Relative Truth Degree of Logic Formulas and Graded Fuzzy Logic

The concept of logic proposition induced functions is proposed in the present paper, then the concept of relative truth degree of propositions with respect to a logic theory Γ is introduced by means of infinite product of evenly distributed probability spaces and integrated semantics respectively w.r.t. discrete and continuous situations, and a graded approximate reasoning theory is established. Next, theory of consistency degrees of finite logic theories is also proposed. Finally, the simple application of graded fuzzy logic in fuzzy inference is given by examples.


Preliminaries
be a countable set, and 0 be a special element not contained in S , and For classical two valued propositional logic system L , the following are axiom schemes.
The deduction rule is modus ponens (MP), i.e., from Theorem 1 In classical propositional logic system L , suppose that ( ), , ( ) In the fuzzy prepositional logic system * L the following are axiom schemes L is also MP, and we have the following

Truth degrees of propositions in L and
as follows: the value of ,( see Ref G.J.Wang.(2003,1998)).
In the following 1 2 ( , ,... ) : is the free algebra generated by and for every valuation .In fact, assume that ( ) Conversely, assume that , then there exists an unique v ∈Ω , such that ( ) Specifically, we can define integrated truth degree of A with respect to diverse implication operators.In reference P. H´ajek(1998), you can the following definition: τ is called integrated truth degree of A with respect to implication operator R.

Graded approximate reasoning theory
Suppose that truth degree of B relative to Γ of the fact that Γ implies B in L , simply, the truth degree Theorem 3 told us from Γ can imply B , then Γ implies B is a tautology, i.e., its truth degree equals 1 in L .
and only 2 of them, i.e., (1,1,0)and (1,1,1) imply that , and B are finite theory and a formula in * L respectively, and * L is complete ( Ref D.W. Pei,etc(2002) It is similar with theorem 3, for integrated truth degrees of formulas, we have: is an almost tautology, i.e., the induced function E of E equals 1 almost every where in then it follows from the definition of integrated truth degree of E that the integral of E equals 1, hence the function E equals 1 almost everywhere in [0,1] m .
Theorem 4 means the theory of truth degrees of propositions is uniform with deduction theorem of finite theories in * L .
This fact is significant, it can be used to discuss deduction theorem by truth theory, and to propose a theory of consistency degrees of finite theories in * L in section 5, except that the concept of truth degrees of formulas in two valued logic has to be substituted by the concept of integrated truth degrees of formulas.

Example 4 If
then it follows from the basic properties of 0 R -implication operator that, ⊗ → is a residuated pair,(see Ref G.J.Wang,etc(2003)),that is, , 1 0, 1 In [9] a pseudo-metric was defined on F(S) in L as follows: where . Furthermore, by the concept of truth degrees of formulae of ( ) F S , in * L , a pseudo-metric can be defined as follows: ( , ) 1 ( ) ( ), , ( )

Consistency degrees of finite theories
It is well known that a theory Γ (i.e., a set of wffs) is consistent if the contradiction 0 is not a conclusion of Γ , i.e., | 0 Γ − does not hold, otherwise Γ is inconsistent.Hence we see that if | 0 Γ − is (fully) true, then Γ is (fully) inconsistent.
Suppose that we can in certain way measure the truth degree, say,α ,of " | 0 Γ − ", then it is natural to call α the inconsistency degree of Γ , and the consistency degree of Γ can then be defined to be 1 −α .In this sub-section we propose the concept of consistency degrees of finite theories in accordance with this idea., (( ) ( 0)) 4 2 p q q p q q τ τ → → = → → → = , hence it follows that 1 ({ , 0}) 1 (( ) ) 4 p q q p q q ξ τ → → = − → → = , and 1 ({ , }) 1 (( ) ( 0)) 2 p q q p q q ξ τ → = − → → → = .The consistency degree of { , } p q Γ = in L is by (i)and (ii) in a finite step of calculus.Members of S are called atomic propositions or atomic formulas (briefly, atoms), that of ) (S F are propositions or formulas or wffs.0 is called contradiction.Moreover, in what follows . . The following theorem holds in L(Ref  A.G. Hamilton.(1979)): called the consistency degree of Γ .
on theorem 2, we can propose a theory of consistency degrees of finite theories in * L by truth degrees of formulas.
and B can contain different atoms , can use extensions A * and B * instead of A and B correspondingly , such that A * and B * contain one and the same group of atoms.e.g., if