Consistency Degrees of Theories in Łukasiewicz Fuzzy and 𝑛-Valued Propositional Logic Systems

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 and 𝑛-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.

1. Introduction

Whether a theory (i.e., a set of formulas) is consistent or not (a theory Γ 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: Incons(Γ)=sup{𝑎∣Γ⊢_𝑎𝐴∧¬𝐴and𝐴∈ℱ_ℒ(Γ)}, (1) 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: 𝜉(Γ)=lim_𝑛→∞1[1−div(Γ)(1−2⌈𝛼_𝑛𝜉,Γ⌉)],_∗1(Γ)=1−2div(Γ)(1+𝑖(Γ)), (2) where 𝛼_𝑛,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 of Łukasewicz 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 that 𝑆={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, Ł_𝑛) 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: 𝐿_𝑛={0,1/(𝑛−1),2/(𝑛−1),…,(𝑛−2)/(𝑛−1),1} and 𝐿=[0,1]. Define on 𝐿_𝑛 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 MV_𝑛-chain and the standard MV-algebra (𝐿,¬,→), respectively.

A homomorphism 𝑣∶𝐹(𝑆)→𝐿_𝑛 (or 𝐿) of type (¬,→) from 𝐹(𝑆) into the valuation set 𝐿_𝑛 (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 𝐿_𝑛={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 Ł_𝑛 and Łuk are in perfect harmony, that is, the standard completeness theorem holds in Ł_𝑛 and Łuk, respectively.

The deduction rule is Modus Ponens (briefly, MP) of which 𝐵 can be deduced from 𝐴 and 𝐴→𝐵. As it is well known, the following abbreviations introduced in Łuk and Ł_𝑛 are popular [1]:𝐴𝐴∨𝐵∶=(𝐵→𝐴)→𝐴,𝐴∧𝐵∶=¬(¬𝐴∨¬𝐵),_𝑘∶=𝐴&𝐴&⋯&𝐴_{𝑘−𝑡𝑖𝑚𝑒𝑠}. (3)

2.2. Truth Degrees of Formulas in Ł_𝑛

Definition 1 (see [7]). Suppose that (𝑋_𝑘,𝐴_𝑘,𝜇_𝑘)(𝑘=1,2,…) are probability measure spaces; let 𝑋=∏^∞_𝑘=1𝑋_𝑘, then ∏^∞_𝑘=1𝐴_𝑘 generates on 𝑋 a 𝜎-algebra 𝐴. 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 ∏^𝑚_𝑘=1𝑋_𝑘,𝐸×∏^∞_𝑘=𝑚+1𝑋_𝑘 is 𝜇-measurable and 𝜇(𝐸×^∞_𝑛=𝑚+1𝑋_𝑛)=(𝜇₁×𝜇₂×⋯×𝜇_m)(𝐸),𝑚=1,2,…, (4)𝜇 is called the infinite product measure of 𝜇₁,𝜇₂,…, and (𝑋,𝐴,𝜇) is called the infinite product of {(𝑋_𝑘,𝐴_𝑘,𝜇_𝑘)}^∞_𝑘=1. (𝑋,𝐴,𝜇) can be abbreviated as 𝑋 if no confusion arises.

Suppose that 𝑛 is a fixed natural number, 𝑛≥2, and (𝑌,𝐵,𝜂) is an evenly distributed probability measure space, where 𝑌={𝑦₁,𝑦₂,…,𝑦_𝑛}, that is, 𝜂(∅)=0,𝜂(𝑌)=1, and 𝜂({𝑦_𝑖})=1/𝑛(𝑖=1,2,…,𝑛). Let (𝑋_𝑘,𝐴_𝑘,𝜇_𝑘)=(𝑌,𝐵,𝜂)(𝑘=1,2,…,), and let (𝑋,𝐴,𝜇) be the infinite product of {(𝑋_𝑘,𝐴_𝑘,𝜇_𝑘)}^∞_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 𝑋_𝑘1=𝐿𝑛={0,,2𝑛−1𝑛−1,…,𝑛−2𝑛−1,1},𝑘=1,2,…, (5) and (𝑋,𝐴,𝜇) can also be written as 𝐿^∞_𝑛.

Let 𝑣∈Ω, then 𝑣 is decided by its restriction 𝑣∣𝑆 because 𝑆 generates the free algebra 𝐹(𝑠). Assume that 𝑣(𝑝_𝑘)=𝑣_𝑘(𝑘=1,2,…), then an infinite-dimensional vector 𝜈=(𝑣₁,𝑣₂,…) in 𝐿^∞_𝑛 is obtained. Conversely, let 𝜈=(𝑣₁,𝑣₂,…) be any element of 𝐿^∞_𝑛, then there exists a unique 𝑣∈Ω such that 𝑣(𝑝_𝑘)=𝑣_𝑘(𝑘=1,2,…). Hence there exists a bijection 𝜑∶Ω→𝐿^∞_𝑛 defined by 𝜑(𝑣)=𝜈.

Definition 2 (see [8]). The abovementioned mapping is called the measured mapping of Ω.

Definition 3 (see [8]). Suppose that 𝐴∈𝐹(𝑆), 𝑛≥2 and define [𝐴]_{𝑖/(𝑛−1)}={𝜈∈𝐿^∞_𝑛∣𝜙⁻¹(𝑖𝜈)(𝐴)=𝑛−1},𝑖=0,1,…,𝑛−1,𝑇(𝐴)=^𝑛−1_𝑖=0𝑖𝑛−1𝜇([𝐴]_{𝑖/(𝑛−1)})=^𝑛−1_𝑖=1𝑖𝑛−1𝜇([𝐴]_{𝑖/(𝑛−1)}), (6) then 𝑇(𝐴) is called the 𝑛-valued truth degree of 𝐴.

Example 1. Let 𝐿₈={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, 𝜇([𝑝→𝑞]_𝑖/7)=8×92×64,𝑖=7,𝑖+164,1≤𝑖≤6,𝑇(𝑝→𝑞)=⁶_𝑖=1𝑖7𝑖+1+648×9=2×6439.48 (7)
Assume that 𝐴=𝐴(𝑝₁,𝑝₂,…,𝑝_𝑚) is a formula generated by atomic formulas 𝑝₁,𝑝₂,…,𝑝_𝑚 through connectives ¬ and →. Substitute 𝑥_𝑖 for 𝑝_𝑖 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 lattice𝐿_𝑛. Then we get a function 𝐴^∗∶𝐿^𝑚_𝑛→𝐿_𝑛 and call 𝐴^∗(𝑥₁,…,𝑥_𝑚) the truth degree function of 𝐴.

Lemma 1 (see [8]). Let 𝐴=𝐴(𝑝₁,…,𝑝_𝑚)∈𝐹(𝑆),𝑛≥2. Then 𝑇(𝐴)=^𝑛−1_𝑖=0𝑖𝑛−1|(𝐴^∗)⁻¹(𝑖/(𝑛−1))|𝑛^𝑚, (8) where 𝐴^∗(𝑥₁,…,𝑥_𝑚)=𝑖/(𝑛−1) if and only if 𝑥_𝑗=𝜐(𝑝_𝑗),𝑗=1,2,…,𝑚, and 𝜐(𝐴(𝑝₁,…,𝑝_𝑚))=𝑖/(𝑛−1),𝑖=0,1,…𝑛−1. Obviously, 𝐿^𝑚_𝑛=∪{(𝐴^∗)⁻¹(𝑖/(𝑛−1))∣𝑖=0,1,…,𝑛−1},|𝐿^𝑚_𝑛|=𝑛^𝑚, and 𝜇([𝐴]_{𝑖/(𝑛−1)})=|(𝐴^∗)⁻¹(𝑖/(𝑛−1))|/𝑛^𝑚.

2.3. Truth Degrees of Formulas in Ł𝑢𝑘

Let 𝐴=𝐴(𝑝₁,𝑝₂,…,𝑝_𝑚) be a formula obtained by connecting the atomic formulas 𝑝₁,𝑝₂,…,𝑝_𝑚 with the connectives ¬ and →, then 𝐴 determines a function 𝐴∶[0,1]^𝑚→[0,1], where 𝐴(𝑥₁,…,𝑥_𝑚) is obtained by connecting the variables 𝑥₁,…,𝑥_𝑚 in [0,1] with the operators ¬ and → on [0,1] in the same way as 𝐴 is constructed from 𝑝₁,…,𝑝_𝑚. 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,…,𝑝_𝑚))=𝐴(𝜐(𝑝₁),𝜐(𝑝₂),…,𝜐(𝑝_𝑚))(𝐴∈𝐹(𝑆),𝜐∈Ω) [9, 10], where 𝐴 is a McNaughton function [11].

Definition 4 (see [9, 10]). Let 𝐴(𝑝₁,𝑝₂,…,𝑝_𝑚) be a formula of 𝐹(𝑆) and 𝑇(𝐴)=¹₀⋯¹_0𝑚𝐴(𝑥₁,…,𝑥_𝑚)𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑚. (9) 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 𝐴(𝑥₁,…,𝑥_𝑚+𝑘)=𝐴(𝑥₁,…,𝑥_𝑚)(𝑥_𝑚+1,…,𝑥_𝑚+𝑘∈[0,1]),𝑇(𝐴)=¹₀⋯¹_0𝑚+𝑘𝐴(𝑥₁,…,𝑥_𝑚+𝑘)𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑚+𝑘. (10) Hence if we are given two formulas 𝐴(𝑝₁,𝑝₂,…,𝑝_𝑛) and 𝐵(𝑞₁,𝑞₂,…,𝑞_𝑘), we can choose an 𝑚 large enough such that both 𝐴 and 𝐵 can be written as 𝐴(𝑥₁,…,𝑥_𝑚) and 𝐵(𝑥₁,…,𝑥_𝑚), respectively.

2.4. Pseudometric of F(S) in Ł_𝑛 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,…,𝐴_𝑛=𝐴 such that, for each 1≤𝑖≤𝑛,𝐴_𝑖 is an axiom, or 𝐴_𝑖∈Γ, or there are 𝑗,𝑘∈{1,…,𝑖−1} such that 𝐴_𝑖 follows from 𝐴_𝑗 and 𝐴_𝑘 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 that 𝐴,𝐵∈𝐹(𝑆), then 𝜉(𝐴,𝐵)=𝑇((𝐴→𝐵)∧(𝐵→𝐴)) (11) is called the resemblance degree between 𝐴 and 𝐵.

Definition 7 (see [9, 10]). Suppose that 𝐴,𝐵∈𝐹(𝑆), and define 𝜌(𝐴,𝐵)=1−𝜉(𝐴,𝐵). (12) Then 𝜌 is a pseudometric on 𝐹(𝑆), and (𝐹(𝑆),𝜌) is called the logic metric space.

Definition 8 (see [9, 10]). Suppose that Γ is a theory; define div(Γ)=sup{𝜌(𝐴,𝐵)∣𝐴,𝐵∈𝐷(Γ)}. (13) 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 thatΓ∪{𝐴}⊢𝐵iffΓ⊢𝐴→𝐵. (14)

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 that Γ is a theory, and 𝐴,𝐵∈𝐹(𝑆), then
(i) in Łuk,Γ∪{𝐴}⊢𝐵 if and only if ∃𝑚∈𝑁,𝑠.𝑡.Γ⊢𝐴^𝑚→𝐵; 
(ii) in Ł_𝑛, Γ∪{𝐴}⊢𝐵 if and only if Γ⊢𝐴^𝑛−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 Ł_𝑛 can be equivalently described as follows Γ⊢𝐴iff∃𝐴₁,…,𝐴_𝑠∈Γ,𝑚₁,…,𝑚_𝑠∈𝑁,s.t.⊢𝐴^𝑚1₁&…&𝐴^𝑚𝑠_𝑠→𝐴,hold. (15)

3. Main Results

It is easy for the reader to check the following lemma.

Lemma 2. Suppose that 𝐴=𝐴(𝑝_𝑖1,𝑝_𝑖2,…,𝑝_𝑖𝑡)∈𝐹(𝑠) and 𝑛≥2, then 𝑇(𝐴)=1 if and only if 𝐴 is a tautology in Ł_𝑛.

Lemma 3. In Ł_𝑛 and Luk, if 𝐴→𝐵 is a logical theorem, then 𝑇(𝐴)≤𝑇(𝐵).

Lemma 4. Let 𝐴,𝐵,𝐶∈𝐹(𝑆) in Ł_𝑛 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 in Łuk 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,…,𝐴_𝑠∈Γ and 𝑚₁,𝑚₂,…,𝑚_𝑠∈𝑁 such that ⊢𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0 holds. By standard completeness theorem, 𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0 is a tautology; then 𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)=1 in Łn and Łuk, respectively. Conversely, if there are a finite sequence of formulas 𝐴₁,𝐴₂,⋯,𝐴_𝑠∈Γ and 𝑚₁,𝑚₂,…,𝑚_𝑠∈𝑁 such that 𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)=1, by Lemma 2 and Remark 1, 𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→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(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0) of all possible formulas of the form 𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0, where 𝐴₁,𝐴₂,…,𝐴_𝑠∈Γ and 𝑚₁,𝑚₂,…,𝑚_𝑠∈𝑁. Moreover, the larger 𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→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(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0) of all formulas with the form 𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0, where 𝐴₁,…,𝐴_𝑠∈Γ and 𝑚₁,…,𝑚_𝑠∈𝑁 to measure the inconsistency of Γ.

Definition 9. Suppose that Γ is a theory, and 0 is a contradiction. Define 𝑗(Γ)=sup{𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)∣𝐴₁,…,𝐴_𝑠∈Γ,𝑠∈𝑁,(𝑚₁,…,𝑚_𝑠)∈𝑁^𝑠},𝑘(Γ)=max{[𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)]∣𝐴₁,…,𝐴_𝑠∈Γ,𝑠∈𝑁,(𝑚₁,…,𝑚_𝑠)∈𝑁^𝑠}, (16)(17) where [𝑥]=0,0≤𝑥<1,1,𝑥=1. (18)

Then 𝑗(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 that Γ is a theory and define Consist1(Γ)=1−2(𝑗(Γ)+𝑘(Γ)), (19) 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 𝐴^𝑚1₁&⋯&𝐴^𝑚𝑠_𝑠→𝑜 does not depend on the order of 𝐴^𝑚𝑖_𝑖'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(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠1_𝑠1→0)≤𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠1_𝑠1&⋯&𝐴^𝑚𝑠2_𝑠2→0).

Theorem 2. Suppose that Γ is a theory, then, in Ł_𝑛, 𝑗(Γ)=sup{𝑇(𝐴₁^𝑛−1&𝐴₂^𝑛−1&⋯&𝐴_𝑠^𝑛−1→0)∣𝐴₁,𝐴₂,…,𝐴_𝑠∈Γ,𝑠∈𝑁}. (20)

Proof. For every 𝐴∈𝐹(𝑠), if 𝑚≥𝑛−1, then it is easy to check that 𝐴^𝑚∼𝐴^𝑛−1. If 𝑚<𝑛−1, then ⊢𝐴^𝑛−1→𝐴^𝑚 holds by axioms. Since ⊢(𝐴^𝑛−1→𝐴^𝑚)→((𝐴^𝑚→0)→(𝐴^𝑛−1→0)), thus, for every 𝑚∈𝑁, ⊢((𝐴^𝑚→0)→(𝐴^𝑛−1→0)) holds; then by Lemma 3, 𝑇(𝐴^𝑚→0)≤𝑇(𝐴^𝑛−1→0). For the same reason, for any 𝐴₁,𝐴₂,…,𝐴_𝑠∈Γ and any 𝑚₁,𝑚₂,…,𝑚_𝑠∈𝑁, we have ⊢𝐴₁^𝑛−1&⋯&𝐴_𝑠^𝑛−1→𝐴^𝑚1₁&⋯&𝐴^𝑚𝑠_𝑠 and 𝑇(𝐴₁^𝑛−1&⋯&𝐴_𝑠^𝑛−1→0)≥𝑇(𝐴^𝑚1₁&⋯&𝐴^𝑚𝑠_𝑠→0). Then 𝑗(Γ)=sup{𝑇(𝐴₁^𝑛−1&𝐴₂^𝑛−1&⋯&𝐴_𝑠^𝑛−1→0)∣𝐴₁,𝐴₂,…,𝐴_𝑠∈Γ,𝑠∈𝑁}.

Theorem 3. Suppose that Γ={𝐴₁,𝐴₂,⋯,𝐴_𝑚}, then,
(i) in Łn,𝑗(Γ)=𝑇(𝐴₁^𝑛−1&⋯&𝐴_𝑚^𝑛−1→0)=1−𝑇(𝐴₁^𝑛−1&⋯&𝐴_𝑚^𝑛−1).  
(ii) In Łuk,𝑗(Γ)=sup{𝑇(𝐴^𝑡1₁&𝐴^𝑡2₂&⋯&𝐴^𝑡𝑚_𝑚→0)∣(𝑡₁,…,𝑡_𝑚)∈𝑁^𝑚}=1−inf{𝑇(𝐴^𝑡1₁&𝐴^𝑡2₂&⋯&𝐴^𝑡𝑚_𝑚)|(𝑡₁,…,𝑡_𝑚)∈𝑁^𝑚}.

Proof. By Definition 9, Theorem 2, and Lemma 4, it is easy to check the theorem above. The proof is left to the reader.

Theorem 4. Suppose that Γ is a theory, and 𝑜 is a contradiction. Then 𝑗(Γ)=sup{𝑇(𝐴→𝑜)∣𝐴∈𝐷(Γ)}. (21)

Proof. For all 𝐴₁,𝐴₂,…𝐴_𝑠∈Γ,(𝑚₁,…,𝑚_𝑠)∈𝑁^𝑠, since 𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠∈𝐷(Γ), thus {𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0∣𝐴₁,𝐴₂,…𝐴_𝑠∈Γ,𝑠∈𝑁,(𝑚₁,…,𝑚_𝑠)∈𝑁^𝑠}⊆{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,…,𝐴_𝑡∈Γ and (𝑚₁,…,𝑚_𝑡)∈𝑁^𝑡 such that ⊢𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑡_𝑡→𝐴, ⊢(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑡_𝑡→𝐴)→((𝐴→𝑜)→(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑡_𝑡→𝑜)) by axioms; thus ⊢(𝐴→𝑜)→(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑡_𝑡→𝑜) by MP. It follows from Lemma 3 that 𝑇(𝐴→𝑜)≤𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑡_𝑡→𝑜); then 𝑗(Γ)≥sup{𝑇(𝐴→𝑜)∣𝐴∈𝐷(Γ)}. This completes the proof.

Theorem 5. Suppose that Γ 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. Since Γ is consistent or inconsistent, and 𝑘(Γ)∈{0,1}, it suffices to prove (i).
Assume that Γ is consistent, that is, 0∉𝐷(Γ). For any given 𝐴₁,𝐴₂,…,𝐴_𝑠∈Γ and 𝑚₁,𝑚₂,…,𝑚_𝑠∈𝑁, then it follows from generalized deduction theorems that (𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠)→0 is not a theorem; then, by Lemma 2 and Remark 1, 𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)<1; thus 𝑘(Γ)=0. Conversely, if 𝑘(Γ)=0, then, for any 𝐴₁,𝐴₂,…,𝐴_s∈Γ and 𝑚₁,𝑚₂,…,𝑚_𝑠∈𝑁, it follows from Definition 9 that 𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)<1; hence 𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→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 that Γ is a theory, then, in Ł_𝑛 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,…,𝐴_𝑠∈Γ and 𝐵₁,…,𝐵_𝑡∈Γ, such that ⊢𝐴₁^𝑛−1&⋯&𝐴_𝑠^𝑛−1→(𝐴→𝐵) and ⊢𝐵₁^𝑛−1&⋯&𝐵_𝑡^𝑛−1→(𝐵→𝐴). Since ⊢𝐴₁^𝑛−1&⋯&𝐴_𝑠^𝑛−1&𝐵₁^𝑛−1&⋯&𝐵_𝑡^𝑛−1→(A→𝐵)&(𝐵→𝐴) and ⊢(𝐴→𝐵)&(𝐵→𝐴)→(𝐴→𝐵)∧(𝐵→𝐴) holds, hence ⊢𝐴₁^𝑛−1&⋯&𝐴_𝑠^𝑛−1&𝐵₁^𝑛−1&⋯&𝐵_𝑡^𝑛−1→(𝐴→𝐵)∧(𝐵→𝐴) by hypothetical syllogism. By axioms, ⊢((𝐴→𝐵)∧(𝐵→𝐴)→𝑜)→(𝐴₁^𝑛−1&⋯&𝐴_𝑠^𝑛−1&𝐵₁^𝑛−1&⋯&𝐵_𝑡^𝑛−1→𝑜), so by Lemma 3 we have 𝜌(𝐴,𝐵)=𝑇((𝐴→𝐵)∧(𝐵→𝐴)→𝑜)≤𝑇(𝐴₁^𝑛−1&⋯&𝐴_𝑠^𝑛−1&𝐵₁^𝑛−1&⋯&𝐵_𝑡^𝑛−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,…,𝐴_𝑡∈Γ, 𝐴₁^𝑛−1&⋯&𝐴_𝑡^𝑛−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(𝐴₁^𝑛−1&⋯&𝐴_𝑡^𝑛−1)=𝑇(𝐴₁^𝑛−1&⋯&𝐴_𝑡^𝑛−1→𝑜), where 𝐵=𝐴₁^𝑛−1&⋯&𝐴_𝑡^𝑛−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 that Γ 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. In Ł_𝑛 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 that consist(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,…,𝐴_𝑠∈Γ₁ and 𝑚₁,𝑚₂,…,𝑚_𝑠∈𝑁 such that 𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)>𝑟 and, for any 𝐵₁,𝐵₂,…,𝐵_𝑡∈Γ₂ and 𝑚₁,𝑚₂,…,𝑚_𝑡∈𝑁, 𝑇(𝐵^𝑚1₁&𝐵^𝑚2₂&⋯&𝐵^𝑚𝑡_𝑡→0)<𝑟; thus 𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)>𝑇(𝐵^𝑚1₁&𝐵^𝑚2₂&⋯&𝐵^𝑚𝑡_𝑡→0), where 𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠∈𝐷(Γ₁) and 𝐵^𝑚1₁&𝐵^𝑚2₂&⋯&𝐵^𝑚𝑡_𝑡∈𝐷(Γ₂). Thus 𝜌(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠,0)=1−𝑇(𝐴^𝑚1₁&𝐴^𝑚2₂&⋯&𝐴^𝑚𝑠_𝑠→0)<1−𝑟<1−𝑇(𝐵^𝑚1₁&𝐵^𝑚2₂&⋯&𝐵^𝑚𝑡_𝑡→0)=𝜌(𝐵^𝑚1₁&𝐵^𝑚2₂&⋯&𝐵^𝑚𝑡_𝑡,0); then 𝑑(𝑜,Γ₁)<𝑑(0,Γ₂) holds.
Conversely, suppose that 𝑑(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 Theorem 7, 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. In 𝐿₈, calculate 𝑗(Γ) and Consist(Γ) for (i)  Γ={𝑝}, and (ii)  Γ=𝑆={𝑝₁,𝑝₂,…}.

Solution
(i) Since [𝑝⁷]_𝑖/7={𝜈∈𝐿^∞₈∣𝜈=𝜑(𝜐),𝜐(𝑝⁷)=𝑖/7,𝜐∈Ω}, and 𝜐(𝑝⁷6)=7𝜐(𝑝)−6,𝜐(𝑝)>7,60,𝜐(𝑝)≤7, (22) hence 𝜇([𝑝⁷]₀)=7/8,𝜇([𝑝⁷]_𝑖/7)=0(𝑖=1,2,…,6),𝜇([𝑝⁷]₁)=1/8,𝑇(𝑝⁷)=∑⁷_𝑖=0(𝑖/7)𝜇([𝑝⁷]_𝑖/7)=1/8. By Theorem 3, 𝑗(Γ)=1−𝑇(𝑝⁷)=7/8,𝑘(Γ)=max{[𝑇(𝑝^𝑚→0)]∣𝑚∈𝑁}=0, and Consist(Γ)=1−(1/2)(7/8)=9/16. 
(ii) By [1], ⊢𝑝⁷₁&𝑝⁷₂&⋯&𝑝⁷_𝑚→𝑝₁∧𝑝₂∧⋯∧𝑝_𝑚; then it follows from Lemma 3 that 𝑇(𝑝⁷₁&𝑝⁷₂&⋯&𝑝⁷_𝑚)≤𝑇(𝑝₁∧𝑝₂∧⋯∧𝑝_𝑚). Let 𝐴=𝑝₁∧𝑝₂∧⋯∧𝑝_𝑚. Then 𝐴^∗(𝑥₁,𝑥₂,…,𝑥_𝑚)=𝑥₁∧𝑥₂∧⋯∧𝑥_𝑚. Hence 𝐴^∗(𝑥₁,𝑥₂,…,𝑥_𝑚)=𝑖/7 if and only if ∃𝑗∈{1,2,…,𝑚},𝑥_𝑗=𝑖/7, and 𝑥_𝑘≥𝑖/7(forall𝑘≠𝑗). Since there are a total of 8^𝑚  𝑚-tuples (𝑥₁,𝑥₂,…,𝑥_𝑚) 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,…,𝑥_𝑚)=𝑖/7, hence 𝜇([𝑝₁∧𝑝₂∧⋯∧𝑝_𝑚]_𝑖/7)=(1/8^𝑚)[(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∧⋯∧𝑝_𝑚)=(1/8^𝑚)∑⁷_𝑖=0(𝑖/7)|(𝐴^∗)⁻¹(𝑖/7)|→0,(𝑚→∞). We have lim_𝑚→∞(𝑇(𝑝⁷₁&𝑝⁷₂&⋯&𝑝⁷_𝑚))=0. It is easy to check that 𝑘(Γ)=0; then 𝑗(Γ)=1−inf{𝑇(𝑝⁷₁&𝑝⁷₂&⋯&𝑝⁷_𝑚)|𝑝₁,𝑝₂,…,𝑝_𝑚∈𝑆,𝑚∈𝑁}=1 and Consis(Γ)=1−1/2=1/2.

Example 3. In Luk, calculate 𝑗(Γ) and Consist(Γ) for (i)Γ={𝑝}, and (ii)Γ=𝑆={𝑝₁,𝑝₂,…}.

Solution
(i) Since x^𝑚=𝑚𝑥−(𝑚−1),𝑥>𝑚−1𝑚,0,𝑥≤𝑚−1𝑚, (23)𝑇(𝑝^𝑚)=∫¹₀𝑥^𝑚𝑑𝑥=∫¹_{(𝑚−1)/𝑚}[𝑚𝑥−(𝑚−1)]𝑑𝑥=1/2𝑚, then 𝑗(Γ)=sup{𝑇(𝑝^𝑚→0)|𝑚∈𝑁}=1−inf{𝑇(𝑝^𝑚)∣𝑚∈𝑁}=1,𝑘(Γ)=max{[𝑇(𝑝^𝑚→0)]∣𝑚∈𝑁}=max{[1−𝑇(𝑝^𝑚)]∣𝑚∈𝑁}=0, and Consist(Γ)=1−(1/2)(𝑗(Γ)+𝑘(Γ))=1/2. 
(ii) Since ⊢𝑝^𝑚1₁&𝑝^𝑚2₂&⋯&𝑝^𝑚𝑠_𝑠→𝑝₁∧𝑝₂∧⋯∧𝑝_𝑠 by [1], then 𝑇(𝑝^𝑚1₁&𝑝^𝑚2₂&⋯&𝑝^𝑚𝑠_𝑠)≤𝑇(𝑝₁∧𝑝₂∧⋯∧𝑝_𝑠). Let 𝛿_𝑠={(𝑥₁,𝑥₂,…,𝑥_𝑠)∈[0,1]^𝑠|𝑥_𝑖>(ln𝑠/𝑠),𝑖=1,2,…,𝑠} and Δ=[0,1]^𝑠. Lebesgue measure 𝑚(𝛿_𝑠)=(1−(ln𝑠)/𝑠)^𝑠. In Δ−𝛿_𝑠, we have 𝑥₁∧𝑥₂⋯∧𝑥_𝑠≤(ln𝑠)/𝑠. Thus 𝑇(𝑝₁∧𝑝₂∧⋯∧𝑝_𝑠)=¹₀⋯¹_0𝑠(𝑥₁∧𝑥₂∧⋯∧𝑥_𝑠)𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑠=_{Δ−𝛿𝑠}(𝑥₁∧𝑥₂∧⋯∧𝑥_𝑠)𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑠+_𝛿𝑠(𝑥₁∧𝑥₂∧⋯∧𝑥_𝑠)𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑠≤ln𝑠𝑠+(1−(ln𝑠𝑠))^𝑠→0,(𝑠→∞). (24) Thus lim_𝑠→∞(𝑇(𝑝^𝑚1₁&p^𝑚2₂&⋯&𝑝^𝑚𝑠_𝑠))=0. It is easy to check that 𝑘(Γ)=0; then 𝑗(Γ)=1−inf{𝑇(𝑝^𝑚1₁&𝑝^𝑚2₂&⋯&𝑝^𝑚𝑠_𝑠)|𝑝₁,𝑝₂,…,𝑝_𝑠∈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 theory Γ 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.

Acknowledgment

This project was supported by the Natural Science Foundation of Fujian Province of China (no. 2006J0221).