The Theory of Membership Degree of Γ-Conclusion in Several n-Valued Logic Systems *

Based on the analysis of the properties of Γ-conclusion by means of deduction theorems, completeness theorems and the theory of truth degree of formulas, the present papers introduces the concept of the membership degree of formulas A is a consequence of Γ (or Γ-conclusion) in Łukasiewicz n-valued propositional logic systems, n-valued propositional logic system and the n-valued propositional logic systems. The condition and related calculations of formulas A being Γ-conclusion were discussed by extent method. At the same time, some properties of membership degree of formulas A is a Γ-conclusion were given. We provide its algorithm of the membership degree of formulas A is a Γ-conclusion by the constructions of theory root. Godel 


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][2][3][4][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][7][8][9].So, a natural question then arises: how to judge whether or not a general formula

Preliminaries
A is a conclusion of a given theory  , or to what extend the formula A 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  A 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.

L
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  1 2 0, , , ,1 1 1 and three popularly used implication operators and the correspond ing t-norms defined as follows: > , = min( , ), , ; and in the corresponding algebras where is the t-norm defined on .Remark 1.It is easy to verify that the following assertions are true: (1) in , for every .

n
(2) in , ) for every and .
= a , for every na  .Definition 3 [7,8].(1) A homomorphism , is called an R-valuation of The set of all R-valuations will be denoted by R  ) [8,13].It is not difficult to verify in the above-mentioned three logic systems that A we shall henceforth mean a deduction of A from the empty set.We shall also write It is easy for the reader to check the following Proposition 1.

and
, then in and . This completes the proof Theorem 3 [8].at .Suppose th ( ) , , , A A  and are a p p p  ulas of ( ) It is easy to verify that 1 (1)

operties o the Roots of Theories
Definition 7 [3].Suppose that

Pr f
 is a theory, If for every ( ) Theorem 5. Suppose that  is a finite theory say , , for ever B  , there exist 1 2 , , , by Hypothe shows that and so is . This shows valent, and so is

Membership Degree of Formulas A Is Γ
In following, let us first take an analysis on the c ions of formulas A ppose 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 1 is a tautology, and , together with the sult e pr similar to that the proof of (1) and so is omitted.by Theorem 5,  is a ro e ot of  by Th orem 5, hence the proof of ( 2) is similar to that the proof of (1).In fact 1 is a root of  by 5, hence for every ( ), . Thus for every , and ) is simi heorem 5, the proof of (2 lar to that the proof of (1) and so is omitted.
) Notice that in 3 G , 1 root of is a  by Theorem 5, the at the proof of ( ) and so is omitted.
Theorem 8. Suppose that  is a infinite theory.Then proof of ( 2) is similar to th 1

T s u p A A A A A
.
(1 or every ( ) B D   , it following from Proposition 1 that there exist a finite string of fo ulas 1 2 , , , , .
is a tautology by completeness theorem, and for every It following form references [14] that A A n n    by Remark 1, the Proof of ( 2) is similar to that the Proof of (1)

 
and so is omitted.
is Prov  ably equivalent, the Proof of (3) (1) and so is omitted is similar to that the Proof of .Theorem 9. Suppose that  is a theory, ar im rators rmulas and (2) In 3 G , assume that 1 2