0’ defined by
0’(A,b) = 0(A,b) ∧ s up 0(A’,b), A’⊆ A A ’ i s
f ini t e
is the maximal compact α-consequence subrelation contained in 0. In [5] Chakraborty proved tat any fuzzy relation R: P(F)xF [---------------L [0,1] can be extended to a minimal 1-consequence relation. Now this result will be generalized to α-consequences: Proposition 1. Any fuzzy relation R:P(F)xF [---------------L [0,1] can be extended to a minimal α-consequence relation 0r. Proof: Let T = {0 i / i∈I} be the family of all the α-consequence relations containing to R; that is, 0i is a α-consequence relations and R(A,b) ≤ 0i(A,b) for all A∈P(F) and b∈F, for every i∈I. T is non empty since the relation
01(A,b) = 1 for every A∈P(F) and b∈F, belong to T. Let us define 0r = n 0i. From proposition 11 of Chakraborty’s paper, 0r is a i∈I α-consequence relation. On the other hand, for every A∈P(F) and b∈F, it is R(A,b) ≤ 0i(A,b), ∀ i∈I; hence R(A,b) ≤ n 0i(A,b) = 0r(A,b), and R ⊆ 0r is i∈I concluded. Finally, if 0 is a α-consequence relation such that R ⊆ 0, then
0 ∈ T and 0r = n 0i ≤ 0.P i∈I 0r is the intersection of all α-consequence extensions of R and shall be called the closure of R under α-reflexivity, graded monotonicity and graded cut. Let be noted that 0r’ is a compact α-consequence relation, but 0 r’ can be
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smaller than R. However, if R is compact, then 0r’= 0r is a minimal compact α-consequence relation containing R. Proposition 2.. For any α-consequence relation 0 , the crisp relation 9 defined by A 9 b if and only if 0(A,b) ≥ α, is
a
classical
consequence
relation,
and
the
operator
C: P(F) [---------------L P(F)
defined by C(A) = { x∈F : 0(A,x) ≥ α }, is a classical consequence operators. Proof: From α-reflexivity follows inclusion and from graded monotony follows monotonicity. If 0(C(A),b) ≥ α, from graded cut follows
0(A,b) ≥ 0(A ∪ C(A),b) ∧
i nf 0(A,x) ≥ 0(C(A),b) ∧ α ≥ α, x∈C(A) hence C(C(A)) ⊆ C(A) and idempotence is concluded.P A question is if the Chakraborty’s notion of consequence is sound. That is, are all monotonic reasoning methods in Multiple Valued Logic covered by it?. If the answer is not, then it will be necessary to define another notion of consequence covering all those cases.
§2. What Approximate Reasoning Consequences are covered by Tarski or Chakraborty consequences?. 2.1.
Most
approximate
reasoning
methods
consist
on
giving
a
inexact
conditional, R: FxF [---------------L [0,1], (a,b) 9----------L R(b/a), and by using Modus Ponens to make inferences: R(b/a) v(a) ------ - v(b) ≥ M(v(a),R(b/a)) where M is a Modus Ponens function and v is a valuation function [3].
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The M-transitivity of R: M(R(y/x),R(z/y)) ≤ R(z/x), together its α-reflexivity: R(a/a) ≥ α, are usually added in order to model the monotonicity of the reasoning. The only properties we will suppose to M are commutativity and M(1,1) = 1. Definition. Let R be a fuzzy relation on F. R is said to be α-reflexive when and only when R(a/a) ≥ α for any a∈F. 1-Reflexive and M-transitive fuzzy relations are called M-preorders. We can associate a indexed family of operators and a graded relation to R. Definition. For each ε in [0,1] we define the operator Cε in the power set of F by Cε(A) = { b∈F : R(b/a) ≥ ε for some a∈A }; Definition. We define the relation 0 from the power set of F to F by
0(A,b) = s u p R(b/a). a∈A ε The questions are: i) Are C Tarski’s consequence operators? and ii) Is 0 a Chakraborty graded consequence relation? Proposition 3. i) R is ε-reflexive iff A ⊆ Cε(A) for all A⊆F. ii) R is M-transitive iff Cε(Cδ(A)) ⊆ CM(ε,δ)(A) for all A⊆F, and any ε,δ>0. iii) When R
is
M-transitive,
C1
is
a
classical
consequence
operator
iff
R
is 1-reflexive. iv) For each ε∈[0,1), when R is M-transitive and ε-reflexive, Cε is a classical consequence operator iff M(x,x) = x for every x∈[0,1] . Proof: i) Let us suppose R is ε-reflexive. If a∈A then R(a/a)≥ε and a∈C ε(A). Conversely, let us suppose A⊆Cε(A). Then a∈Cε({a}) and R(a/a) ≥ ε. ii) Let us suppose R is M-transitive. If a∈C ε(C δ(A)) then there exist a b∈Cδ(A) such that R(a/b)≥ε. As b∈Cδ(A), there exist a c∈A such that R(b/c)≥δ. Hence R(a/c) ≥ M(R(a/b),R(b/c) ≥ M(ε,δ) and a∈CM(ε,δ).
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Conversely, let us suppose Cε(Cδ(A)) ⊆ CM(ε,δ)(A) for all A⊆F, and any ε,δ>0. Taking ε=R(c/b), δ=R(b/a) and A={a}, follows c∈CM(ε,δ)({a}) and R(c/a) ≥ M(R(c/b),R(b/a). iii) and iv) are immediately followed from i) and ii). P
Proposition Min-transitive
4.
i)
iff
0
R
is
ε-reflexive
verifies cut.
iii)
iff
0
R is
is
ε-reflexive.
α-reflexive
and
iff 0 is a α-consequence relation. Proof: i) is obvious. ii) Let us suppose that R is Min-Transitive. It is R(c/b) ∧ R(b/a) ≤ R(c/a), for every b in B and a in A. Taking the supremum relatively to a, R(c/b) ∧ 0(A,b)) ≤ 0(A,c), and R(c/b) ∧ i n f 0(A,x)) ≤ R(c/b) ∧ 0(A,b)) ≤ 0(A,c); x∈B hence
0(A,c) ≥ s u p R(c/b) ∧ 0(A,b), b∈B 0(A,c) = s u p R(c/a) ≥ s u p R(c/a) ∧ 0(A,b), a∈A a∈A and
0(A,c) ≥ 0(A∪B,c) ∧ i n f 0(A,x), x∈B Thus 0 verifies cut. Conversely, let us suppose 0 verifies cut. Then
0({a},c) ≥ 0({a}∪{b},c) ∧ i nf 0({a},x), x∈{b} hence
0({a},c) ≥ 0({b},c) ∧ 0({a},b), and R(c/a) ≥ R(c/b) ∧ R(b/c),
7
ii)
R
is
Min-transitive
hence R is Min-transitive. iii) It is immediately followed from i) and ii).P 2.2. Examples. 1.- Let it F a Boolean Algebra of propositions. The ordinary conditional ⊃ on
F - ⊃(a,b) = 1 if a ⊃ b is a tautology, and ⊃(a,b) = 0 in other case -, is 1-reflexive and Min-transitive, hence Cε is a consequence operator for each ε>0 and 0 is a α-consequence relation for each α≥0. In fact, C ε = C 1 for each ε>0 and 0 is the consequence relation associated to the Modus Ponens inference rule. Obviously, the ordinary consequence 9 is an extension of 0, which is also a α-consequence relation for each α≥0. 2.- Let it be (F,p) a probabilized Boolean Algebra and W the Luckasiewicz t-norm - W(x,y) = Max (0,x+y-1)-. The relation I p defined as I p(b/a) = p(a’+b) is a W-preorder, but not a M-preorder for every M such that M ≥ π -π(x,y) = x*y-
(see
[1]).
Relation
Ip
is
basic
in
Nilsson’s
Probabilistic
Logic [7]. Thus, C 1 is the only consequence operator and 0 is not an α-consequence relation for every α in [0,1]. 3.-
G.
Pólya
used,
in
Plausible
Reasoning
[9],
the
relation
p*(b/a) = p(ba)/p(a) (Conditional Probability) defined on F+ = {x∈F : p(a)≥0} that is not a M-preorder for any M such that M(1,x) ≠ 0 (see [1]). Moreover, p* is 1-reflexive, can be easely checked that C1 is a consequence operator (see [1]), and 0 is not an α-consequence relation for every α in [0,1]. 4.- Given a necessity measure N on a Boolean Algebra F, the relation RN(a,b) = = N(a’+b) is a M-fuzzy preorder for any function M≤Min: M(N(a’+b),N(b’+c) ≤ Min(N(a’+b),N(b’+c)) = N([a’+b][b’+c]) = N(a’b’+a’c+bc) ≤ ≤ N(a’+c). Thus Cε is a consequence operator for every α-consequence relation for every α in [0,1].
8
ε in
[0,1] and 0 is a
5.- If v is a fuzzy subset of F (truth assignation), the relations RV1 (b/a) = Max (1-v(a),v(b)), Rv2(b/a) = Min (v(a),v(b)); are widely used in expert systems. Rw1 is W-transitive (W is the Lukasiewicz t-norm); and Rv2 is Min-transitive. Rv1 is only 1/2 reflexive and Rv2 is only 0-reflexive. Thus Rv1 and Rv2 are Cε consequence operators for ε=1 and ε=0 respectively and 0 is not a α-consequence relation for any α>0. 6.- On the other hand, if * is a continuous t-norm [10], the residuated implication I*v (b/a) = sup { z / v(a)*z ≤ v(b) }, is a *-preorder. In particular, for minimum, product and Luckasiewicz t-norms are respectively obtained
( IMv i n = { 1 9 v(b)
i f v( a )≤v(b) otherwi s e,
IPv r o d
=
( if v(a)≤ v(b) { 1 9 v(b)/v(a) otherwise,
IWv (b/a) = Min (1, 1-v(a)+v(b)). Thus, C1 is a consequence operator, C ε is a consequence operator for 1>ε≥0 when and only when * is the minimum and 0 is a α-consequence relation if and only if * is the minimum. u-----------------------------------------------------------------i-------------------------------------i-----------------------------i-----------------------------i------------------------------i---------------------------i----------------------i-------------------------------------------------------------------------------------o p p p p p p p p I p p* ppp R N ppp R v1 ppp R v2 ppp I *v p p p ⊃ p p p p p p p p p p p p p p p p p p p p p p p - - - - - - - - - - - -- p - - - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - p - - - - - - - - - - - - - - p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 p p Tar sk i ’ s p ε p p p p p p C (*≠M i n) p p p C p 1 p 1 p ε pp C 1 pp C 0 pp - - - - - - - - - - - - - -pp p p p C p C p C p p p p p p p p p pConsequence p p p p p p p Cε p p p ∀ ε ≥0 p p p ∀ ε ≥0p p p p p p p p p p p p ∀ ε ≥ 0 , (*=M i n) p p p p p p p p p p p - - - - - - - - - - - -- p - - - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - p - - - - - - - - - - - - - - p p p p p p p p p p p p p p p p p p p p p p p p p p p p pChakr abor t y’sp p p p p p p N o ne ( * ≠ M i n)p p p p p p p p p p p p ∀ α ≥0 p N one p N one p ∀α≥0p Nonep α=0p - - - - - - - - - - - - - - p p p p p p p p p p pConsequence p p p p p p p ∀ α≥0 ( * = M i n)p p p p p p p p p p p p p p p p p p p p p p p p p p m-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------.
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§3. Fuzzy Consequences As it was showed in the section 1, several examples of Approximate Reasoning methods do not match with none of the above notions of consequence. An alternative notion of consequence, which will permit to model those consequence relations, was introduced by Pavelka in [8] by extending the classical
exact
notion
of
consequence’s
operator
to
a
general
notion
of
approximate consequences from approximate premises. Let us denote by « the fuzzy inclusion, i.e., A«B iff A(x)≤B(x) for every x in F and every A,B in
[0,1]F.
Definition. A fuzzy consequence operator on F is defined as a mapping C from the fuzzy power set of F into itself verifying A« C(A),
C1’) FUZZY INCLUSION:
C2’) FUZZY MONOTONICITY:If A« B« F, then C(A)« C(B). C3’) IDEMPOTENCE:
C(C(A))=C(A);
for any fuzzy subsets A,B of F. If A« C(B) it is said that A is a fuzzy set of approximate consequences of the fuzzy set B of approximate premises. Now, the compacity axiom is: C4’) FUZZY COMPACITICITY:
C(A)(y) ≤
su p C(A1G)(y), G⊆ F G
f i ni te
where A1G is the fuzzy subset of F defined by
( A1G (y) = { A(y), if y∈G 9 0 otherw i se. Theorem 5. i) Let C be a fuzzy consequence operator on F. The restriction C of
C to the (non fuzzy) power set of F, defined by
10
C(A) = { x∈F : C(A)(x) = 1 }, is a classical consequence operator. Moreover, C satisfies the compacticity axiom whenever C satisfies the fuzzy compacticity axiom. ii) The family of all consequence operators on F is a complete lattice under the ordering
C1 ≤ C2 iff C1(A) « C2(A), for each A ∈ [0,1]F. Proof: see [4]. Here, [Inf Ci(A)](x) = Inf Ci(A)(x) i
i
sup Ci = Inf { C : Ci ⊆ C , for all i }.P i
i
Proposition 6. For any fuzzy consequence operator C, the operator C’ defined by
C’(A)(y) = C(A)(y) ∧
s up C(A1G)(y), G⊆ F G
f i ni te
is the maximal compact consequence operator contained in C, i.e. such that
C’(A) « C(A) for any A « F. Proof: As fuzzy consequence operators are closed by infima, we only need to prove that C’(y) =
s up C(A1G)(y) is a fuzzy consequence operator. Fuzzy G⊆ F
G
i s
f ini t e
inclusion and fuzzy monotony follows immediately. On the other hand, from
C(C(A1G))(y) = C(A1G)(y) follows idempotence. Finally, fuzzy compacticity and maximality are evident.P Now a new notion of consequence relation will be introduced as an extension of 1-consequence relations. It will play a similar role with respect fuzzy consequence operators to that played by Gentzen’s consequences relation with respect Tarski consequence operators.
11
Definition.
A
fuzzy
∝: Pd(F)xF [---------------L [0,1],
consequence
relation
∝
(Pd(F) represents the fuzzy
is poset
a
fuzzy of
relation
F), satisfying
the conditions: G1’’) FUZZY REFLEXIVITY:
∝(A,y) ≥ A(y),
G2’’) FUZZY MONOTONICITY : If A’« A then ∝(A’,y) ≤ ∝(A,y), ∝(A,y) ≥ ∝(A∪[A)B],y),
G3’’) FUZZY CUT:
for any fuzzy subset A, any subset B and any element y of F, being
( [A)B](x) = { ∝(A , x) , i f x∈B 9 0 , ot herwise ; ∝(A,y) means the degree in which y is considered to be a consequence of the fuzzy set of premises A. In two valued logic, when we say that from the premises x and x⊃y, y can be deduced, meaning that if x⊃y is true and x is true then y is true. On the other hand, in multivalued logic exists rules as: if x with truth value v(x) and x⊃y with truth value v(x⊃y) then y with truth value v(y). Thus, the premises and the conclusions have associated a truth value; in this sense we talk about a fuzzy set of premises A: F [----------L [0,1], and about a truth value of consequences C(A)(y). This concept is a generalization of bivaluated logic: in bivaluated logic it is deduced from subset of premises, in multivaluated logic it is deduced from fuzzy set of premises. Fuzzy cut means that if the fuzzy set of premises is extended with some fuzzy consequences of A, then the fuzzy consequences are not enlarged. Definition. A fuzzy relation R from the fuzzy power set of F to F is said to be compact if and only if satisfies G4’’) COMPACTICITY: R(A,y) ≤
su p R((A1G),y). G⊆ F G
12
i s
f i n i te
Theorem 7. For any operator C satisfying C1’)-C3’), the relation ∝ defined by ∝(A,x)
=
C(A)(x),
satisfies
G1’’)-G3’’).
Conversely,
for
any
relation
∝
satisfying G1’’)-G3’’), the operator C defined by C(A)(x) = ∝(A,x) satisfies C1’)-C3’). Moreover, C4’) is equivalent to G4’’). Proof: Straightforward. P Proposition 8. For any fuzzy consequence relation ∝, the relation ∝’ defined by
∝’(A,y) = ∝(A,y) ∧
s up R((A1G),y), G ⊆ F G
i s
f i n it e
is the maximal compact fuzzy consequence subrelation contained in ∝. Proof: It is consequence of the proposition 5 and theorem 6.P Proposition 9. Any fuzzy relation R from the fuzzy power set of F to F can be extended to a minimal fuzzy consequence relation ∝r, that is exactly the intersection of all single consequence extensions of R, such new relation shall be called the closure of R under reflexivity, monotonicity and cut. Proof: As fuzzy consequence operators are closed by infimum, also fuzzy consequence relation are closed by infimum. Defining ∝ by ∝(A,b) = 1 for every fuzzy subset A and every element b in F, we have a fuzzy consequence relation which contains R. Thus, we can consider the infimum of all fuzzy consequence relations containing R and this will be the minimal ∝r.P Remark. Let be noted that ∝r’ is a compact fuzzy consequence relation, but ∝r’ can be smaller than R. However, if R is compact, then ∝r’= ∝r is a minimal compact fuzzy consequence relation containing R.
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§4. What Approximate Reasoning Consequences are covered by Fuzzy Consequence Relations?. Let R a fuzzy relation on F. Definition. We define the relation ∝ from the fuzzy power set of F to F by ∝(A,b) = s u p M(A(a),R(b/a)). a∈F Proposition 10. Being M a continuous t-norm: i) R is 1-reflexive iff ∝ verifies
reflexivity.
ii)
R
is
M-transitive
iff
∝
verifies
cut.
iii)
R
1-reflexive and M-transitive iff ∝ is a fuzzy consequence relation. Proof: i) Let us suppose R is 1-Reflexive. Then R(a/a) = 1 and ∝(A,b) = s u p M(A(a),R(b/a)) ≥ M(A(b),R(b/b)) = M(A(b),1) = A(b). a∈F Conversely, if ∝ verifies reflexivity, then C({a})∝a≥∝({a},a)=1 and R(a/a) = 1. ii) Let us suppose R is M-transitive. Then M(M(A(a),R(c/b)),R(b/a)) ≤ M(A(a),R(c/a)) for every b in B and a in A. Taking the supremum on a we have M(R(c/b),∝(A,b)) ≤ ∝(A,c), and taking supremum on b ∝(A,c) ≥ ∝(A∪{x/∝(A,x}x∈B , c). Conversely, let us suppose ∝ verifies cut, then ∝({a},c) ≥ ∝({a}∪{x/∝({a},x}x∈{b}) , c) and ∝({a},c) ≥ ∝({a}∪{b/∝({a},b}) , c), hence ∝({a},c) ≥ sup { M(1,R(c/a)),M(∝({a},b),R(c/b))} and ∝({a},c) ≥ M(∝({a},b), R(c/b)) that is,
14
is
R(c/a) ≥ M(R(b/a),R(c/b).P In consequence, we can say that fuzzy consequence relations models exactly those conditionals that are fuzzy preorders. Thus, only if the conditional is a fuzzy preorder, we can assure the soundness of the fuzzy consequence relations. We can now complete the scheme of consequence’s notions by: u-----------------------------------------------------------------i-----------------------------i------------------------------i--------------------------------i--------------------------------i------------------------------i----------------------------i-----------------------------------------------------------------------------o p p p p p p p I p ppp p * ppp R N ppp R v1 ppp R v2 ppp I *v p p p ⊃ p p p p p p p p p p p p p p p p p p p p p - - - - - - - - - - - -- p - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - - - - - - - - - p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 p p Tar sk i p p p p p p p C (*≠M i n) p p p ε pp C 1 pp C 1 pp C ε pp C 1 pp C 0 pp - - - - - - - - - - - -pp p p C p p p p p p p p ε p pConsequences p p p p p p p C p p p ∀ ε ≥ 0p p p∀ ε ≥ 0 p p p p p p p p p p p p ∀ ε ≥ 0 , ( * = M i n)p p p p p p p p p p p - - - - - - - - - - - -- p - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - - - - p p p p p p p p p p p p p p p p p p p p p p p p p p p p pChakr abor t y p p p p p p p N o n e ( * ≠ M i n)p p p p p p p p p p p p ∀ α ≥ 0p N o nep N o n e p ∀α ≥ 0 p No n ep α = 0 p p p p p p p p p p p p c onsequences p p p p p p p - - - - - - - - - - - -p p p p p p p p p p p p p p p p p p p p p p p p p p p ∀α≥ 0 ( * = M i n)p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p - - - - - - - - - - - -- p - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - - p - - - - - - - - - - - - p p p p p p p p p p p p p p p p p p p p p p p p p p p p p Fuzzy p p p p p p p p p p p p p p p p p p p Y e s p Yes p No pp Yespp Nopp No pp Yes p p p p p p pConsequences p p p p p p p p p p p p p p p m----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------.
Finally,
let’s
study
the
relations
between
the
different
notions
of
consequence associated to a fuzzy relation. Let us consider the consequences Cε related to the α-cut of R and the consequences Cε obtained as α-cuts of the fuzzy consequence operator related to ∝. They are defined on P(F) by Cε(A) = { b∈F : R(b/a) ≥ ε for some a∈A }, and Cε(A) = C(A)ε = { b∈F : C(A)(b) ≥ ε } = { b∈F : s u p M(A(a),R(b/a)) ≥ ε }. a∈A
15
Proposition 11. Let us suppose that M(1,x) = x and M(0,x) = 0, ∀x∈[0,1]. Then, i) Cε(A) ⊆ Cε(A). ii) Cε(A) ⊆ Cδ(A) for every δ
