The Theory of Membership Degree of Γ-Conclusion in Several n-Valued Logic Systems ()
1. Introduction
Fuzzy logic is the theoretical foundation of fuzzy control. Spurred by the success in its applications, especially in fuzzy control, fuzzy logic has aroused the interest of many famous scholars, a series of important results have been created in documents [1-5]. For the sake of reasoning, we have to choose a subset 
of well-formed formulas, which can reflect come essential properties, as the axioms of the logical system and we then deduce the so-called 
-conclusion through some reasonable inference rules [6-9]. So, a natural question then arises: how to judge whether or not a general formula 
is a conclusion of a given theory
, or to what extend the formula 
is a conclusion of
? It is basic problem to judge one thing belong to one kind in artificial intelligence. As is well known, human reasoning is approximate rather than precise in nature. we basic starting point is to establish graded version of basic logical notions. In order to establish a solid foundation for fuzzy reasoning, professor G. J. Wang proposed the concept of root of theory [3], J. C. Zhang proposed the concept of generalized root of theory [10,11], in propositional logic systems. The graded description and properties of formulas 
being 
-conclusion were discussed. And provide its algorithm of membership degree of formulas A is a 
-conclusion, by the constructions of theory root in the above-mentioned logic systems.
2. Preliminaries
It is well known that different implication operators and valuation lattices 
(i.e., the set of truth degrees for logic) determine different logic systems (see [12]). Here valuation lattices is 
and three popularly used implication operators and the correspond ing t-norms defined as follows:
These three implication operators 
and 
are called Łukasiewicz implication operator
, Gödel implication operator
, and the 
-implication operator
, respectively. The t-norm, which corresponds to 
-implication operator
, is called also Nilpotent Minimumtnorm [6]. If we fix a t-norm 
above we then fix a propositional calculus (whose set of truth values is
): 
is taken for the truth function of the strong conjunction &, the residuum 
of 
becomes the truth function of the implication operator and R(.,0) is the truth function of the negation. In more details, we have the following definitions.
Definition 1 [7,8]. The propositional calculus 
given by a t-norm 
has the set 
of propositional variables 
and connectives
. The set 
of well-formed formulas in 
is defined inductively as follows: each propositional variable is a formula; if
, 
are formulas, then
, 
and 
are all formulas.
Definition 2 [8,9,13]. The formal deductive systems of 
given by 
corresponding to 
and
, are called Łukasiewicz n-valued logic systems
, 
n-valued logic systems
, and the 
nvalued logic systems
, respectively.
Define in the above-mentioned logic systems
(1)
and in the corresponding algebras 
(2)
where 
is the t-norm defined on
.
Remark 1. It is easy to verify that the following assertions are true:
(1) in
, 
for every 
.
(2) in
, 
for every 
and 
.
(3) in 
, for every
.
Definition 3 [7,8]. (1) A homomorphism 
of type 
from 
into the valuation lattice
, i.e. 
, is called an R-valuation of
. The set of all R-valuations will be denoted by
.
(2) A formula 
is called a tautology w.r.t. 
if 
holds.
Remark 2 [8,13]. It is not difficult to verify in the above-mentioned three logic systems that 
, and 
for every valuation 
. Moreover, one can check in 
and 
that 
and 
are logically equivalent.
Definition 4 [8]. Assume that 
is a formula generated by propositional variables 
through connectives
. Substitute 
for 
in 
and keep the logic connectives in 
unchanged but explain them as the corresponding operators defined on the valuation lattice
. The we get a function 
and call 
the truth degree function of
.
Definition 5 [7,8]. (1) A subset of 
is called a theory.
(2) Let 
be a theory,
. A deduction of 
from
, in symbols, 
, is a finite sequence of formulas 
such that for each 
is an axiom of
, or
, or there are 
such that 
follows from 
and 
by MP. Equivalently, we say that 
is a conclusion of 
(or 
-conclusion). The set of all conclusions of 
is denoted by
. By a proof of 
we shall henceforth mean a deduction of 
from the empty set. We shall also write 
in place of 
and call 
a theorem.
It is easy for the reader to check the following Proposition 1.
Proposition 1. Let 
be a theory and 
If 
then there exist a finite subset of 
say, 
such that
.
Theorem 1 (Generalized deduction theorems) [7, 8,12]. Suppose that 
is a theory, 
, then
(1) in 
,
iff 
s. t.
.
(2) in
,
iff
.
(3) in
,
iff
.
Definition 6 [8,13]. Suppose that 
is a formula of 
containing m atomic formulas
, and 
be the truth degree function of
. Then
is called the truth degree of
, where 
is the cardinal of set
.
Theorem 2. Suppose that 
and
, then in 
and 
iff 
is a tautology i.e.,
.
Proof. Assume that
. Since
then
. By definite, 
, thus 
i.e., 
, 
, then 
is a tautology. Conversely, assume that A is a tautology i.e.,
, then 
, so
. This completes the proof.
Theorem 3 [8]. Suppose that
, then in
, 
iff 
is a tautology, i.e.,
.
Theorem 4. Suppose that
. If for every
, then
.
Proof. Suppose that 
and 
are all a formulas of 
containing 
atomic formulas
, it follows from 
that
and
hence
.
It is easy to verify that
then
.
3. Properties of the Roots of Theories
Definition 7 [3]. Suppose that 
is a theory,
. If for every 
we have
, then 
is called the root of
.
Theorem 5. Suppose that 
is a finite theory, say
, then
(1) in 
is a root of
;
(2) in
,
is a root of
;
(3) in
, 
is a root of
.
Proof. (1) It following form references [4] that
, for every
, there exist 
such that
by Theorem 1. It is easy to check that 
by Remark 1, it following from 
that 
where 
, thus 
by Hypothetical, this shows that 
is a root of
.
(2) It following form references [4] that 
, for every
, it following from Theorem 1 that 
, since 
and 
are provably equivalent, and so is
. This shows 
is a root of
.
(3) It following from references [4] that 
for every
, we get 
by Theorem 1, it is easy to verify that 
and 
are provably equivalent, hence 
and 
are provably equivalent, and so is
. This shows that 
is a root of
.
4. Membership Degree of Formulas A Is Γ-Conclusion
In following, let us first take an analysis on the conditions of formulas A is a 
-conclusion in
. Suppose that 
is a theory and A is a 
-conclusion , it follows from Proposition 1 and Theorem 1 that there exit a finite string of formulas 
and 
such that 
holds, i.e., the formula 
is a theorem of
, let
, hence 
is a tautology, it follows from Theorem 2 that
. Conversely, if there exist a 
-conclusion 
such that
, then following from Theorem 2 that 
is a tautology, thus 
is a theorem of
, i.e., 
holds and
, we have that 
by MP, i.e., 
is a 
-conclusion. Moreover, the larger the membership degree of such formulas are, the more closer A is to be 
-conclusion. Hence it is natural and reasonable for us using the supremum of truth degree of all formulas with the form 
where 
to measure A is a 
-conclusion.
Definition 8. Suppose that 
is a theory,
. Define
then 
is called the membership degree of formulas A is a 
-conclusion.
It is easy to verify that 
and following Proposition 2 by Definition 8.
Proposition 2. In
, 
and
If A is a 
-conclusion, then
.
Theorem 6. In
, 
and
, if 
is a finite theory, say
, then A is a 
-conclusion iff
.
Proof. The necessity part by proposition 2, it is only necessary to prove the sufficiency. Let
. For every number 
there exist a formulas 
such that 
by Definition 8.
(1) In
, it follows from Theorem 5 that 
is a root of 
and 
hold. Hence for every 
we have 
it follows from properties of implication operators that
, since 
is arbitrary, we have
, thus 
is a tautology, and 
is a theorem , together with the result
, then 
by MP, i.e., 
(2) In
, notice that 
is a root of 
by Theorem 5, hence the proof of (2) is similar to that the proof of (1) and so is omitted.
(3) In
, notice that 
is a root of 
by Theorem 5, hence the proof of (2) is similar to that the proof of (1). In fact 
is a theorem by Definition 7, hence 
we have 
and 
, thus 
, 
holds, then 
is a theorem , together with the result 
we have 
by MP. The proof is completed.
Theorem 7. Suppose that
, then
(1) in
,
;
(2) in
,
;
(3) in
,
.
Proof. (1) Since 
is a root of 
by Theorem 5, hence for every 
we have
. Thus for every
, 
, and 
holds, then 
by Theorem 4. It follows from 
that 
i.e.,
.
(2) Notice that in
, 
is a root of 
by Theorem 5, the proof of (2) is similar to that the proof of (1) and so is omitted.
(3) Notice that in
, 
is a root of 
by Theorem 5, the proof of (2) is similar to that the proof of (1) and so is omitted.
Theorem 8. Suppose that 
is a infinite theory. Then
(1) in
,
;
(2) 
,
;
(3) in
,
.
Proof. (1) For every
, it following from Proposition 1 that there exist a finite string of formulas 
such that 
It follows from Theorem 1 that 
is a theorem, hence 
is a tautology by completeness theorem, and for every
, 
, we have
by Theorem 4.
It following form references [14] that 
, then
.
(2) Notice that in
, 
by Remark 1, the Proof of (2) is similar to that the Proof of (1) and so is omitted.
(3) Notice that in
, 
and 
is Provably equivalent, the Proof of (3) is similar to that the Proof of (1) and so is omitted.
Theorem 9. Suppose that 
is a theory, 
and
, then 
Proof. (1) If 
we get
, then 
(2) If 
we get 
and
, for any given positive number 
such that 
and 
there exists formulas 
such that 
and 
. It follows from properties of Regular implication operators that 
and 
It is easy to verify that 
and 
are provably equivalent (i.e., logically equivalent), hence 
. It follows from the theory of truth degrees of formulas and 
that 
.
Bucas 
and 
are provably equivalent (i.e., logically equivalent), hence 
, it is easy to verify that 
then 
by the definition of the membership degree of formulas.
Example 1. Suppose that 
In
, 
and
, compute 
respectively.
Solution. (1) In
, assume that 
Since 
and
thus
and
We have 
and
hence
then
.
(2) In
, assume that 
Since 
, and
thus
then
then
.
(3) In
, assume that 
Since 
, and
thus
then 
Example 2. Suppose that 
, in
, compute 
Solution. (1) Assume that 
Since
, and
thus
, then p2
is a 
-conclusion.
NOTES