# Consistency Degrees of Theories in Łukasiewicz Fuzzy andn-Valued Propositional Logic Systems

Authors: Jiancheng Zhang
Journal: Applied Computational Intelligence and Soft Computing
 (2010)

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## Abstract

By means of theory of truth degrees of formulas, according to deduction theorems
and completeness theorems, the new concepts of consistency degrees and polar index for general theories in Łukasewicz fuzzy andn-valued propositional logic systems are introduced. Moreover, sufficient and necessary conditions for a theory Γ to be consistent, inconsistent, and fully divergent are obtained. Finally, some important properties of truth degrees of formulas are proposed.


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## Body

## 1. Introduction

Whether a theory (i.e., a set of formulas) is consistent or not (a theoryΓ is inconsistent if Γ⊢o¯, otherwise it is consistent, where o¯ is a contradiction [1]) is one of the crucial questions in any logic system. The reason is that, in classical logic, a contradictory theory (i.e., a theory which is not consistent) turns into a useless theory in which everything is provable. Quite surprising is that the same result holds also in fuzzy and many-valued logic systems (it was proved in [1] that Γ is inconsistent if and only if Γ⊢A for each A). It is clear that theories Γ containing a contradiction are inconsistent, but inconsistent theories Γ need not contain a contradiction. For example, let Γ={p,¬p}, where p is any atomic formula of S, since p→(¬p→o¯) is a theorem, then Γ is an inconsistent theory not containing any contradiction. In [4], a systematic and exhaustive treatment of fuzzy logic (FLn) is proposed where both semantics and syntax are evaluated gradationally and even axioms there need not be fully true. Hence to propose a graded theory for expounding consistent degrees of fuzzy theories seems not easy to be accomplished as it is pointed out in [4] that “even the attempt to introduce some kind of degrees of consistency mostly does not work.”  The concept of degree of inconsistency of a fuzzy theory Γ has been introduced in [4] as follows: (1)Incons(Γ)=sup{a∣Γ⊢aA∧¬AandA∈ℱℒ(Γ)},
where ⊢a is an inference sign (see [5]). It was proven that a fuzzy theory is consistent if and only if Incons(Γ)≤1/2. As was pointed out in [2], the truth valuation D(A∧¬A) of a formula A∧¬A need not be zero in multivalued logic even though it looks like a contradiction. Hence, it seems that to define Incons(Γ) by means of Γ⊢aA∧¬A is not as good as to analyse the disparities of formulas of Γ-conclusions.How to measure the extent to which a theory is consistent is also one of the crucial questions in logic systems. For trying to grade the extent of consistency of different theories, many authors have proposed different methods in fuzzy (continuous-valued) logic systems and have obtained many good results [2, 3]. The concept of consistency degree of a fuzzy theory Γ has been introduced in [2, 3] as follows: (2)ξ(Γ)=limn→∞[1-div(Γ)(1-12⌈αn,Γ⌉)],ξ*(Γ)=1-12div(Γ)(1+i(Γ)),
where αn,Γ=(d(Γ(n+2))-d(Γ(n)))/d(Γ(n+2)), div(Γ)=sup{ρ(A,B)∣A,B∈D(Γ)}, Γ(n)={A∈ℱ∣ there exists a deduction of A from Γ of length n}, d(Γ)=sup{ρ(A,B)∣A,B∈Γ}, i(Γ)=max{[ρ(A,B)]∣A,B∈D(Γ)}, ⌈x⌉ is the smallest integer at least as large as x (for any real number x), and ρ(A,B) and D(Γ) were defined by Definitions 5 and 7.The consistency degrees formula of theories ofŁukasewicz fuzzy propositional logic systems is obtained only in the condition of finite theories Γ (Γ is a finite set of formulas) in [2]; it was pointed out in the conclusion of [2] that “How to define a reasonable concept of consistency degrees for infinite theories of  Luk and how to extend the result obtained in the present paper to the more general fuzzy logic with graded syntax given in [5] would be attractive research topics.” In the condition of infinite theories, the consistency degrees formula is obtained by the definitions of divergence degree in the references [2, 3]. But the character of divergence degrees is different from the character of consistency degree. It is natural for us to find another measure to describe consistency degree. In this paper, from logical point of view and based on deduction theorems, completeness theorems, and the concept of truth degrees of formulas, we introduce a more natural and reasonable definition of consistency degrees of theories in Łukasewiczn-valued and fuzzy propositional logic systems.


## 2. Preliminaries

### 2.1. Logic Systems:Łn and Łuk

Suppose thatS={p1,p2,…} is a countable set, let ¬ and → be unary and binary operations, respectively, and let F(S) be the free algebra of type (¬,→) generated by S, that is, F(S) is the smallest set containing S and closed under the operation ¬ and →. Elements of F(S) are called propositions or formulas and those of S are called atomic propositions or atomic formulas. In Łukasiewicz fuzzy propositional logic (briefly, Luk), there are four axiom schemes as follows: A→(B→A), (A→B)→((B→C)→(A→C)), (¬A→¬B)→(B→A), ((A→B)→B)→((B→A)→A). The axioms of Łukasiewiczn-valued propositional logic (briefly, Łn) are those of Luk plus (A→nB)→(A→n-1B),(A≡(A→s-2¬B))→n-1A for any 2≤s≤n-1 such that s is not a divisor of n-1. Where A≡B instead of (A→B)&(B→A), A&B is ¬(A→¬B) (see [6]).Here are two kinds of valuation sets:Ln={0,1/(n-1),2/(n-1),…,(n-2)/(n-1),1} and L=[0,1]. Define on Ln and L a unary operator and a binary operator as follows: ¬x=1-x,x→y=(1-x+y)∧1. These two sets of valuation are just the standard MVn-chain and the standard MV-algebra (L,¬,→), respectively.A homomorphismv:F(S)→Ln (or L) of type (¬,→) from F(S) into the valuation set Ln (or L), that is, v(¬A)=¬v(A),v(A→B)=v(A)→v(B), is called a valuation of F(S). The set of all valuations will be denoted by Ω.Łukasiewicz propositional logic system is called the Łukasiewiczn-valued propositional logic system if the set of valuation is Ln={0,1/(n-1),2/(n-1),…,(n-2)/(n-1),1}. Łukasiewicz propositional logic system is called the Łukasiewicz Fuzzy propositional logic system if the set of valuation is L=[0,1]. It has been proved that the semantics and syntax of Łn and Łuk are in perfect harmony, that is, the standard completeness theorem holds in Łn and Łuk, respectively.The deduction rule is Modus Ponens (briefly, MP) of whichB can be deduced from A and A→B. As it is well known, the following abbreviations introduced in Łuk and Łn are popular [1]:(3)A∨B∶=(B→A)→A,A∧B∶=¬(¬A∨¬B),Ak∶=A&A&⋯&A︸k-times.


### 2.2. Truth Degrees of Formulas inŁn

Definition 1 (see [7]).
Suppose that(Xk,Ak,μk)(k=1,2,…) are probability measure spaces; let X=∏k=1∞Xk, then ∏k=1∞Ak generates on X a σ-algebraA. There exists on X a measure μ satisfying the following conditions.
(i)  A is the set consisting of all μ-measurable sets.
(ii) For any measurable subset E of ∏k=1mXk,E×∏k=m+1∞Xk is μ-measurable and
(4)μ(E×∏n=m+1∞Xn)=(μ1×μ2×⋯×μm)(E),m=1,2,…,μ is called the infinite product measure of μ1,μ2,…, and (X,A,μ) is called the infinite product of {(Xk,Ak,μk)}k=1∞.  (X,A,μ) can be abbreviated as X if no confusion arises.Suppose thatn is a fixed natural number, n≥2, and (Y,B,η) is an evenly distributed probability measure space, where Y={y1,y2,…,yn}, that is, η(∅)=0,η(Y)=1, and η({yi})=1/n(i=1,2,…,n). Let (Xk,Ak,μk)=(Y,B,η)(k=1,2,…,), and let (X,A,μ) be the infinite product of {(Xk,Ak,μk)}k=1∞.In the following,Y will be written as Y=Ln={0,1/(n-1),2/(n-1),…,(n-2)/(n-1),1}; hence we have (5)Xk=Ln={0,1n-1,2n-1,…,n-2n-1,1},k=1,2,…,
and (X,A,μ) can also be written as Ln∞.Letv∈Ω, then v is decided by its restriction v∣S because S generates the free algebra F(s). Assume that v(pk)=vk(k=1,2,…), then an infinite-dimensional vector ν⃗=(v1,v2,…) in Ln∞ is obtained. Conversely, let ν⃗=(v1,v2,…) be any element of Ln∞, then there exists a unique v∈Ω such that v(pk)=vk(k=1,2,…). Hence there exists a bijection φ:Ω→Ln∞ defined by φ(v)=ν⃗.Definition 2 (see [8]).
The abovementioned mapping is called the measured mapping ofΩ.Definition 3 (see [8]).
Suppose thatA∈F(S), n≥2 and define
(6)[A]i/(n-1)={ν⃗∈Ln∞∣ϕ-1(ν⃗)(A)=in-1},i=0,1,…,n-1,T(A)=∑i=0n-1in-1μ([A]i/(n-1))=∑i=1n-1in-1μ([A]i/(n-1)),
then T(A) is called the n-valued truth degree of A.Example 1.
LetL8={0,1/7,2/7,…,6/7,1}. Consider the truth degree T(p→q). Notice that υ(p→q)=υ(p)→υ(q)=(1-υ(p)+υ(q))∧1. Let υ(p)=j/7,υ(q)=k/7, and υ(p→q)=i/7, then (1-(j/7)+(k/7))∧1=i/7,i=1,2,…,7⋯(*). If i=7, then 0≤j≤k≤7 by (*); there are a total of (8×9)/2 pairs (υ(p),υ(q)) such that υ(p→q)=1. If 1≤i≤6, then j=7-i+k by (*); there are a total of i+1 pairs (j,k) such that j=7-i+k. Thus,
(7)μ([p→q]i/7)={8×92×64,i=7,i+164,1≤i≤6,T(p→q)=∑i=16i7i+164+8×92×64=3948.
Assume thatA=A(p1,p2,…,pm) is a formula generated by atomic formulas p1,p2,…,pm through connectives ¬ and →. Substitute xi for pi in A(i=1,2,…,m) and keep the logic connectives in A unchanged but explain them as the corresponding operators defined on the valuation latticeLn. Then we get a function A*:Lnm→Ln and call A*(x1,…,xm) the truth degree function of A.Lemma 1 (see [8]).
LetA=A(p1,…,pm)∈F(S),n≥2. Then
(8)T(A)=∑i=0n-1in-1|(A*)-1(i/(n-1))|nm,
where A*(x1,…,xm)=i/(n-1) if and only if xj=υ(pj),j=1,2,…,m, and υ(A(p1,…,pm))=i/(n-1),i=0,1,…n-1. Obviously, Lnm=∪{(A*)-1(i/(n-1))∣i=0,1,…,n-1},|Lnm|=nm, and μ([A]i/(n-1))=|(A*)-1(i/(n-1))|/nm.


### 2.3. Truth Degrees of Formulas inŁuk

LetA=A(p1,p2,…,pm) be a formula obtained by connecting the atomic formulas p1,p2,…,pm with the connectives ¬ and →, then A determines a function A¯:[0,1]m→[0,1], where A¯(x1,…,xm) is obtained by connecting the variables x1,…,xm in [0,1] with the operators ¬ and → on [0,1] in the same way as A is constructed from p1,…,pm. For example, if A(p1,p2,p3)=p1→(¬p2→p3), then A¯(x1,x2,x3)=x1→(¬x2→x3)=(1-x1+(x2+x3)∧1)∧1. It is clear that υ(A(p1,p2,p3))=A¯(υ(p1),υ(p2),υ(p3)) and in general υ(A(p1,…,pm))=A¯(υ(p1),υ(p2),…,υ(pm))(A∈F(S),υ∈Ω) [9, 10], where A¯ is a McNaughton function [11].Definition 4 (see [9, 10]).
LetA(p1,p2,…,pm) be a formula of F(S) and
(9)T(A)=∫01⋯∫01︸mA¯(x1,…,xm)dx1dx2⋯dxm.
Then T(A) is called the truth degree of A.Remark 1.
(1) It is easy to verify that T(A)=1 if and only if A is a (logical) theorem, and T(A)=0 if and only if A is a contradiction (see [9]).
(2) Notice that A¯ can also be considered as a function of m+k variables, where
(10)A¯(x1,…,xm+k)=A¯(x1,…,xm)(xm+1,…,xm+k∈[0,1]),T(A)=∫01⋯∫01︸m+kA¯(x1,…,xm+k)dx1dx2⋯dxm+k.
Hence if we are given two formulas A(p1,p2,…,pn) and B(q1,q2,…,qk), we can choose an m large enough such that both A¯ and B¯ can be written as A¯(x1,…,xm) and B¯(x1,…,xm), respectively.


### 2.4. Pseudometric of F(S) inŁn and Łuk

Definition 5.
(i) A subset Γ of F(S) is called a theory.
(ii) Let Γ be a theory; A∈F(S). A deduction of A from Γ, in symbols, Γ⊢A, is a finite sequence of formulas A1,…,An=A such that, for each 1≤i≤n,Ai is an axiom, or Ai∈Γ, or there are j,k∈{1,…,i-1} such that Ai follows from Aj and Ak by MP. Equivalently, we say that A is a conclusion of Γ (or Γ-conclusion). The set of all conclusions of Γ is denoted by D(Γ). By a proof of A we will henceforth mean a deduction of A from the empty set. We will also write A in place of ∅⊢A and call A a theorem.Definition 6 (see [9, 10]).
Suppose thatA,B∈F(S), then
(11)ξ(A,B)=T((A→B)∧(B→A))
is called the resemblance degree between A and B.Definition 7 (see [9, 10]).
Suppose thatA,B∈F(S), and define
(12)ρ(A,B)=1-ξ(A,B).
Then ρ is a pseudometric on F(S), and (F(S),ρ) is called the logic metric space.Definition 8 (see [9, 10]).
Suppose thatΓ is a theory; define
(13)div(Γ)=sup{ρ(A,B)∣A,B∈D(Γ)}.
Then div(Γ) is called the divergence degree of Γ.  Γ is said to be fully divergent if div(Γ)=1.


### 2.5. Generalized Deduction Theorems

Deduction theorem is one of the most important theorems in classical mathematical logic, and it says that(14)Γ∪{A}⊢BiffΓ⊢A→B.The deduction theorem is no longer valid in fuzzy logic systems in general. Fortunately, it has been proved that there exist different weak forms of the deduction theorems (called Generalized deduction theorems) in different logic systems, respectively [1, 12].Theorem 1.
Suppose thatΓ is a theory, and A,B∈F(S), then
(i) in Łuk,Γ∪{A}⊢B if and only if  ∃m∈N,s.t.Γ⊢Am→B;
(ii) in Łn, Γ∪{A}⊢B if and only if  Γ⊢An-1→B.Remark 2.
It is easy to verify that(A&B)→C and A→(B→C) are provably equivalent; hence by the definition of deduction, the generalized deduction theorem in Łuk and Łn can be equivalently described as follows
(15)Γ⊢Aiff∃A1,…,As∈Γ,m1,…,ms∈N,s.t.⊢A1m1&…&Asms→A,hold.


## 2.1. Logic Systems:Łn and Łuk

Suppose thatS={p1,p2,…} is a countable set, let ¬ and → be unary and binary operations, respectively, and let F(S) be the free algebra of type (¬,→) generated by S, that is, F(S) is the smallest set containing S and closed under the operation ¬ and →. Elements of F(S) are called propositions or formulas and those of S are called atomic propositions or atomic formulas. In Łukasiewicz fuzzy propositional logic (briefly, Luk), there are four axiom schemes as follows: A→(B→A), (A→B)→((B→C)→(A→C)), (¬A→¬B)→(B→A), ((A→B)→B)→((B→A)→A). The axioms of Łukasiewiczn-valued propositional logic (briefly, Łn) are those of Luk plus (A→nB)→(A→n-1B),(A≡(A→s-2¬B))→n-1A for any 2≤s≤n-1 such that s is not a divisor of n-1. Where A≡B instead of (A→B)&(B→A), A&B is ¬(A→¬B) (see [6]).Here are two kinds of valuation sets:Ln={0,1/(n-1),2/(n-1),…,(n-2)/(n-1),1} and L=[0,1]. Define on Ln and L a unary operator and a binary operator as follows: ¬x=1-x,x→y=(1-x+y)∧1. These two sets of valuation are just the standard MVn-chain and the standard MV-algebra (L,¬,→), respectively.A homomorphismv:F(S)→Ln (or L) of type (¬,→) from F(S) into the valuation set Ln (or L), that is, v(¬A)=¬v(A),v(A→B)=v(A)→v(B), is called a valuation of F(S). The set of all valuations will be denoted by Ω.Łukasiewicz propositional logic system is called the Łukasiewiczn-valued propositional logic system if the set of valuation is Ln={0,1/(n-1),2/(n-1),…,(n-2)/(n-1),1}. Łukasiewicz propositional logic system is called the Łukasiewicz Fuzzy propositional logic system if the set of valuation is L=[0,1]. It has been proved that the semantics and syntax of Łn and Łuk are in perfect harmony, that is, the standard completeness theorem holds in Łn and Łuk, respectively.The deduction rule is Modus Ponens (briefly, MP) of whichB can be deduced from A and A→B. As it is well known, the following abbreviations introduced in Łuk and Łn are popular [1]:(3)A∨B∶=(B→A)→A,A∧B∶=¬(¬A∨¬B),Ak∶=A&A&⋯&A︸k-times.


## 2.2. Truth Degrees of Formulas inŁn

Definition 1 (see [7]).
Suppose that(Xk,Ak,μk)(k=1,2,…) are probability measure spaces; let X=∏k=1∞Xk, then ∏k=1∞Ak generates on X a σ-algebraA. There exists on X a measure μ satisfying the following conditions.
(i)  A is the set consisting of all μ-measurable sets.
(ii) For any measurable subset E of ∏k=1mXk,E×∏k=m+1∞Xk is μ-measurable and
(4)μ(E×∏n=m+1∞Xn)=(μ1×μ2×⋯×μm)(E),m=1,2,…,μ is called the infinite product measure of μ1,μ2,…, and (X,A,μ) is called the infinite product of {(Xk,Ak,μk)}k=1∞.  (X,A,μ) can be abbreviated as X if no confusion arises.Suppose thatn is a fixed natural number, n≥2, and (Y,B,η) is an evenly distributed probability measure space, where Y={y1,y2,…,yn}, that is, η(∅)=0,η(Y)=1, and η({yi})=1/n(i=1,2,…,n). Let (Xk,Ak,μk)=(Y,B,η)(k=1,2,…,), and let (X,A,μ) be the infinite product of {(Xk,Ak,μk)}k=1∞.In the following,Y will be written as Y=Ln={0,1/(n-1),2/(n-1),…,(n-2)/(n-1),1}; hence we have (5)Xk=Ln={0,1n-1,2n-1,…,n-2n-1,1},k=1,2,…,
and (X,A,μ) can also be written as Ln∞.Letv∈Ω, then v is decided by its restriction v∣S because S generates the free algebra F(s). Assume that v(pk)=vk(k=1,2,…), then an infinite-dimensional vector ν⃗=(v1,v2,…) in Ln∞ is obtained. Conversely, let ν⃗=(v1,v2,…) be any element of Ln∞, then there exists a unique v∈Ω such that v(pk)=vk(k=1,2,…). Hence there exists a bijection φ:Ω→Ln∞ defined by φ(v)=ν⃗.Definition 2 (see [8]).
The abovementioned mapping is called the measured mapping ofΩ.Definition 3 (see [8]).
Suppose thatA∈F(S), n≥2 and define
(6)[A]i/(n-1)={ν⃗∈Ln∞∣ϕ-1(ν⃗)(A)=in-1},i=0,1,…,n-1,T(A)=∑i=0n-1in-1μ([A]i/(n-1))=∑i=1n-1in-1μ([A]i/(n-1)),
then T(A) is called the n-valued truth degree of A.Example 1.
LetL8={0,1/7,2/7,…,6/7,1}. Consider the truth degree T(p→q). Notice that υ(p→q)=υ(p)→υ(q)=(1-υ(p)+υ(q))∧1. Let υ(p)=j/7,υ(q)=k/7, and υ(p→q)=i/7, then (1-(j/7)+(k/7))∧1=i/7,i=1,2,…,7⋯(*). If i=7, then 0≤j≤k≤7 by (*); there are a total of (8×9)/2 pairs (υ(p),υ(q)) such that υ(p→q)=1. If 1≤i≤6, then j=7-i+k by (*); there are a total of i+1 pairs (j,k) such that j=7-i+k. Thus,
(7)μ([p→q]i/7)={8×92×64,i=7,i+164,1≤i≤6,T(p→q)=∑i=16i7i+164+8×92×64=3948.
Assume thatA=A(p1,p2,…,pm) is a formula generated by atomic formulas p1,p2,…,pm through connectives ¬ and →. Substitute xi for pi in A(i=1,2,…,m) and keep the logic connectives in A unchanged but explain them as the corresponding operators defined on the valuation latticeLn. Then we get a function A*:Lnm→Ln and call A*(x1,…,xm) the truth degree function of A.Lemma 1 (see [8]).
LetA=A(p1,…,pm)∈F(S),n≥2. Then
(8)T(A)=∑i=0n-1in-1|(A*)-1(i/(n-1))|nm,
where A*(x1,…,xm)=i/(n-1) if and only if xj=υ(pj),j=1,2,…,m, and υ(A(p1,…,pm))=i/(n-1),i=0,1,…n-1. Obviously, Lnm=∪{(A*)-1(i/(n-1))∣i=0,1,…,n-1},|Lnm|=nm, and μ([A]i/(n-1))=|(A*)-1(i/(n-1))|/nm.


## 2.3. Truth Degrees of Formulas inŁuk

LetA=A(p1,p2,…,pm) be a formula obtained by connecting the atomic formulas p1,p2,…,pm with the connectives ¬ and →, then A determines a function A¯:[0,1]m→[0,1], where A¯(x1,…,xm) is obtained by connecting the variables x1,…,xm in [0,1] with the operators ¬ and → on [0,1] in the same way as A is constructed from p1,…,pm. For example, if A(p1,p2,p3)=p1→(¬p2→p3), then A¯(x1,x2,x3)=x1→(¬x2→x3)=(1-x1+(x2+x3)∧1)∧1. It is clear that υ(A(p1,p2,p3))=A¯(υ(p1),υ(p2),υ(p3)) and in general υ(A(p1,…,pm))=A¯(υ(p1),υ(p2),…,υ(pm))(A∈F(S),υ∈Ω) [9, 10], where A¯ is a McNaughton function [11].Definition 4 (see [9, 10]).
LetA(p1,p2,…,pm) be a formula of F(S) and
(9)T(A)=∫01⋯∫01︸mA¯(x1,…,xm)dx1dx2⋯dxm.
Then T(A) is called the truth degree of A.Remark 1.
(1) It is easy to verify that T(A)=1 if and only if A is a (logical) theorem, and T(A)=0 if and only if A is a contradiction (see [9]).
(2) Notice that A¯ can also be considered as a function of m+k variables, where
(10)A¯(x1,…,xm+k)=A¯(x1,…,xm)(xm+1,…,xm+k∈[0,1]),T(A)=∫01⋯∫01︸m+kA¯(x1,…,xm+k)dx1dx2⋯dxm+k.
Hence if we are given two formulas A(p1,p2,…,pn) and B(q1,q2,…,qk), we can choose an m large enough such that both A¯ and B¯ can be written as A¯(x1,…,xm) and B¯(x1,…,xm), respectively.


## 2.4. Pseudometric of F(S) inŁn and Łuk

Definition 5.
(i) A subset Γ of F(S) is called a theory.
(ii) Let Γ be a theory; A∈F(S). A deduction of A from Γ, in symbols, Γ⊢A, is a finite sequence of formulas A1,…,An=A such that, for each 1≤i≤n,Ai is an axiom, or Ai∈Γ, or there are j,k∈{1,…,i-1} such that Ai follows from Aj and Ak by MP. Equivalently, we say that A is a conclusion of Γ (or Γ-conclusion). The set of all conclusions of Γ is denoted by D(Γ). By a proof of A we will henceforth mean a deduction of A from the empty set. We will also write A in place of ∅⊢A and call A a theorem.Definition 6 (see [9, 10]).
Suppose thatA,B∈F(S), then
(11)ξ(A,B)=T((A→B)∧(B→A))
is called the resemblance degree between A and B.Definition 7 (see [9, 10]).
Suppose thatA,B∈F(S), and define
(12)ρ(A,B)=1-ξ(A,B).
Then ρ is a pseudometric on F(S), and (F(S),ρ) is called the logic metric space.Definition 8 (see [9, 10]).
Suppose thatΓ is a theory; define
(13)div(Γ)=sup{ρ(A,B)∣A,B∈D(Γ)}.
Then div(Γ) is called the divergence degree of Γ.  Γ is said to be fully divergent if div(Γ)=1.


## 2.5. Generalized Deduction Theorems

Deduction theorem is one of the most important theorems in classical mathematical logic, and it says that(14)Γ∪{A}⊢BiffΓ⊢A→B.The deduction theorem is no longer valid in fuzzy logic systems in general. Fortunately, it has been proved that there exist different weak forms of the deduction theorems (called Generalized deduction theorems) in different logic systems, respectively [1, 12].Theorem 1.
Suppose thatΓ is a theory, and A,B∈F(S), then
(i) in Łuk,Γ∪{A}⊢B if and only if  ∃m∈N,s.t.Γ⊢Am→B;
(ii) in Łn, Γ∪{A}⊢B if and only if  Γ⊢An-1→B.Remark 2.
It is easy to verify that(A&B)→C and A→(B→C) are provably equivalent; hence by the definition of deduction, the generalized deduction theorem in Łuk and Łn can be equivalently described as follows
(15)Γ⊢Aiff∃A1,…,As∈Γ,m1,…,ms∈N,s.t.⊢A1m1&…&Asms→A,hold.


## 3. Main Results

It is easy for the reader to check the following lemma.Lemma 2.
Suppose thatA=A(pi1,pi2,…,pit)∈F(s) and n≥2, then T(A)=1 if and only if A is a tautology in Łn.Lemma 3.
InŁn and  Luk, if A→B is a logical theorem, then T(A)≤T(B).Lemma 4.
LetA,B,C∈F(S) in Łn and  Luk, then the following properties hold.
(i) If A is a theorem, then T(A→B)=T(B).
(ii) one has  T(A→o¯)=1-T(A).Now we are ready to define the concept of consistency degree for finite and infinite theories. Let us first take an analysis on the inconsistency of a theory inŁuk and Łn. Suppose that Γ is a theory and Γ is inconsistent, then the contradiction 0¯ is a conclusion of Γ; that is to say, Γ⊢0¯ holds. It follows from Remark 2 that there exits a finite string of formulas A1,A2,…,As∈Γ and m1,m2,…,ms∈N such that ⊢A1m1&A2m2&⋯&Asms→0¯ holds. By standard completeness theorem, A1m1&A2m2&⋯&Asms→0¯ is a tautology; then T(A1m1&A2m2&⋯&Asms→0¯)=1 in Łn and Łuk, respectively. Conversely, if there are a finite sequence of formulas A1,A2,⋯,As∈Γ and m1,m2,…,ms∈N such that T(A1m1&A2m2&⋯&Asms→0¯)=1, by Lemma 2 and Remark 1, A1m1&A2m2&⋯&Asms→0¯ is a theorem; hence by the generalized deduction theorem, Γ⊢0¯ holds, and so Γ is inconsistent. From the above analysis, in order to judge whether a given theory Γ is inconsistent or not, it suffices to calculate T(A1m1&A2m2&⋯&Asms→0¯) of all possible formulas of the form A1m1&A2m2&⋯&Asms→0¯, where A1,A2,…,As∈Γ and m1,m2,…,ms∈N. Moreover, the larger T(A1m1&A2m2&⋯&Asms→0¯) of such formulas is, the more closer Γthat is to be inconsistent (see Theorem 6). Note that there may be an infinite number of such formulas for a given theory Γ even though Γ is finite; hence it is natural and reasonable for us using the supremum of T(A1m1&A2m2&⋯&Asms→0¯) of all formulas with the form A1m1&A2m2&⋯&Asms→0¯, where A1,…,As∈Γ and m1,…,ms∈N to measure the inconsistency of Γ.Definition 9.
Suppose thatΓ is a theory, and 0¯ is a contradiction. Define
(16)j(Γ)=sup{T(A1m1&A2m2&⋯&Asms→0¯)∣A1,…,As∈Γ,s∈N,(m1,…,ms)∈NsA1m10¯},(17)k(Γ)=max{[T(A1m1&A2m2&⋯&Asms→0¯)]∣A1,…,As∈Γ,s∈N,(m1,…,ms)∈Ns[T(A1m1&A2m2&⋯&Asms→0¯)]},
where
(18)[x]={0,0≤x<1,1,x=1.Thenj(Γ) is called the first polar index of Γ and k(Γ) is called the second polar index of Γ in Łuk and Łn, respectively.Definition 10.
Suppose thatΓ is a theory and define
(19)Consist(Γ)=1-12(j(Γ)+k(Γ)),
where Consist(Γ) is called the consistency degree of Γ in Łuk and Łn, respectively.Remark 3.
(i) Since, for all A,B,C∈F(S),A&B→C and B&A→C are logically equivalent, the definition of A1m1&⋯&Asms→o¯ does not depend on the order of Aimi's.
(ii) It is easy to verify that, for all A,B,C∈F(S),T(A&B→C)≥T(A→C); hence if s1≤s2, then T(A1m1&A2m2&⋯&As1ms1→0¯)≤T(A1m1&A2m2&⋯&As1ms1&⋯&As2ms2→0¯).Theorem 2.
Suppose thatΓ is a theory, then, in Łn,
(20)j(Γ)=sup{T(A1n-1&A2n-1&⋯&Asn-1→0¯)∣A1,A2,…,As∈Γ,s∈N}.Proof.
For everyA∈F(s), if m≥n-1, then it is easy to check that Am~An-1. If m<n-1, then ⊢An-1→Am holds by axioms. Since ⊢(An-1→Am)→((Am→0¯)→(An-1→0¯)), thus, for every m∈N, ⊢((Am→0¯)→(An-1→0¯)) holds; then by Lemma 3, T(Am→0¯)≤T(An-1→0¯). For the same reason, for any A1,A2,…,As∈Γ and any m1,m2,…,ms∈N, we have ⊢A1n-1&⋯&Asn-1→A1m1&⋯&Asms and T(A1n-1&⋯&Asn-1→0¯)≥T(A1m1&⋯&Asms→0¯). Then j(Γ)=sup{T(A1n-1&A2n-1&⋯&Asn-1→0¯)∣A1,A2,…,As∈Γ,s∈N}.Theorem 3.
Suppose thatΓ={A1,A2,⋯,Am}, then,
(i) in Łn,j(Γ)=T(A1n-1&⋯&Amn-1→0¯)=1-T(A1n-1&⋯&Amn-1).
(ii) In Łuk,j(Γ)=sup{T(A1t1&A2t2&⋯&Amtm→0¯)∣(t1,…,tm)∈Nm}=1-inf{T(A1t1&A2t2&⋯&Amtm)|(t1,…,tm)∈Nm}.Proof.
By Definition9, Theorem 2, and Lemma 4, it is easy to check the theorem above. The proof is left to the reader.Theorem 4.
Suppose thatΓ is a theory, and o¯ is a contradiction. Then
(21)j(Γ)=sup{T(A→o¯)∣A∈D(Γ)}.Proof.
For allA1,A2,…As∈Γ,(m1,…,ms)∈Ns, since A1m1&A2m2&⋯&Asms∈D(Γ), thus {A1m1&A2m2&⋯&Asms→0¯∣A1,A2,…As∈Γ,s∈N,(m1,…,ms)∈Ns}⊆{A→o¯∣A∈D(Γ)}; then j(Γ)≤sup{T(A→o¯)∣A∈D(Γ)}. Conversely, for all A∈D(Γ), it follows from Remark 2 that ∃A1,…,At∈Γ and (m1,…,mt)∈Nt such that ⊢A1m1&A2m2&⋯&Atmt→A, ⊢(A1m1&A2m2&⋯&Atmt→A)→((A→o¯)→(A1m1&A2m2&⋯&Atmt→o¯)) by axioms; thus ⊢(A→o¯)→(A1m1&A2m2&⋯&Atmt→o¯) by MP. It follows from Lemma 3 that T(A→o¯)≤T(A1m1&A2m2&⋯&Atmt→o¯); then j(Γ)≥sup{T(A→o¯)∣A∈D(Γ)}. This completes the proof.Theorem 5.
Suppose thatΓ is a theory. Then the following are given.
(i)Γ is consistent in Łuk and Łn if and only if k(Γ)=0 in Łuk and Łn, respectively.
(ii)Γ is inconsistent in Łuk and Łn if and only if k(Γ)=1 in Łuk and Łn, respectively.Proof.
SinceΓ is consistent or inconsistent, and k(Γ)∈{0,1}, it suffices to prove (i).
Assume thatΓ is consistent, that is, 0¯∉D(Γ). For any given A1,A2,…,As∈Γ and m1,m2,…,ms∈N, then it follows from generalized deduction theorems that (A1m1&A2m2&⋯&Asms)→0¯ is not a theorem; then, by Lemma 2 and Remark 1, T(A1m1&A2m2&⋯&Asms→0¯)<1; thus k(Γ)=0. Conversely, if k(Γ)=0, then, for any A1,A2,…,As∈Γ and m1,m2,…,ms∈N, it follows from Definition 9 that T(A1m1&A2m2&⋯&Asms→0¯)<1; hence A1m1&A2m2&⋯&Asms→0¯ is not a theorem by Lemma 2 and Remark 1. Thus 0¯∉D(Γ) by generalized deduction theorems; then, Γ is consistent.Theorem 6.
Suppose thatΓ is a theory, then, in Łn and Łuk,
(i)  Γ is consistent if and only if 1/2≤Consist(Γ)≤1,
(ii)  Γ is inconsistent if and only if Consist(Γ)=0,
(iii)  Γ is consistent and full divergent if and only if Consist(Γ)=1/2.Proof.
(a) In Łn,(i) by Theorem 5, Γ is consistent if and only if k(Γ)=0; hence (i) holds.
(ii) If Γ is inconsistent, then k(Γ)=1 by Theorem 5, thus j(Γ)=1 and Consist(Γ)=0. Conversely, if Consist(Γ)=0, then k(Γ)+j(Γ)=2; hence k(Γ)=1 and Γ is inconsistent by Theorem 5.
(iii) If Γ is consistent and full divergent, then div(Γ)=1 and k(Γ)=0 by Theorem 5. For all ρ(A,B)∈{ρ(A,B)|A,B∈D(Γ)}, since ρ(A,B)=1-T((A→B)∧(B→A)) and A→B,B→A∈D(Γ), it follows from generalized deduction theorems that there exist A1,…,As∈Γ and B1,…,Bt∈Γ, such that ⊢A1n-1&⋯&Asn-1→(A→B) and ⊢B1n-1&⋯&Btn-1→(B→A). Since ⊢A1n-1&⋯&Asn-1&B1n-1&⋯&Btn-1→(A→B)&(B→A) and ⊢(A→B)&(B→A)→(A→B)∧(B→A) holds, hence ⊢A1n-1&⋯&Asn-1&B1n-1&⋯&Btn-1→(A→B)∧(B→A) by hypothetical syllogism. By axioms, ⊢((A→B)∧(B→A)→o¯)→(A1n-1&⋯&Asn-1&B1n-1&⋯&Btn-1→o¯), so by Lemma 3 we have ρ(A,B)=T((A→B)∧(B→A)→o¯)≤T(A1n-1&⋯&Asn-1&B1n-1&⋯&Btn-1→o¯); thus div(Γ)≤j(Γ) and j(Γ)=1; then Consist(Γ)=1/2. Conversely, if Consist(Γ)=1/2, then k(Γ)+j(Γ)=1; thus k(Γ)=0 and j(Γ)=1. For any A1,…,At∈Γ, A1n-1&⋯&Atn-1∈D(Γ) by axioms. If A is a theorem, since ρ(A,B)=1-T((A→B)∧(B→A))=1-T(A→B)=1-T(B)=1-T(A1n-1&⋯&Atn-1)=T(A1n-1&⋯&Atn-1→o¯), where B=A1n-1&⋯&Atn-1, thus j(Γ)≤div(Γ) by Definitions 9 and 8, then div(Γ)=1, so Γ is consistent and full divergent.
(b) The proof of Łuk is analogous to that of (a) and so it is omitted.Definition 11.
Suppose thatΓ is a theory, and define d(0¯,Γ)=inf{ρ(0¯,A)∣A∈D(Γ)}. It can be verified that ρ(0¯,A)=1-T(A→0¯) and d(0¯,Γ)=1-sup{T(A→0¯)∣A∈D(Γ)}.Theorem 7.
InŁn and Łuk, if Γ1 is consistent, or Γ1 is inconsistent but consist(Γ2)≠1/2, then consist(Γ1)<consist(Γ2) if and only if d(0¯,Γ1)<d(0¯,Γ2).Proof.
Suppose thatconsist(Γ1)<consist(Γ2). First of all, we prove that j(Γ1)>j(Γ2), if Γ1 is consistent, and k(Γ1)=0 by Theorem 5; hence 1-(1/2)j(Γ1)<1-(1/2)(j(Γ2)+k(Γ2)) and j(Γ1)>(j(Γ2)+k(Γ2)), so k(Γ2)=0, and then j(Γ1)>j(Γ2). If Γ1 is inconsistent but consist(Γ2)≠1/2, it follows from Theorems 5 and 6 that k(Γ1)=1 and consist(Γ1)=0, hence consist(Γ2)>0 and j(Γ1)=1; it follows from Theorem 6 that Γ2 is consistent and consist(Γ2)>1/2; thus we have j(Γ2)<1, and then j(Γ1)>j(Γ2). Secondly, we prove that d(0¯,Γ1)<d(0¯,Γ2). Hence there exists r∈(1,0) such that j(Γ1)>r>j(Γ2). By definition, there exist A1,A2,…,As∈Γ1 and m1,m2,…,ms∈N such that T(A1m1&A2m2&⋯&Asms→0¯)>r and, for any B1,B2,…,Bt∈Γ2 and m1,m2,…,mt∈N, T(B1m1&B2m2&⋯&Btmt→0¯)<r; thus T(A1m1&A2m2&⋯&Asms→0¯)>T(B1m1&B2m2&⋯&Btmt→0¯), where A1m1&A2m2&⋯&Asms∈D(Γ1) and B1m1&B2m2&⋯&Btmt∈D(Γ2). Thus ρ(A1m1&A2m2&⋯&Asms,0¯)=1-T(A1m1&A2m2&⋯&Asms→0¯)<1-r<1-T(B1m1&B2m2&⋯&Btmt→0¯)=ρ(B1m1&B2m2&⋯&Btmt,0¯); then d(o¯,Γ1)<d(0¯,Γ2) holds.
Conversely, suppose thatd(0¯,Γ1)<d(0¯,Γ2); since there exists r∈(0,1) such that d(0¯,Γ1)<r<d(0¯,Γ2), it follows from Definition 11 that ∃B∈D(Γ1), and for all A∈D(Γ2) such that 1-T(A→o¯)>r>1-T(B→o¯), thus T(B→o¯)>1-r>T(A→o¯), and j(Γ1)>1-r≥j(Γ2). Then consist(Γ1)<consist(Γ2). This completes the proof.Remark 4.
From Theorem7, the definition of consistency degree is reasonable. If the consistency degree is higher, the distance between every conclusion of Γ and contradiction should be longer.Example 2.
InL8, calculate j(Γ) and Consist(Γ) for (i)  Γ={p}, and (ii)  Γ=S={p1,p2,…}.Solution
(i) Since [p7]i/7={ν⃗∈L8∞∣ν⃗=φ(υ),υ(p7)=i/7,υ∈Ω}, and
(22)υ(p7)={7υ(p)-6,υ(p)>67,0,υ(p)≤67,
hence μ([p7]0)=7/8,μ([p7]i/7)=0(i=1,2,…,6),μ([p7]1)=1/8,T(p7)=∑i=07(i/7)μ([p7]i/7)=1/8. By Theorem 3, j(Γ)=1-T(p7)=7/8,k(Γ)=max{[T(pm→0¯)]∣m∈N}=0, and Consist(Γ)=1-(1/2)(7/8)=9/16.
(ii) By [1], ⊢p17&p27&⋯&pm7→p1∧p2∧⋯∧pm; then it follows from Lemma 3 that T(p17&p27&⋯&pm7)≤T(p1∧p2∧⋯∧pm). Let A=p1∧p2∧⋯∧pm. Then A*(x1,x2,…,xm)=x1∧x2∧⋯∧xm. Hence A*(x1,x2,…,xm)=i/7 if and only if ∃j∈{1,2,…,m},xj=i/7, and xk≥i/7(forallk≠j). Since there are a total of 8m  m-tuples (x1,x2,…,xm) and among them only (8-i)7+(8-i-1)(8-i)6+(8-i-1)2(8-i)5+⋯+(8-i-1)5(8-i)2+(8-i-1)6(8-i)+(8-i-1)7 imply that A*(x1,x2,…,xm)=i/7, hence μ([p1∧p2∧⋯∧pm]i/7)=(1/8m)[(8-i)7+(8-i-1)(8-i)6+(8-i-1)2(8-i)5+⋯+(8-i-1)5(8-i)2+(8-i-1)7]; thus T(p1∧p2∧⋯∧pm)=(1/8m)∑i=07(i/7)|(A*)-1(i/7)|→0,(m→∞). We have limm→∞(T(p17&p27&⋯&pm7))=0. It is easy to check that k(Γ)=0; then j(Γ)=1-inf{T(p17&p27&⋯&pm7)|p1,p2,…,pm∈S,m∈N}=1 and Consis(Γ)=1-1/2=1/2.Example 3.
In  Luk, calculatej(Γ) and Consist(Γ) for (i)Γ={p}, and (ii)Γ=S={p1,p2,…}.Solution
(i) Since
(23)xm={mx-(m-1),x>m-1m,0,x≤m-1m,T(pm)=∫01xmdx=∫(m-1)/m1[mx-(m-1)]dx=1/2m, then j(Γ)=sup{T(pm→0¯)|m∈N}=1-inf{T(pm)∣m∈N}=1,k(Γ)=max{[T(pm→0¯)]∣m∈N}=max{[1-T(pm)]∣m∈N}=0, and Consist(Γ)=1-(1/2)(j(Γ)+k(Γ))=1/2.
(ii) Since ⊢p1m1&p2m2&⋯&psms→p1∧p2∧⋯∧ps by [1], then T(p1m1&p2m2&⋯&psms)≤T(p1∧p2∧⋯∧ps). Let δs={(x1,x2,…,xs)∈[0,1]s|xi>(lns/s),i=1,2,…,s} and Δ=[0,1]s. Lebesgue measure m(δs)=(1-(lns)/s)s. In Δ-δs, we have x1∧x2⋯∧xs≤(lns)/s. Thus
(24)T(p1∧p2∧⋯∧ps)=∫01⋯∫01︸s(x1∧x2∧⋯∧xs)dx1dx2⋯dxs=∫Δ-δs(x1∧x2∧⋯∧xs)dx1dx2⋯dxs+∫δs(x1∧x2∧⋯∧xs)dx1dx2⋯dxs≤lnss+(1-(lnss))s→0,(s→∞).
Thus lims→∞(T(p1m1&p2m2&⋯&psms))=0. It is easy to check that k(Γ)=0; then j(Γ)=1-inf{T(p1m1&p2m2&⋯&psms)|p1,p2,…,ps∈S,s∈N}=1 and Consist(Γ)=1/2.Remark 5.
We compared the consistency degree concepts in [2, 3] with the definitions of this paper. From [2, 3], we see that the concept of consistency degree ξ(Γ) and ξ*(Γ) of a theory Γ based on the concept of the divergence degree, moreover, though [2] has given the concept of consistency degree based on the condition of finite theories while the concept of consistency degree Consist(Γ) of Γ is clearly is based on the truth degrees of formulas and deduction theorems. From the logical point of view, the definition of Consist(Γ) and j(Γ) is more natural and reasonable. It is a more theoretical idea to understand while comparing to the facilitating calculation in this paper. Therefore, it is important and essential to give out the definition and characterization of the consistency degree in this paper.


## 4. Concluding Remarks

Based on deduction theorems, completeness theorems, and by means of the theory of truth degrees of formulas, the present paper defines a new polar index of a theoryΓ and a new concept of consistency degrees of theories in Łukasiewicz fuzzy and n-valued propositional logic system, which is more natural and reasonable from the logical point of view. Moreover, some important properties of truth degree of formula are proposed. Finally, sufficient and necessary conditions for a theory to be consistent, inconsistent, and fully divergent are obtained.



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